Notation by work
Signal Reconstruction
All notation · Glossary · Source coverage
435 symbol spellings in 800 mathematical passages across 3 selected source files. Subscripts and superscripts have their own entries alongside their base symbols.
The same symbol can have several meanings here. A passage classified as a definition may define another part of its expression. Read the linked context before assigning a meaning.
All 435 entries are available below.
*Use in source · 10 passages
Search aliases: *
Use in source · Signal_Reconstruction_Traditional.tex, line 218
Compute P_aP_b=V^-1E_aE_bV=δ_abP_a and ∑_aP_a=V^-1IV=I. In the unitary case the pulled-back projections are orthogonal: P_a^*=P_a. Their images are mutually orthogonal, and SG-01 writes x as their finite orthogonal sum. The Pythagorean identity gives the energy formula. Without unitarity and orthogonality the algebraic identities remain valid while that norm identity need not hold.
Exact mathematical passage
P_a^*=P_a
+Use in source · 135 passages
Search aliases: +
Use in source · Signal_Reconstruction_Traditional.tex, line 227
smallDomain and hypotheses. Absolutely summable sequences on Z. Fix an integer N≥1. Let h,x∈ℓ^1( Z) and y=h*x. Define x_a[q]=x[Nq+a] and y_b[q]=y[Nq+b], for 0≤ a,b<N. [SG-05 Polyphase analysis and convolution kernels]
Exact mathematical passage
x_a[q]=x[Nq+a]
\mathbf1Use in source · 2 passages
Search aliases: 1 mathbf
Use in source · Signal_Reconstruction_Traditional.tex, line 437
smallDomain and hypotheses. The interval packet j=1_[-a,a) on T. Assume 0<a<π, let λ_M=M+1/2, and use the right edge coordinate x_M(z)=a-z/λ_M. [SG-12 The scaled interval-edge Gibbs profile]
Exact mathematical passage
j=\mathbf1_{[-a,a)}
\mathbf1_{[-a,a)}Use in source · 1 passage
Search aliases: 1_[-a,a) mathbf
Use in source · Signal_Reconstruction_Traditional.tex, line 437
smallDomain and hypotheses. The interval packet j=1_[-a,a) on T. Assume 0<a<π, let λ_M=M+1/2, and use the right edge coordinate x_M(z)=a-z/λ_M. [SG-12 The scaled interval-edge Gibbs profile]
Exact mathematical passage
j=\mathbf1_{[-a,a)}
\mathbf1_{[0,T/2)}Use in source · 1 passage
Search aliases: 1_[0,T/2) mathbf
Use in source · Signal_Reconstruction_Traditional.tex, line 1339
G_T _1=T/4. For a shift by d∈[0,T/2], on an interval of length T-d the difference has magnitude d/T, and on an interval of length d it has magnitude 1-d/T. Hence its L^1 norm is (T-d)d/T+d(1-d/T)=2d(1-d/T). Apply the triangle inequality term by term. Periodic translation invariance makes the calculation depend only on circular shift distance. To refute coefficient one, take j=1_[0,T/2)-tfrac12, widehat j=1_[T/4,T/2)-tfrac14. Both have jumps (1,-1), with one jump shifted by T/4. Their norm differ…
Exact mathematical passage
j=\mathbf1_{[0,T/2)}-\tfrac12,\qquad \widehat j=\mathbf1_{[T/4,T/2)}-\tfrac14.
\mathbf1_{[T/4,T/2)}Use in source · 1 passage
Search aliases: 1_[T/4,T/2) mathbf
Use in source · Signal_Reconstruction_Traditional.tex, line 1339
G_T _1=T/4. For a shift by d∈[0,T/2], on an interval of length T-d the difference has magnitude d/T, and on an interval of length d it has magnitude 1-d/T. Hence its L^1 norm is (T-d)d/T+d(1-d/T)=2d(1-d/T). Apply the triangle inequality term by term. Periodic translation invariance makes the calculation depend only on circular shift distance. To refute coefficient one, take j=1_[0,T/2)-tfrac12, widehat j=1_[T/4,T/2)-tfrac14. Both have jumps (1,-1), with one jump shifted by T/4. Their norm differ…
Exact mathematical passage
j=\mathbf1_{[0,T/2)}-\tfrac12,\qquad \widehat j=\mathbf1_{[T/4,T/2)}-\tfrac14.
=Use in source · 270 passages
Search aliases: =
Use in source · Signal_Reconstruction_Traditional.tex, line 84
The default periodic domain is T= R/(2π Z), with widehat f(n)=frac12π∫_0^2πf(x)e^-inx dx. Signal identities inferred from Fourier coefficients are identities of L^1 classes. For a piecewise-constant edge we select the right-continuous representative. This convention fixes the values at jumps, which Fourier integrals alone cannot determine. When an L^1 norm occurs it is the unnormalized integral over a period. On the unit circle we use chord distance |z-w| and circular angular distance d_ T(τ,σ).
Exact mathematical passage
\mathbb T=\mathbb R/(2\pi\mathbb Z)
AUse in source · 35 passages
Search aliases: A
Use in source · Signal_Reconstruction_Traditional.tex, line 520
smallDomain and hypotheses. Finite real or complex Euclidean spaces. Let S⊆ F^m be a subspace, P its orthogonal projector, R=I-P, D∈ F^m× d, and A=RD. Let s≥0 be an integer. [SG-14 Residualized sparse injectivity]
Exact mathematical passage
A=RD
\mathcal AUse in source · 8 passages
Search aliases: A mathcal
Use in source · Signal_Reconstruction_Traditional.tex, line 128
The maps A x=(P_ax)_a, S(x_a)_a=∑_ax_a are inverse isomorphisms between X and ⊕_aX_a. In particular, x=∑_aP_ax uniquely.
Exact mathematical passage
\mathcal A x=(P_ax)_a,\qquad \mathcal S(x_a)_a=\sum_ax_a
aUse in source · 98 passages
Search aliases: a
Use in source · Signal_Reconstruction_Traditional.tex, line 124
smallDomain and hypotheses. Vector spaces over a field. Let P_1,…,P_m∈End(X) satisfy P_aP_b=δ_abP_a and ∑_aP_a=I_X. Set X_a=P_aX. [SG-01 Complete packet analysis and synthesis]
Exact mathematical passage
P_aP_b=\delta_{ab}P_a
a_*Use in source · 4 passages
Search aliases: a_*
Use in source · Signal_Reconstruction_Traditional.tex, line 594
smallDomain and hypotheses. Finite real dictionaries with positive amplitudes. Let B have full column rank, P be the orthogonal projector onto its range, R=I-P, and let Q be a finite mask library with Rq≠0 for every q. Suppose y=Bθ_*+a_*q_*, with a_*>0 and q_*∈ Q. [SG-16 Profiled one-packet discovery and its finite gap]
Exact mathematical passage
y=B\theta_*+a_*q_*
a_0Use in source · 1 passage
Search aliases: a_0
Use in source · Signal_Reconstruction_Traditional.tex, line 159
T=∑_a,bP_bTP_a, (ST)_ca=∑_bS_cbT_ba. The block representation is an algebra isomorphism to End(⊕_aX_a), with multiplication by contraction of intermediate packets. For every integer k≥1, iterating the product gives P_a_kT^kP_a_0 =∑_a_1,…,a_k-1 (P_a_kTP_a_k-1)⋯(P_a_1TP_a_0).
Exact mathematical passage
P_{a_k}T^kP_{a_0} =\sum_{a_1,\ldots,a_{k-1}} (P_{a_k}TP_{a_{k-1}})\cdots(P_{a_1}TP_{a_0}).
a_1Use in source · 1 passage
Search aliases: a_1
Use in source · Signal_Reconstruction_Traditional.tex, line 159
T=∑_a,bP_bTP_a, (ST)_ca=∑_bS_cbT_ba. The block representation is an algebra isomorphism to End(⊕_aX_a), with multiplication by contraction of intermediate packets. For every integer k≥1, iterating the product gives P_a_kT^kP_a_0 =∑_a_1,…,a_k-1 (P_a_kTP_a_k-1)⋯(P_a_1TP_a_0).
Exact mathematical passage
P_{a_k}T^kP_{a_0} =\sum_{a_1,\ldots,a_{k-1}} (P_{a_k}TP_{a_{k-1}})\cdots(P_{a_1}TP_{a_0}).
a_{c\ell}Use in source · 1 passage
Search aliases: a_cℓ ell
Use in source · Signal_Reconstruction_Traditional.tex, line 1396
smallDomain and hypotheses. A finite collection of exponential moment channels. Let m_n^(c)=∑_ℓ=1^ra_cℓz_ℓ^n, with distinct common unit-modulus nodes. At each node at least one channel amplitude is nonzero. Form the stacked linear recurrence system for a monic polynomial of degree r and assume its coefficient matrix has column rank r. [SG-39 Shared-location multichannel recovery]
Exact mathematical passage
m_n^{(c)}=\sum_{\ell=1}^ra_{c\ell}z_\ell^n
a_kUse in source · 1 passage
Search aliases: a_k
Use in source · Signal_Reconstruction_Traditional.tex, line 159
T=∑_a,bP_bTP_a, (ST)_ca=∑_bS_cbT_ba. The block representation is an algebra isomorphism to End(⊕_aX_a), with multiplication by contraction of intermediate packets. For every integer k≥1, iterating the product gives P_a_kT^kP_a_0 =∑_a_1,…,a_k-1 (P_a_kTP_a_k-1)⋯(P_a_1TP_a_0).
Exact mathematical passage
P_{a_k}T^kP_{a_0} =\sum_{a_1,\ldots,a_{k-1}} (P_{a_k}TP_{a_{k-1}})\cdots(P_{a_1}TP_{a_0}).
a_{k-1}Use in source · 1 passage
Search aliases: a_k-1
Use in source · Signal_Reconstruction_Traditional.tex, line 159
T=∑_a,bP_bTP_a, (ST)_ca=∑_bS_cbT_ba. The block representation is an algebra isomorphism to End(⊕_aX_a), with multiplication by contraction of intermediate packets. For every integer k≥1, iterating the product gives P_a_kT^kP_a_0 =∑_a_1,…,a_k-1 (P_a_kTP_a_k-1)⋯(P_a_1TP_a_0).
Exact mathematical passage
P_{a_k}T^kP_{a_0} =\sum_{a_1,\ldots,a_{k-1}} (P_{a_k}TP_{a_{k-1}})\cdots(P_{a_1}TP_{a_0}).
A_MUse in source · 9 passages
Search aliases: A_M
Use in source · Signal_Reconstruction_Traditional.tex, line 1282
smallDomain and hypotheses. A normed signal space. Let f=s+j, let widehat j belong to the same space, and let A_M be a bounded linear operator with norm at most C_M. Define widehat f_M=A_M(f-widehat j)+widehat j. [SG-36 Resolved synthesis splits the two error sources]
Exact mathematical passage
A_M
\mathcal A_PUse in source · 2 passages
Search aliases: A_P mathcal
Use in source · Signal_Reconstruction_Traditional.tex, line 187
P_bTP_a=∑_σ∈ F_b∑_ρ∈ F_aQ_σ TQ_ρ. The coordinate changes J= A_Q S_P and C= A_P S_Q satisfy CJ=I and JC=I on their packet coordinate spaces. Refinement is exact before any child packet is discarded.
Exact mathematical passage
C=\mathcal A_P\mathcal S_Q
\mathcal A_QUse in source · 2 passages
Search aliases: A_Q mathcal
Use in source · Signal_Reconstruction_Traditional.tex, line 186
P_bTP_a=∑_σ∈ F_b∑_ρ∈ F_aQ_σ TQ_ρ. The coordinate changes J= A_Q S_P and C= A_P S_Q satisfy CJ=I and JC=I on their packet coordinate spaces. Refinement is exact before any child packet is discarded.
Exact mathematical passage
J=\mathcal A_Q\mathcal S_P
a_\ellUse in source · 20 passages
Search aliases: a_ℓ ell
Use in source · Signal_Reconstruction_Traditional.tex, line 664
Let U_K,L consist of real periodic signals f=s+j, where s∈ S_K and j is right-continuous, piecewise constant, mean zero, and has at most L distinct active jumps. An active jump has nonzero amplitude. For an actual jump count r, write Dj=∑_ℓ=1^r a_ℓδ_τ_ℓ, z_ℓ=e^-iτ_ℓ, ∑_ℓ a_ℓ=0. The sum-zero condition is required by periodicity. A single nonzero active jump cannot occur, so the nonconstant real class starts at r=2. The case r=0 is allowed and will be handled explicitly. The smooth-edge representa…
Exact mathematical passage
Dj=\sum_{\ell=1}^r a_\ell\delta_{\tau_\ell},\quad z_\ell=e^{-i\tau_\ell},\quad\sum_\ell a_\ell=0.
\arcsinUse in source · 4 passages
Search aliases: arcsin
Use in source · Signal_Reconstruction_Traditional.tex, line 922
ly the triangle inequality to the coefficient error polynomial and use |z|≤|z_ℓ|+ρ. On the circle, |z-z_ℓ|=ρ and |z-z_j|≥|z_ℓ-z_j|-ρ, giving the lower product bound. For unit true nodes and a recovered root inside a disk with ρ<1, radial normalization changes its chord error to at most 2ρ by the triangle inequality. Its angular location error is then at most 2arcsinρ. The disk radius is a complex-root error, not directly an angular error.
Exact mathematical passage
2\arcsin\rho
\argUse in source · 1 passage
Search aliases: arg
Use in source · Signal_Reconstruction_Traditional.tex, line 747
The unique monic annihilator Q(z)=∏_ℓ=1^r(z-z_ℓ)=∑_k=0^r-1q_kz^k+z^r is recovered from H_r,rq=-v, v=(m_n_0+r+i)_i=0^r-1. Its roots are exactly the nodes. Once the nodes are known, r consecutive moments determine the coefficients uniquely by a Vandermonde solve. For unit-modulus nodes the corresponding locations are τ_ℓ=-arg z_ℓpmod2π.
Exact mathematical passage
\tau_\ell=-\arg z_\ell\pmod{2\pi}
BUse in source · 6 passages
Search aliases: B
Use in source · Signal_Reconstruction_Traditional.tex, line 171
Insert I=∑_aP_a on both sides of T to obtain reconstruction. Insert that identity between S and T to obtain the product formula. Equivalently, the block map is T↦ A T S; SG-01 shows that its inverse is B↦ S B A and that it preserves multiplication. The path formula follows by inserting the same identity between every successive pair of operators. These are finite sums, so no convergence hypothesis is hidden in the insertion.
Exact mathematical passage
B\mapsto\mathcal S B\mathcal A
bUse in source · 48 passages
Search aliases: b
Use in source · Signal_Reconstruction_Traditional.tex, line 124
smallDomain and hypotheses. Vector spaces over a field. Let P_1,…,P_m∈End(X) satisfy P_aP_b=δ_abP_a and ∑_aP_a=I_X. Set X_a=P_aX. [SG-01 Complete packet analysis and synthesis]
Exact mathematical passage
P_aP_b=\delta_{ab}P_a
b_0Use in source · 1 passage
Search aliases: b_0
Use in source · Signal_Reconstruction_Traditional.tex, line 1255
for the coarse phase error. The strict branch margin selects the associated root by SG-32. Normalize the aliased estimate and use SG-33: its root error is N^-1 times an error of order N^-p-1, giving N^-p-2. On the unit circle, d_ T=2arcsin(|widehat z-z|/2), which has the same order for small errors. For the amplitude use widehat a=widehat b_0 overline u^-q_0 and a=b_0u^-q_0. Since q_0 is fixed and both powers have unit modulus, SG-26's telescoping power estimate bounds the power error by q_0|ove…
Exact mathematical passage
a=b_0u^{-q_0}
b_1Use in source · 1 passage
Search aliases: b_1
Use in source · Signal_Reconstruction_Traditional.tex, line 260
Exactly d unit-modulus nodes satisfy the same data. Hence the physical node is unique precisely when d=1. If integers c,b_1,…,b_k obey Nc+∑_jr_jb_j=d, then u^c∏_jw_j^b_j=z^d. For d=1 this is an explicit exact de-aliasing formula.
Exact mathematical passage
c,b_1,\ldots,b_k
b_jUse in source · 2 passages
Search aliases: b_j
Use in source · Signal_Reconstruction_Traditional.tex, line 261
Exactly d unit-modulus nodes satisfy the same data. Hence the physical node is unique precisely when d=1. If integers c,b_1,…,b_k obey Nc+∑_jr_jb_j=d, then u^c∏_jw_j^b_j=z^d. For d=1 this is an explicit exact de-aliasing formula.
Exact mathematical passage
Nc+\sum_jr_jb_j=d
b_kUse in source · 1 passage
Search aliases: b_k
Use in source · Signal_Reconstruction_Traditional.tex, line 260
Exactly d unit-modulus nodes satisfy the same data. Hence the physical node is unique precisely when d=1. If integers c,b_1,…,b_k obey Nc+∑_jr_jb_j=d, then u^c∏_jw_j^b_j=z^d. For d=1 this is an explicit exact de-aliasing formula.
Exact mathematical passage
c,b_1,\ldots,b_k
b_MUse in source · 3 passages
Search aliases: b_M
Use in source · Signal_Reconstruction_Traditional.tex, line 1356
smallDomain and hypotheses. The rate family of SG-35 with T=2π. Assume a fixed finite jump count, bounded jump amplitudes, the hypotheses of SG-35, and a bounded linear smooth reconstruction A_M acting on L^1( T). Require widehat j to be the canonical packet of real matched estimated amplitudes whose sum is zero. Let b_M= s-A_Ms _1 and C_M= A_M _L^1→ L^1. [SG-38 A conditional global L1 reconstruction rate]
Exact mathematical passage
b_M=\|s-A_Ms\|_1
b_{\min}Use in source · 1 passage
Search aliases: b_min min
Use in source · Signal_Reconstruction_Traditional.tex, line 1231
smallDomain and hypotheses. A selected sequence of decimation scales. Let p≥0, fix q_0 and the finite number of channel samples and offsets, and take a sequence N→∞. Assume distinct aliased nodes, uniform rank-threshold gaps/local inverse bounds for the aligned Prony and coefficient fits, |b_ℓ,0|≥ b_min>0, uniformly bounded amplitudes, and the strict branch margin of SG-32. Suppose the moment defect is O(N^-p-1). [SG-35 Conditional asymptotic rates after calibrated lifting]
Exact mathematical passage
|b_{\ell,0}|\ge b_{\min}>0
b_{\ell,0}Use in source · 8 passages
Search aliases: b_ℓ,0 ell
Use in source · Signal_Reconstruction_Traditional.tex, line 1211
All high indices exceed K, so the derivative moments have no smooth defect. The base channel is an exponential sum in the distinct u_ℓ with coefficients b_ℓ,0=a_ℓ u_ℓ^q_0≠0. Its 2L_max+1 moments form an (L_max+1) square Hankel matrix; SG-19 gives its rank r. If r=0, the edge is zero and the low modes finish the recovery. Otherwise SG-20 recovers the u_ℓ and then the base coefficients. The adjacent channel supplies at least r moments of ∑_ℓ b_ℓ,0z_ℓ u_ℓ^q. Using the same recovered nodes, its Vand…
Exact mathematical passage
b_{\ell,0}=a_\ell u_\ell^{q_0}\ne0
b_{\ell,1}Use in source · 3 passages
Search aliases: b_ℓ,1 ell
Use in source · Signal_Reconstruction_Traditional.tex, line 1217
0≠0. Its 2L_max+1 moments form an (L_max+1) square Hankel matrix; SG-19 gives its rank r. If r=0, the edge is zero and the low modes finish the recovery. Otherwise SG-20 recovers the u_ℓ and then the base coefficients. The adjacent channel supplies at least r moments of ∑_ℓ b_ℓ,0z_ℓ u_ℓ^q. Using the same recovered nodes, its Vandermonde system recovers b_ℓ,1=b_ℓ,0z_ℓ. Division gives z_ℓ=b_ℓ,1/b_ℓ,0, and a_ℓ=b_ℓ,0u_ℓ^-q_0. SG-21 constructs j, and subtraction from the low modes constructs s. The t…
Exact mathematical passage
b_{\ell,1}=b_{\ell,0}z_\ell
b_{\ell,r}Use in source · 2 passages
Search aliases: b_ℓ,r ell
Use in source · Signal_Reconstruction_Traditional.tex, line 976
m_(q_0+q)N+r=∑_ℓ b_ℓ,ru_ℓ^q+e_(q_0+q)N+r, u_ℓ=z_ℓ^N, b_ℓ,r=a_ℓ u_ℓ^q_0z_ℓ^r. For each matched edge b_ℓ,r+1=z_ℓ b_ℓ,r. If |widehat s(n)|≤ C_s(1+|n|)^-p-2 with p≥0, then for nonnegative q,r |e_(q_0+q)N+r|≤2π C_s(q_0N)^-p-1.
Exact mathematical passage
m_{(q_0+q)N+r}=\sum_\ell b_{\ell,r}u_\ell^q+e_{(q_0+q)N+r}, \quad u_\ell=z_\ell^N,\quad b_{\ell,r}=a_\ell u_\ell^{q_0}z_\ell^r.
b_{\ell,r+1}Use in source · 1 passage
Search aliases: b_ℓ,r+1 ell
Use in source · Signal_Reconstruction_Traditional.tex, line 980
m_(q_0+q)N+r=∑_ℓ b_ℓ,ru_ℓ^q+e_(q_0+q)N+r, u_ℓ=z_ℓ^N, b_ℓ,r=a_ℓ u_ℓ^q_0z_ℓ^r. For each matched edge b_ℓ,r+1=z_ℓ b_ℓ,r. If |widehat s(n)|≤ C_s(1+|n|)^-p-2 with p≥0, then for nonnegative q,r |e_(q_0+q)N+r|≤2π C_s(q_0N)^-p-1.
Exact mathematical passage
b_{\ell,r+1}=z_\ell b_{\ell,r}
CUse in source · 29 passages
Search aliases: C
Use in source · Signal_Reconstruction_Traditional.tex, line 187
P_bTP_a=∑_σ∈ F_b∑_ρ∈ F_aQ_σ TQ_ρ. The coordinate changes J= A_Q S_P and C= A_P S_Q satisfy CJ=I and JC=I on their packet coordinate spaces. Refinement is exact before any child packet is discarded.
Exact mathematical passage
C=\mathcal A_P\mathcal S_Q
\mathbb CUse in source · 3 passages
Search aliases: C mathbb
Use in source · Signal_Reconstruction_Traditional.tex, line 256
smallDomain and hypotheses. Unit-modulus exponential nodes. Let z∈ C, |z|=1, and suppose u=z^N and w_j=z^r_j are known, with N≥1 and integer offsets r_j. Let d=gcd(N,r_1,…,r_k). [SG-06 The arithmetic of residual alias ambiguity]
Exact mathematical passage
z\in\mathbb C
cUse in source · 33 passages
Search aliases: c
Use in source · Signal_Reconstruction_Traditional.tex, line 152
T=∑_a,bP_bTP_a, (ST)_ca=∑_bS_cbT_ba. The block representation is an algebra isomorphism to End(⊕_aX_a), with multiplication by contraction of intermediate packets. For every integer k≥1, iterating the product gives P_a_kT^kP_a_0 =∑_a_1,…,a_k-1 (P_a_kTP_a_k-1)⋯(P_a_1TP_a_0).
Exact mathematical passage
T=\sum_{a,b}P_bTP_a,\qquad (ST)_{ca}=\sum_bS_{cb}T_{ba}.
\mathbb C^rUse in source · 1 passage
Search aliases: C^r mathbb
Use in source · Signal_Reconstruction_Traditional.tex, line 725
Expand a matrix entry to obtain the factorization. The leading r× r Vandermonde submatrix has determinant ∏_ℓ<k(z_k-z_ℓ), which is nonzero. Thus both rectangular Vandermonde factors have full column rank, and their transposes have full row rank. The diagonal factor is invertible. Its product with V_w^T is surjective onto C^r, and subsequent multiplication by the injective V_h has image dimension r. For an empty exponential sum every entry is zero and the rank is zero.
Exact mathematical passage
\mathbb C^r
c_0Use in source · 4 passages
Search aliases: c_0
Use in source · Signal_Reconstruction_Traditional.tex, line 1053
S(c)=R|b|^2z≠0, Φ(c):=S(c)/|S(c)|=z. Two channels already give z=c_1/c_0 exactly. More channels provide repeated adjacent relations, without an unconditional deterministic averaging gain.
Exact mathematical passage
z=c_1/c_0
c_1Use in source · 1 passage
Search aliases: c_1
Use in source · Signal_Reconstruction_Traditional.tex, line 1053
S(c)=R|b|^2z≠0, Φ(c):=S(c)/|S(c)|=z. Two channels already give z=c_1/c_0 exactly. More channels provide repeated adjacent relations, without an unconditional deterministic averaging gain.
Exact mathematical passage
z=c_1/c_0
C_MUse in source · 5 passages
Search aliases: C_M
Use in source · Signal_Reconstruction_Traditional.tex, line 1282
smallDomain and hypotheses. A normed signal space. Let f=s+j, let widehat j belong to the same space, and let A_M be a bounded linear operator with norm at most C_M. Define widehat f_M=A_M(f-widehat j)+widehat j. [SG-36 Resolved synthesis splits the two error sources]
Exact mathematical passage
C_M
C_qUse in source · 2 passages
Search aliases: C_q
Use in source · Signal_Reconstruction_Traditional.tex, line 605
For an unrestricted scalar amplitude the least-squares profile is widehat a(q)=frac⟨ Rq,Ry⟩ Rq _2^2, J(q)= Ry-widehat a(q)Rq _2^2. The true mask has widehat a(q_*)=a_* and J(q_*)=0. It is the unique zero-residual positive-amplitude mask exactly when no other residualized mask lies on its positive ray. Define the closed rays C_q={aRq:a≥0}. If Γ=inf_q≠ q_*dist(Ry,C_q)>0, with the convention inf∅=+∞, then every perturbed residualized datum Ry+e with e _2<Γ/2 selects q_* uniquely by nearest-ray dist…
Exact mathematical passage
C_q=\{aRq:a\ge0\}
c_rUse in source · 5 passages
Search aliases: c_r
Use in source · Signal_Reconstruction_Traditional.tex, line 1047
smallDomain and hypotheses. One matched unit-circle coefficient progression. Let R≥1 be an integer counting adjacent products, and let c_r=bz^r for 0≤ r≤ R, with b≠0 and |z|=1. Define S(c)=∑_r=0^R-1overlinec_rc_r+1. [SG-29 Exact phase from adjacent offset products]
Exact mathematical passage
c_r=bz^r
c_{r+1}Use in source · 4 passages
Search aliases: c_r+1
Use in source · Signal_Reconstruction_Traditional.tex, line 1047
smallDomain and hypotheses. One matched unit-circle coefficient progression. Let R≥1 be an integer counting adjacent products, and let c_r=bz^r for 0≤ r≤ R, with b≠0 and |z|=1. Define S(c)=∑_r=0^R-1overlinec_rc_r+1. [SG-29 Exact phase from adjacent offset products]
Exact mathematical passage
S(c)=\sum_{r=0}^{R-1}\overline{c_r}c_{r+1}
C_sUse in source · 5 passages
Search aliases: C_s
Use in source · Signal_Reconstruction_Traditional.tex, line 981
m_(q_0+q)N+r=∑_ℓ b_ℓ,ru_ℓ^q+e_(q_0+q)N+r, u_ℓ=z_ℓ^N, b_ℓ,r=a_ℓ u_ℓ^q_0z_ℓ^r. For each matched edge b_ℓ,r+1=z_ℓ b_ℓ,r. If |widehat s(n)|≤ C_s(1+|n|)^-p-2 with p≥0, then for nonnegative q,r |e_(q_0+q)N+r|≤2π C_s(q_0N)^-p-1.
Exact mathematical passage
|\widehat s(n)|\le C_s(1+|n|)^{-p-2}
c_\ellUse in source · 6 passages
Search aliases: c_ℓ ell
Use in source · Signal_Reconstruction_Traditional.tex, line 706
smallDomain and hypotheses. Finite exponential moments over C. Let m_n_0+k=∑_ℓ=1^r c_ℓ z_ℓ^k, with distinct nonzero nodes and nonzero coefficients. Let H_h,w=(m_n_0+i+j)_0≤ i<h, 0≤ j<w, with h,w≥ r. [SG-19 Hankel rank detects the actual order]
Exact mathematical passage
m_{n_0+k}=\sum_{\ell=1}^r c_\ell z_\ell^k
\cosUse in source · 1 passage
Search aliases: cos
Use in source · Signal_Reconstruction_Traditional.tex, line 1665
Take a bandlimited background s(x)=frac14+frac25cos(2x)-frac13sin(3x)+frac15cos(5x) and a mean-zero edge with locations and jumps (τ_ℓ)=(0,π/2,π,3π/2), (a_ℓ)=(1,-2,-1,2). The physical nodes are (1,-i,-1,i). Their derivative moments are m_n=1-2(-i)^n-(-1)^n+2i^n, so in residues modulo four they are m_4q=0, m_4q+1=2+4i, m_4q+2=0, m_4q+3=2-4i. Since the smooth degree is five, the first clean consecutive moment is n_0=6. With an order bound four, coefficients at n=6,…,13 determine the edge. The Hank…
Exact mathematical passage
s(x)=\frac14+\frac25\cos(2x)-\frac13\sin(3x)+\frac15\cos(5x)
DUse in source · 28 passages
Search aliases: D
Use in source · Signal_Reconstruction_Traditional.tex, line 359
smallDomain and hypotheses. Periodic integrable signals with one known edge shape. Let S_K be the trigonometric polynomials of degree at most K. Let m∈ L^1( T) and a finite nonempty set F⊂{n:|n|>K} satisfy D_F=∑_n∈ F|widehat m(n)|^2>0. [SG-09 Out-of-band matched identification]
Exact mathematical passage
D_F=\sum_{n\in F}|\widehat m(n)|^2>0
dUse in source · 51 passages
Search aliases: d
Use in source · Signal_Reconstruction_Traditional.tex, line 85
The default periodic domain is T= R/(2π Z), with widehat f(n)=frac12π∫_0^2πf(x)e^-inx dx. Signal identities inferred from Fourier coefficients are identities of L^1 classes. For a piecewise-constant edge we select the right-continuous representative. This convention fixes the values at jumps, which Fourier integrals alone cannot determine. When an L^1 norm occurs it is the unnormalized integral over a period. On the unit circle we use chord distance |z-w| and circular angular distance d_ T(τ,σ).
Exact mathematical passage
\widehat f(n)=\frac1{2\pi}\int_0^{2\pi}f(x)e^{-inx}\,dx.
d_{\mathbb T}Use in source · 3 passages
Search aliases: d_ T mathbb
Use in source · Signal_Reconstruction_Traditional.tex, line 93
identities inferred from Fourier coefficients are identities of L^1 classes. For a piecewise-constant edge we select the right-continuous representative. This convention fixes the values at jumps, which Fourier integrals alone cannot determine. When an L^1 norm occurs it is the unnormalized integral over a period. On the unit circle we use chord distance |z-w| and circular angular distance d_ T(τ,σ).
Exact mathematical passage
d_{\mathbb T}(\tau,\sigma)
D_FUse in source · 3 passages
Search aliases: D_F
Use in source · Signal_Reconstruction_Traditional.tex, line 359
smallDomain and hypotheses. Periodic integrable signals with one known edge shape. Let S_K be the trigonometric polynomials of degree at most K. Let m∈ L^1( T) and a finite nonempty set F⊂{n:|n|>K} satisfy D_F=∑_n∈ F|widehat m(n)|^2>0. [SG-09 Out-of-band matched identification]
Exact mathematical passage
D_F=\sum_{n\in F}|\widehat m(n)|^2>0
D_F^{-1}Use in source · 1 passage
Search aliases: D_F^-1
Use in source · Signal_Reconstruction_Traditional.tex, line 363
On X= S_K+span{m}, the functional ℓ_F(g)=D_F^-1∑_n∈ Foverlinewidehat m(n)widehat g(n) satisfies ℓ_F(s+am)=a. The decomposition is direct. If the measured coefficients have errors η_n, then |widetilde a-a|≤ D_F^-1/2(∑_n∈ F|η_n|^2)^1/2.
Exact mathematical passage
\ell_F(g)=D_F^{-1}\sum_{n\in F}\overline{\widehat m(n)}\widehat g(n)
D_F^{-1/2}Use in source · 1 passage
Search aliases: D_F^-1/2
Use in source · Signal_Reconstruction_Traditional.tex, line 368
On X= S_K+span{m}, the functional ℓ_F(g)=D_F^-1∑_n∈ Foverlinewidehat m(n)widehat g(n) satisfies ℓ_F(s+am)=a. The decomposition is direct. If the measured coefficients have errors η_n, then |widetilde a-a|≤ D_F^-1/2(∑_n∈ F|η_n|^2)^1/2.
Exact mathematical passage
|\widetilde a-a|\le D_F^{-1/2}\Bigl(\sum_{n\in F}|\eta_n|^2\Bigr)^{1/2}.
D_MUse in source · 5 passages
Search aliases: D_M
Use in source · Signal_Reconstruction_Traditional.tex, line 456
Direct integration gives widehat j(0)=a/π and widehat j(n)=sin(na)/(π n) for n≠0. The Dirichlet kernel D_M(t)=sin(λ_Mt)/sin(t/2) yields S_Mj(x)=frac12π∫_-a^aD_M(x-t) dt, (S_Mj)'(x)=fracD_M(x+a)-D_M(x-a)2π. At x=a, the coefficient formula gives S_Mj(a)=frac aπ+frac1π∑_n=1^Mfracsin(2na)n ⟶frac12. For completeness, the sine-series limit follows by summation by parts from bounded partial sums of e^2ina and by taking radial limits in ∑_n≥1r^ne^2ina/n=-log(1-re^2ia); its imaginary part tends to (π-2a)…
Exact mathematical passage
D_M(t)=\sin(\lambda_Mt)/\sin(t/2)
d_TUse in source · 1 passage
Search aliases: d_T
Use in source · Signal_Reconstruction_Traditional.tex, line 1309
smallDomain and hypotheses. Mean-zero periodic packets of period T>0. Let j,widehat j have matched real jump data (a_ℓ,τ_ℓ) and (widehat a_ℓ,widehatτ_ℓ), with both amplitude sums zero. Use their canonical representatives and let d_ℓ=d_T(τ_ℓ,widehatτ_ℓ)≤ T/2. [SG-37 Sharp additive L1 control for normalized jump packets]
Exact mathematical passage
d_\ell=d_T(\tau_\ell,\widehat\tau_\ell)\le T/2
d_\ellUse in source · 3 passages
Search aliases: d_ℓ ell
Use in source · Signal_Reconstruction_Traditional.tex, line 1309
smallDomain and hypotheses. Mean-zero periodic packets of period T>0. Let j,widehat j have matched real jump data (a_ℓ,τ_ℓ) and (widehat a_ℓ,widehatτ_ℓ), with both amplitude sums zero. Use their canonical representatives and let d_ℓ=d_T(τ_ℓ,widehatτ_ℓ)≤ T/2. [SG-37 Sharp additive L1 control for normalized jump packets]
Exact mathematical passage
d_\ell=d_T(\tau_\ell,\widehat\tau_\ell)\le T/2
\degUse in source · 1 passage
Search aliases: deg
Use in source · Signal_Reconstruction_Traditional.tex, line 1419
smallDomain and hypotheses. Polynomial-exponential sequences. Let m_n=∑_ℓ=1^rz_ℓ^nP_ℓ(n) with distinct nonzero nodes and deg P_ℓ≤ d. Let E be the forward shift. For an offset ratio assume N≥1 and the leading coefficient of P_ℓ at degree d is nonzero. [SG-40 Confluent annihilation and the leading offset phase]
Exact mathematical passage
\deg P_\ell\le d
\operatorname{diag}Use in source · 1 passage
Search aliases: diag operatorname
Use in source · Signal_Reconstruction_Traditional.tex, line 710
Writing V_h=(z_ℓ^i)_i,ℓ and V_w=(z_ℓ^j)_j,ℓ, H_h,w=V_hdiag(c_ℓ)V_w^ T, rankH_h,w=r. For r=0 the Hankel matrix is zero. The transpose is algebraic and is not a conjugate transpose.
Exact mathematical passage
H_{h,w}=V_h\operatorname{diag}(c_\ell)V_w^{\mathsf T}, \qquad\operatorname{rank}H_{h,w}=r.
\dimUse in source · 1 passage
Search aliases: dim
Use in source · Signal_Reconstruction_Traditional.tex, line 1015
The singular-value perturbation inequality [source reference] gives |σ_j(H)-σ_j(H_0)|≤ E _2 for 1≤ j≤ k. It holds equally over the complex field: in the min--max characterization σ_j(A)=min_dim V=w-j+1max_x∈ V, x =1 Ax . the triangle inequality bounds each restricted maximum for A+E by that for A plus E _2. Minimize and then interchange the two matrices. For j≤ r this yields σ_j(H)≥σ_r(H_0)-ϵ>t. For j>r, when such indices exist, it gives σ_j(H)≤ϵ<t. For rank zero the same upper bound applies to …
Exact mathematical passage
\sigma_j(A)=\min_{\dim V=w-j+1}\max_{x\in V,\,\|x\|=1}\|Ax\|.
\operatorname{dist}Use in source · 1 passage
Search aliases: dist operatorname
Use in source · Signal_Reconstruction_Traditional.tex, line 606
For an unrestricted scalar amplitude the least-squares profile is widehat a(q)=frac⟨ Rq,Ry⟩ Rq _2^2, J(q)= Ry-widehat a(q)Rq _2^2. The true mask has widehat a(q_*)=a_* and J(q_*)=0. It is the unique zero-residual positive-amplitude mask exactly when no other residualized mask lies on its positive ray. Define the closed rays C_q={aRq:a≥0}. If Γ=inf_q≠ q_*dist(Ry,C_q)>0, with the convention inf∅=+∞, then every perturbed residualized datum Ry+e with e _2<Γ/2 selects q_* uniquely by nearest-ray dist…
Exact mathematical passage
\Gamma=\inf_{q\ne q_*}\operatorname{dist}(Ry,C_q)>0,
EUse in source · 26 passages
Search aliases: E
Use in source · Signal_Reconstruction_Traditional.tex, line 204
smallDomain and hypotheses. Vector spaces with an invertible transform. Let V:X→ Y be an isomorphism and (E_a) a finite complete packet family on Y. [SG-04 Pulled-back transform packets]
Exact mathematical passage
(E_a)
eUse in source · 38 passages
Search aliases: e
Use in source · Signal_Reconstruction_Traditional.tex, line 85
The default periodic domain is T= R/(2π Z), with widehat f(n)=frac12π∫_0^2πf(x)e^-inx dx. Signal identities inferred from Fourier coefficients are identities of L^1 classes. For a piecewise-constant edge we select the right-continuous representative. This convention fixes the values at jumps, which Fourier integrals alone cannot determine. When an L^1 norm occurs it is the unnormalized integral over a period. On the unit circle we use chord distance |z-w| and circular angular distance d_ T(τ,σ).
Exact mathematical passage
\widehat f(n)=\frac1{2\pi}\int_0^{2\pi}f(x)e^{-inx}\,dx.
e^{-inx}Use in source · 3 passages
Search aliases: e^-inx
Use in source · Signal_Reconstruction_Traditional.tex, line 85
The default periodic domain is T= R/(2π Z), with widehat f(n)=frac12π∫_0^2πf(x)e^-inx dx. Signal identities inferred from Fourier coefficients are identities of L^1 classes. For a piecewise-constant edge we select the right-continuous representative. This convention fixes the values at jumps, which Fourier integrals alone cannot determine. When an L^1 norm occurs it is the unnormalized integral over a period. On the unit circle we use chord distance |z-w| and circular angular distance d_ T(τ,σ).
Exact mathematical passage
\widehat f(n)=\frac1{2\pi}\int_0^{2\pi}f(x)e^{-inx}\,dx.
e^{-in\delta}Use in source · 2 passages
Search aliases: e^-inδ delta
Use in source · Signal_Reconstruction_Traditional.tex, line 322
Their unordered location sets have permutation-matched bottleneck distance at least π-δ, while their measured derivative moments satisfy |m_n-widetilde m_n| =|(1-e^-inπ)(1-e^-inδ)| ≤2|n|δ (n∈ F). There is no Lipschitz inverse for locations with one constant on all distinct two-jump configurations, even at this fixed order and with known amplitudes.
Exact mathematical passage
|m_n-\widetilde m_n| =|(1-e^{-in\pi})(1-e^{-in\delta})| \le2|n|\delta\qquad(n\in F).
e^{-in\pi}Use in source · 2 passages
Search aliases: e^-inπ pi
Use in source · Signal_Reconstruction_Traditional.tex, line 322
Their unordered location sets have permutation-matched bottleneck distance at least π-δ, while their measured derivative moments satisfy |m_n-widetilde m_n| =|(1-e^-inπ)(1-e^-inδ)| ≤2|n|δ (n∈ F). There is no Lipschitz inverse for locations with one constant on all distinct two-jump configurations, even at this fixed order and with known amplitudes.
Exact mathematical passage
|m_n-\widetilde m_n| =|(1-e^{-in\pi})(1-e^{-in\delta})| \le2|n|\delta\qquad(n\in F).
e^{-in\tau_\ell}Use in source · 1 passage
Search aliases: e^-inτ_ℓ tau ell
Use in source · Signal_Reconstruction_Traditional.tex, line 697
For a smooth periodic test function φ, integrate by parts over each constant segment of j. Boundary terms telescope at each jump to a_ℓφ(τ_ℓ), which is exactly the displayed distribution. Evaluate the distribution on e^-inx: ⟨ Dj,e^-inx⟩ =-∫ j(x)(-in)e^-inx dx =2π inwidehat j(n). The Dirac sum gives ∑_ℓ a_ℓ e^-inτ_ℓ. For n>K, the smooth polynomial contributes zero. Without bandlimiting it contributes the stated defect. At n=0 the derivative moment is zero, in agreement with ∑ a_ℓ=0; it cannot de…
Exact mathematical passage
\sum_\ell a_\ell e^{-in\tau_\ell}
e^{-it}Use in source · 1 passage
Search aliases: e^-it
Use in source · Signal_Reconstruction_Traditional.tex, line 337
Every circular distance from a point of {0,δ} to a point of {π,π+δ} is at least π-δ. This bounds every matching. The moments of the first packet are 1-e^-inδ; translating by π multiplies them by e^-inπ. Subtraction gives the formula. The elementary inequality |1-e^-it|≤|t| proves the bound. Every fixed finite norm of the measurement difference tends to zero as δ→0, whereas the location distance stays bounded away from zero. A common finite Lipschitz constant is therefore impossible. This argumen…
Exact mathematical passage
|1-e^{-it}|\le|t|
e^{-i\tau_\ell}Use in source · 2 passages
Search aliases: e^-iτ_ℓ tau ell
Use in source · Signal_Reconstruction_Traditional.tex, line 664
Let U_K,L consist of real periodic signals f=s+j, where s∈ S_K and j is right-continuous, piecewise constant, mean zero, and has at most L distinct active jumps. An active jump has nonzero amplitude. For an actual jump count r, write Dj=∑_ℓ=1^r a_ℓδ_τ_ℓ, z_ℓ=e^-iτ_ℓ, ∑_ℓ a_ℓ=0. The sum-zero condition is required by periodicity. A single nonzero active jump cannot occur, so the nonconstant real class starts at r=2. The case r=0 is allowed and will be handled explicitly. The smooth-edge representa…
Exact mathematical passage
Dj=\sum_{\ell=1}^r a_\ell\delta_{\tau_\ell},\quad z_\ell=e^{-i\tau_\ell},\quad\sum_\ell a_\ell=0.
e^{-\rho}Use in source · 1 passage
Search aliases: e^-ρ rho
Use in source · Signal_Reconstruction_Traditional.tex, line 418
The reconstruction error is exactly f- R_Mf=s-S_Ms, and f- R_Mf _∞ ≤frac2Ce^-ρ(M+1)1-e^-ρ. The bound is uniform across the jumps of j.
Exact mathematical passage
\|f-\mathcal R_Mf\|_\infty \le\frac{2Ce^{-\rho(M+1)}}{1-e^{-\rho}}.
e^{-\rho n}Use in source · 1 passage
Search aliases: e^-ρ n rho
Use in source · Signal_Reconstruction_Traditional.tex, line 430
Cancel j from the defining identity. Absolute summability of the Fourier coefficients gives an everywhere uniformly convergent series for s. The omitted positive and negative frequencies are bounded by 2C∑_n>Me^-ρ n, which is the displayed geometric series. No limiting trigonometric approximation of j occurs in this estimate.
Exact mathematical passage
2C\sum_{n>M}e^{-\rho n}
e^{-\rho(M+1)}Use in source · 1 passage
Search aliases: e^-ρ(M+1) rho
Use in source · Signal_Reconstruction_Traditional.tex, line 418
The reconstruction error is exactly f- R_Mf=s-S_Ms, and f- R_Mf _∞ ≤frac2Ce^-ρ(M+1)1-e^-ρ. The bound is uniform across the jumps of j.
Exact mathematical passage
\|f-\mathcal R_Mf\|_\infty \le\frac{2Ce^{-\rho(M+1)}}{1-e^{-\rho}}.
e^{-\rho|n|}Use in source · 1 passage
Search aliases: e^-ρ|n| rho
Use in source · Signal_Reconstruction_Traditional.tex, line 414
smallDomain and hypotheses. Periodic signals with a supplied exact edge packet. Let f=s+j with |widehat s(n)|≤ Ce^-ρ|n|, where C≥0 and ρ>0. Use the canonical uniformly convergent Fourier-series representative of s. Set R_Mf=S_Ms+j. [SG-11 Resolved error for an analytic background]
Exact mathematical passage
|\widehat s(n)|\le Ce^{-\rho|n|}
e^{2ia}Use in source · 1 passage
Search aliases: e^2ia
Use in source · Signal_Reconstruction_Traditional.tex, line 468
_Mt)/sin(t/2) yields S_Mj(x)=frac12π∫_-a^aD_M(x-t) dt, (S_Mj)'(x)=fracD_M(x+a)-D_M(x-a)2π. At x=a, the coefficient formula gives S_Mj(a)=frac aπ+frac1π∑_n=1^Mfracsin(2na)n ⟶frac12. For completeness, the sine-series limit follows by summation by parts from bounded partial sums of e^2ina and by taking radial limits in ∑_n≥1r^ne^2ina/n=-log(1-re^2ia); its imaginary part tends to (π-2a)/2 for 0<2a<2π. Now differentiate the scaled function: frac ddzS_Mj(x_M(z))= fracD_M(-z/λ_M)-D_M(2a-z/λ_M)2πλ_M. Th…
Exact mathematical passage
\sum_{n\ge1}r^ne^{2ina}/n=-\log(1-re^{2ia})
e^{2ina}Use in source · 2 passages
Search aliases: e^2ina
Use in source · Signal_Reconstruction_Traditional.tex, line 467
The Dirichlet kernel D_M(t)=sin(λ_Mt)/sin(t/2) yields S_Mj(x)=frac12π∫_-a^aD_M(x-t) dt, (S_Mj)'(x)=fracD_M(x+a)-D_M(x-a)2π. At x=a, the coefficient formula gives S_Mj(a)=frac aπ+frac1π∑_n=1^Mfracsin(2na)n ⟶frac12. For completeness, the sine-series limit follows by summation by parts from bounded partial sums of e^2ina and by taking radial limits in ∑_n≥1r^ne^2ina/n=-log(1-re^2ia); its imaginary part tends to (π-2a)/2 for 0<2a<2π. Now differentiate the scaled function: frac ddzS_Mj(x_M(z))= fracD…
Exact mathematical passage
e^{2ina}
e^{ikt}Use in source · 1 passage
Search aliases: e^ikt
Use in source · Signal_Reconstruction_Traditional.tex, line 498
The averaging kernel is F_M(t)=frac1M+1|∑_k=0^Me^ikt|^2≥0, frac12π∫_0^2πF_M(t) dt=1. Its convolution with j is therefore a weighted mean of numbers in [0,1]. The resulting finite trigonometric polynomial is continuous. A nonconstant interval indicator is discontinuous, so the mean cannot equal its selected representative at all points or have its exact one-sided jump.
Exact mathematical passage
F_M(t)=\frac1{M+1}\left|\sum_{k=0}^Me^{ikt}\right|^2\ge0, \qquad\frac1{2\pi}\int_0^{2\pi}F_M(t)\,dt=1.
e_{(q_0+q)N+r}Use in source · 2 passages
Search aliases: e_(q_0+q)N+r
Use in source · Signal_Reconstruction_Traditional.tex, line 976
m_(q_0+q)N+r=∑_ℓ b_ℓ,ru_ℓ^q+e_(q_0+q)N+r, u_ℓ=z_ℓ^N, b_ℓ,r=a_ℓ u_ℓ^q_0z_ℓ^r. For each matched edge b_ℓ,r+1=z_ℓ b_ℓ,r. If |widehat s(n)|≤ C_s(1+|n|)^-p-2 with p≥0, then for nonnegative q,r |e_(q_0+q)N+r|≤2π C_s(q_0N)^-p-1.
Exact mathematical passage
m_{(q_0+q)N+r}=\sum_\ell b_{\ell,r}u_\ell^q+e_{(q_0+q)N+r}, \quad u_\ell=z_\ell^N,\quad b_{\ell,r}=a_\ell u_\ell^{q_0}z_\ell^r.
E_aUse in source · 4 passages
Search aliases: E_a
Use in source · Signal_Reconstruction_Traditional.tex, line 204
smallDomain and hypotheses. Vector spaces with an invertible transform. Let V:X→ Y be an isomorphism and (E_a) a finite complete packet family on Y. [SG-04 Pulled-back transform packets]
Exact mathematical passage
(E_a)
E_bUse in source · 1 passage
Search aliases: E_b
Use in source · Signal_Reconstruction_Traditional.tex, line 216
Compute P_aP_b=V^-1E_aE_bV=δ_abP_a and ∑_aP_a=V^-1IV=I. In the unitary case the pulled-back projections are orthogonal: P_a^*=P_a. Their images are mutually orthogonal, and SG-01 writes x as their finite orthogonal sum. The Pythagorean identity gives the energy formula. Without unitarity and orthogonality the algebraic identities remain valid while that norm identity need not hold.
Exact mathematical passage
P_aP_b=V^{-1}E_aE_bV=\delta_{ab}P_a
e_jUse in source · 1 passage
Search aliases: e_j
Use in source · Signal_Reconstruction_Traditional.tex, line 1617
The first statement is the defining subtraction identity. Retaining r can carry every component omitted from q, so its data cost is not bounded by a small number of packet coefficients. For an image f∈ R^m, the standard pixel vectors satisfy f=∑_j=1^mf_je_j and form a complete finite basis. Those coefficients are all the pixels themselves. A program using the truth residual thus certifies faithful storage of its available input, rather than uniqueness under a smaller measurement map.
Exact mathematical passage
f=\sum_{j=1}^mf_je_j
e_nUse in source · 3 passages
Search aliases: e_n
Use in source · Signal_Reconstruction_Traditional.tex, line 683
Dj=∑_ℓ a_ℓδ_τ_ℓ, m_n:=2π inwidehat f(n)=∑_ℓ=1^ra_ℓ z_ℓ^n. For a non-bandlimited integrable background the same formula holds with an additive defect e_n=2π inwidehat s(n).
Exact mathematical passage
e_n=2\pi in\widehat s(n)
E_QUse in source · 2 passages
Search aliases: E_Q
Use in source · Signal_Reconstruction_Traditional.tex, line 893
smallDomain and hypotheses. Monic complex polynomials of the same degree. Let r≥1 and Q(z)=∏_ℓ=1^r(z-z_ℓ) have distinct roots and widetilde Q=Q+E_Q. Choose 0<ρ<tfrac12min_ℓ≠ k|z_ℓ-z_k| when r≥2, or any ρ>0 when r=1. [SG-25 Conditional simple-root disk localization]
Exact mathematical passage
\widetilde Q=Q+E_Q
e_rUse in source · 2 passages
Search aliases: e_r
Use in source · Signal_Reconstruction_Traditional.tex, line 1072
smallDomain and hypotheses. A noisy progression as in SG-29. Let c_r=bz^r+e_r for 0≤ r≤ R, with R≥1, b≠0, |z|=1, and |e_r|≤ϵ. Put A=R|b|^2 and Δ=R(2|b|ϵ+ϵ^2). Assume Δ<A. [SG-30 Deterministic offset-phase perturbation]
Exact mathematical passage
c_r=bz^r+e_r
e_{r+1}Use in source · 1 passage
Search aliases: e_r+1
Use in source · Signal_Reconstruction_Traditional.tex, line 1085
Expand each product: overlinec_rc_r+1=|b|^2z +overlinebz^re_r+1+overlinee_rbz^r+1 +overlinee_re_r+1. The three error terms have total magnitude at most 2|b|ϵ+ϵ^2. Summing gives S(c)=Az+E with |E|≤Δ. Therefore |S(c)|≥ A-Δ>0. For any nonzero v and unit z, |frac v|v|-z| ≤|frac v|v|-frac vA|+|frac vA-z| =frac|A-|v||A+frac|v-Az|A ≤frac2|v-Az|A. Apply this inequality to v=S(c) and substitute the definitions. The number of products cancels, as appropriate for coherent adversarial errors.
Exact mathematical passage
\overline{c_r}c_{r+1}=|b|^2z +\overline{bz^r}e_{r+1}+\overline{e_r}bz^{r+1} +\overline{e_r}e_{r+1}.
\operatorname{End}Use in source · 3 passages
Search aliases: End operatorname
Use in source · Signal_Reconstruction_Traditional.tex, line 124
smallDomain and hypotheses. Vector spaces over a field. Let P_1,…,P_m∈End(X) satisfy P_aP_b=δ_abP_a and ∑_aP_a=I_X. Set X_a=P_aX. [SG-01 Complete packet analysis and synthesis]
Exact mathematical passage
P_1,\ldots,P_m\in\operatorname{End}(X)
FUse in source · 17 passages
Search aliases: F
Use in source · Signal_Reconstruction_Traditional.tex, line 180
smallDomain and hypotheses. Finite complete packet systems. Let (Q_ρ) refine (P_a), with a partition of fine labels into F_a and P_a=∑_ρ∈ F_aQ_ρ. [SG-03 Refinement preserves represented operators]
Exact mathematical passage
F_a
\mathbb FUse in source · 2 passages
Search aliases: F mathbb
Use in source · Signal_Reconstruction_Traditional.tex, line 520
smallDomain and hypotheses. Finite real or complex Euclidean spaces. Let S⊆ F^m be a subspace, P its orthogonal projector, R=I-P, D∈ F^m× d, and A=RD. Let s≥0 be an integer. [SG-14 Residualized sparse injectivity]
Exact mathematical passage
\mathcal S\subseteq\mathbb F^m
fUse in source · 32 passages
Search aliases: f
Use in source · Signal_Reconstruction_Traditional.tex, line 85
The default periodic domain is T= R/(2π Z), with widehat f(n)=frac12π∫_0^2πf(x)e^-inx dx. Signal identities inferred from Fourier coefficients are identities of L^1 classes. For a piecewise-constant edge we select the right-continuous representative. This convention fixes the values at jumps, which Fourier integrals alone cannot determine. When an L^1 norm occurs it is the unnormalized integral over a period. On the unit circle we use chord distance |z-w| and circular angular distance d_ T(τ,σ).
Exact mathematical passage
\widehat f(n)=\frac1{2\pi}\int_0^{2\pi}f(x)e^{-inx}\,dx.
\mathbb F^mUse in source · 1 passage
Search aliases: F^m mathbb
Use in source · Signal_Reconstruction_Traditional.tex, line 520
smallDomain and hypotheses. Finite real or complex Euclidean spaces. Let S⊆ F^m be a subspace, P its orthogonal projector, R=I-P, D∈ F^m× d, and A=RD. Let s≥0 be an integer. [SG-14 Residualized sparse injectivity]
Exact mathematical passage
\mathcal S\subseteq\mathbb F^m
\mathbb F^{m\times d}Use in source · 1 passage
Search aliases: F^m× d mathbb times
Use in source · Signal_Reconstruction_Traditional.tex, line 520
smallDomain and hypotheses. Finite real or complex Euclidean spaces. Let S⊆ F^m be a subspace, P its orthogonal projector, R=I-P, D∈ F^m× d, and A=RD. Let s≥0 be an integer. [SG-14 Residualized sparse injectivity]
Exact mathematical passage
D\in\mathbb F^{m\times d}
f_0Use in source · 2 passages
Search aliases: f_0
Use in source · Signal_Reconstruction_Traditional.tex, line 297
If Mf_0=Mf_1 but J(f_0)≠ J(f_1), no map R:Y→ J recovers the correct edge for both. If S∩ J≠{0}, the sum decomposition f=s+j is not unique. In particular, finitely many point samples on a periodic interval cannot recover every arbitrary interval packet.
Exact mathematical passage
Mf_0=Mf_1
f_1Use in source · 2 passages
Search aliases: f_1
Use in source · Signal_Reconstruction_Traditional.tex, line 297
If Mf_0=Mf_1 but J(f_0)≠ J(f_1), no map R:Y→ J recovers the correct edge for both. If S∩ J≠{0}, the sum decomposition f=s+j is not unique. In particular, finitely many point samples on a periodic interval cannot recover every arbitrary interval packet.
Exact mathematical passage
Mf_0=Mf_1
F_aUse in source · 3 passages
Search aliases: F_a
Use in source · Signal_Reconstruction_Traditional.tex, line 180
smallDomain and hypotheses. Finite complete packet systems. Let (Q_ρ) refine (P_a), with a partition of fine labels into F_a and P_a=∑_ρ∈ F_aQ_ρ. [SG-03 Refinement preserves represented operators]
Exact mathematical passage
F_a
F_bUse in source · 1 passage
Search aliases: F_b
Use in source · Signal_Reconstruction_Traditional.tex, line 183
P_bTP_a=∑_σ∈ F_b∑_ρ∈ F_aQ_σ TQ_ρ. The coordinate changes J= A_Q S_P and C= A_P S_Q satisfy CJ=I and JC=I on their packet coordinate spaces. Refinement is exact before any child packet is discarded.
Exact mathematical passage
P_bTP_a=\sum_{\sigma\in F_b}\sum_{\rho\in F_a}Q_\sigma TQ_\rho.
f_jUse in source · 1 passage
Search aliases: f_j
Use in source · Signal_Reconstruction_Traditional.tex, line 1617
The first statement is the defining subtraction identity. Retaining r can carry every component omitted from q, so its data cost is not bounded by a small number of packet coefficients. For an image f∈ R^m, the standard pixel vectors satisfy f=∑_j=1^mf_je_j and form a complete finite basis. Those coefficients are all the pixels themselves. A program using the truth residual thus certifies faithful storage of its available input, rather than uniqueness under a smaller measurement map.
Exact mathematical passage
f=\sum_{j=1}^mf_je_j
F_MUse in source · 1 passage
Search aliases: F_M
Use in source · Signal_Reconstruction_Traditional.tex, line 498
The averaging kernel is F_M(t)=frac1M+1|∑_k=0^Me^ikt|^2≥0, frac12π∫_0^2πF_M(t) dt=1. Its convolution with j is therefore a weighted mean of numbers in [0,1]. The resulting finite trigonometric polynomial is continuous. A nonconstant interval indicator is discontinuous, so the mean cannot equal its selected representative at all points or have its exact one-sided jump.
Exact mathematical passage
F_M(t)=\frac1{M+1}\left|\sum_{k=0}^Me^{ikt}\right|^2\ge0, \qquad\frac1{2\pi}\int_0^{2\pi}F_M(t)\,dt=1.
GUse in source · 8 passages
Search aliases: G
Use in source · Signal_Reconstruction_Traditional.tex, line 445
Uniformly for z in compact real intervals, S_Mj(x_M(z))⟶frac12+frac1πSi(z), Si(z)=∫_0^zfracsin tt dt. The first positive maximum of the limiting profile is 1+G, where G=fracSi(π)π-frac12 =0.089489872236….
Exact mathematical passage
1+G
gUse in source · 6 passages
Search aliases: g
Use in source · Signal_Reconstruction_Traditional.tex, line 363
On X= S_K+span{m}, the functional ℓ_F(g)=D_F^-1∑_n∈ Foverlinewidehat m(n)widehat g(n) satisfies ℓ_F(s+am)=a. The decomposition is direct. If the measured coefficients have errors η_n, then |widetilde a-a|≤ D_F^-1/2(∑_n∈ F|η_n|^2)^1/2.
Exact mathematical passage
\ell_F(g)=D_F^{-1}\sum_{n\in F}\overline{\widehat m(n)}\widehat g(n)
G_TUse in source · 5 passages
Search aliases: G_T
Use in source · Signal_Reconstruction_Traditional.tex, line 771
smallDomain and hypotheses. A period of length T>0. Let distinct locations τ_ℓ∈ R/(T Z) and real amplitudes a_ℓ satisfy ∑_ℓ a_ℓ=0. Define the mean-zero periodic sawtooth G_T(x)=1/2-x/T for 0<x<T, with the right-continuous endpoint value. [SG-21 Canonical periodic integration of jump data]
Exact mathematical passage
G_T(x)=1/2-x/T
\gcdUse in source · 1 passage
Search aliases: gcd
Use in source · Signal_Reconstruction_Traditional.tex, line 256
smallDomain and hypotheses. Unit-modulus exponential nodes. Let z∈ C, |z|=1, and suppose u=z^N and w_j=z^r_j are known, with N≥1 and integer offsets r_j. Let d=gcd(N,r_1,…,r_k). [SG-06 The arithmetic of residual alias ambiguity]
Exact mathematical passage
d=\gcd(N,r_1,\ldots,r_k)
HUse in source · 19 passages
Search aliases: H
Use in source · Signal_Reconstruction_Traditional.tex, line 706
smallDomain and hypotheses. Finite exponential moments over C. Let m_n_0+k=∑_ℓ=1^r c_ℓ z_ℓ^k, with distinct nonzero nodes and nonzero coefficients. Let H_h,w=(m_n_0+i+j)_0≤ i<h, 0≤ j<w, with h,w≥ r. [SG-19 Hankel rank detects the actual order]
Exact mathematical passage
H_{h,w}=(m_{n_0+i+j})_{0\le i<h,\ 0\le j<w}
hUse in source · 29 passages
Search aliases: h
Use in source · Signal_Reconstruction_Traditional.tex, line 227
smallDomain and hypotheses. Absolutely summable sequences on Z. Fix an integer N≥1. Let h,x∈ℓ^1( Z) and y=h*x. Define x_a[q]=x[Nq+a] and y_b[q]=y[Nq+b], for 0≤ a,b<N. [SG-05 Polyphase analysis and convolution kernels]
Exact mathematical passage
h,x\in\ell^1(\mathbb Z)
H^{-1}Use in source · 4 passages
Search aliases: H^-1
Use in source · Signal_Reconstruction_Traditional.tex, line 867
smallDomain and hypotheses. The invertible square Prony system of SG-20. Write Hq=-v, and suppose widetilde H=H+E, widetilde v=v+e. Assume H^-1 _2 E _2<1. [SG-24 Annihilator coefficient perturbation]
Exact mathematical passage
\|H^{-1}\|_2\|E\|_2<1
H_0Use in source · 6 passages
Search aliases: H_0
Use in source · Signal_Reconstruction_Traditional.tex, line 1002
smallDomain and hypotheses. Finite rectangular complex matrices. Let H=H_0+E, with H_0 of rank r≥1, k=min(h,w)≥ r, and ϵ= E _2<σ_r(H_0)/2. Supply a threshold t with ϵ<t<σ_r(H_0)-ϵ. [SG-28 Order recovery with a supplied singular threshold]
Exact mathematical passage
H=H_0+E
h_{ba}Use in source · 3 passages
Search aliases: h_ba
Use in source · Signal_Reconstruction_Traditional.tex, line 231
Interleaving the N channels recovers x exactly. With h_ba[k]=h[Nk+b-a], the output channels satisfy y_b[q]=∑_a=0^N-1∑_t∈ Zh_ba[q-t]x_a[t]. Thus convolution becomes an N× N transfer-kernel system.
Exact mathematical passage
h_{ba}[k]=h[Nk+b-a],
H_{h,w}Use in source · 2 passages
Search aliases: H_h,w
Use in source · Signal_Reconstruction_Traditional.tex, line 706
smallDomain and hypotheses. Finite exponential moments over C. Let m_n_0+k=∑_ℓ=1^r c_ℓ z_ℓ^k, with distinct nonzero nodes and nonzero coefficients. Let H_h,w=(m_n_0+i+j)_0≤ i<h, 0≤ j<w, with h,w≥ r. [SG-19 Hankel rank detects the actual order]
Exact mathematical passage
H_{h,w}=(m_{n_0+i+j})_{0\le i<h,\ 0\le j<w}
H_{L,L}Use in source · 1 passage
Search aliases: H_L,L
Use in source · Signal_Reconstruction_Traditional.tex, line 812
Use SG-18 to form 2L exact high derivative moments. With n_0=K+1, form H_L,L using the first 2L-1 of them. SG-19 says its rank is the actual r. If r=0, j=0 and the low coefficients directly give s. Otherwise the first 2r moments are available. SG-20 recovers the nodes and amplitudes. SG-21 constructs the unique normalized edge. For each |n|≤ K, subtract its coefficient from widehat f(n) to obtain widehat s(n); all other smooth coefficients are zero by the model. Their finite trigonometric sum gi…
Exact mathematical passage
H_{L,L}
H_{r,r}Use in source · 2 passages
Search aliases: H_r,r
Use in source · Signal_Reconstruction_Traditional.tex, line 741
The unique monic annihilator Q(z)=∏_ℓ=1^r(z-z_ℓ)=∑_k=0^r-1q_kz^k+z^r is recovered from H_r,rq=-v, v=(m_n_0+r+i)_i=0^r-1. Its roots are exactly the nodes. Once the nodes are known, r consecutive moments determine the coefficients uniquely by a Vandermonde solve. For unit-modulus nodes the corresponding locations are τ_ℓ=-arg z_ℓpmod2π.
Exact mathematical passage
H_{r,r}q=-v,\qquad v=(m_{n_0+r+i})_{i=0}^{r-1}.
IUse in source · 19 passages
Search aliases: I
Use in source · Signal_Reconstruction_Traditional.tex, line 124
smallDomain and hypotheses. Vector spaces over a field. Let P_1,…,P_m∈End(X) satisfy P_aP_b=δ_abP_a and ∑_aP_a=I_X. Set X_a=P_aX. [SG-01 Complete packet analysis and synthesis]
Exact mathematical passage
\sum_aP_a=I_X
\mathcal IUse in source · 2 passages
Search aliases: I mathcal
Use in source · Signal_Reconstruction_Traditional.tex, line 1197
The Fourier coefficients at the high indices I_0={(q_0+q)N:0≤ q≤2L_max}, I_1={(q_0+q)N+1:0≤ q<L_max} determine the unknown edge order, every physical node, and every jump amplitude exactly. The high-frequency budget is 3L_max+1. Low modes -K≤ n≤ K then give the complete signal. No supplied jump neighborhood is required.
Exact mathematical passage
\mathcal I_0=\{(q_0+q)N:0\le q\le2L_{\max}\},\qquad \mathcal I_1=\{(q_0+q)N+1:0\le q<L_{\max}\}
iUse in source · 27 passages
Search aliases: i
Use in source · Signal_Reconstruction_Traditional.tex, line 85
The default periodic domain is T= R/(2π Z), with widehat f(n)=frac12π∫_0^2πf(x)e^-inx dx. Signal identities inferred from Fourier coefficients are identities of L^1 classes. For a piecewise-constant edge we select the right-continuous representative. This convention fixes the values at jumps, which Fourier integrals alone cannot determine. When an L^1 norm occurs it is the unnormalized integral over a period. On the unit circle we use chord distance |z-w| and circular angular distance d_ T(τ,σ).
Exact mathematical passage
\widehat f(n)=\frac1{2\pi}\int_0^{2\pi}f(x)e^{-inx}\,dx.
i^nUse in source · 1 passage
Search aliases: i^n
Use in source · Signal_Reconstruction_Traditional.tex, line 1674
Take a bandlimited background s(x)=frac14+frac25cos(2x)-frac13sin(3x)+frac15cos(5x) and a mean-zero edge with locations and jumps (τ_ℓ)=(0,π/2,π,3π/2), (a_ℓ)=(1,-2,-1,2). The physical nodes are (1,-i,-1,i). Their derivative moments are m_n=1-2(-i)^n-(-1)^n+2i^n, so in residues modulo four they are m_4q=0, m_4q+1=2+4i, m_4q+2=0, m_4q+3=2-4i. Since the smooth degree is five, the first clean consecutive moment is n_0=6. With an order bound four, coefficients at n=6,…,13 determine the edge. The Hank…
Exact mathematical passage
m_n=1-2(-i)^n-(-1)^n+2i^n,
\mathcal I_0Use in source · 2 passages
Search aliases: I_0 mathcal
Use in source · Signal_Reconstruction_Traditional.tex, line 1197
The Fourier coefficients at the high indices I_0={(q_0+q)N:0≤ q≤2L_max}, I_1={(q_0+q)N+1:0≤ q<L_max} determine the unknown edge order, every physical node, and every jump amplitude exactly. The high-frequency budget is 3L_max+1. Low modes -K≤ n≤ K then give the complete signal. No supplied jump neighborhood is required.
Exact mathematical passage
\mathcal I_0=\{(q_0+q)N:0\le q\le2L_{\max}\},\qquad \mathcal I_1=\{(q_0+q)N+1:0\le q<L_{\max}\}
\mathcal I_1Use in source · 2 passages
Search aliases: I_1 mathcal
Use in source · Signal_Reconstruction_Traditional.tex, line 1197
The Fourier coefficients at the high indices I_0={(q_0+q)N:0≤ q≤2L_max}, I_1={(q_0+q)N+1:0≤ q<L_max} determine the unknown edge order, every physical node, and every jump amplitude exactly. The high-frequency budget is 3L_max+1. Low modes -K≤ n≤ K then give the complete signal. No supplied jump neighborhood is required.
Exact mathematical passage
\mathcal I_0=\{(q_0+q)N:0\le q\le2L_{\max}\},\qquad \mathcal I_1=\{(q_0+q)N+1:0\le q<L_{\max}\}
I_XUse in source · 3 passages
Search aliases: I_X
Use in source · Signal_Reconstruction_Traditional.tex, line 124
smallDomain and hypotheses. Vector spaces over a field. Let P_1,…,P_m∈End(X) satisfy P_aP_b=δ_abP_a and ∑_aP_a=I_X. Set X_a=P_aX. [SG-01 Complete packet analysis and synthesis]
Exact mathematical passage
\sum_aP_a=I_X
\infUse in source · 2 passages
Search aliases: inf
Use in source · Signal_Reconstruction_Traditional.tex, line 606
For an unrestricted scalar amplitude the least-squares profile is widehat a(q)=frac⟨ Rq,Ry⟩ Rq _2^2, J(q)= Ry-widehat a(q)Rq _2^2. The true mask has widehat a(q_*)=a_* and J(q_*)=0. It is the unique zero-residual positive-amplitude mask exactly when no other residualized mask lies on its positive ray. Define the closed rays C_q={aRq:a≥0}. If Γ=inf_q≠ q_*dist(Ry,C_q)>0, with the convention inf∅=+∞, then every perturbed residualized datum Ry+e with e _2<Γ/2 selects q_* uniquely by nearest-ray dist…
Exact mathematical passage
\Gamma=\inf_{q\ne q_*}\operatorname{dist}(Ry,C_q)>0,
\inf_{q\ne q_*}Use in source · 1 passage
Search aliases: inf_q≠ q_* inf ne
Use in source · Signal_Reconstruction_Traditional.tex, line 606
For an unrestricted scalar amplitude the least-squares profile is widehat a(q)=frac⟨ Rq,Ry⟩ Rq _2^2, J(q)= Ry-widehat a(q)Rq _2^2. The true mask has widehat a(q_*)=a_* and J(q_*)=0. It is the unique zero-residual positive-amplitude mask exactly when no other residualized mask lies on its positive ray. Define the closed rays C_q={aRq:a≥0}. If Γ=inf_q≠ q_*dist(Ry,C_q)>0, with the convention inf∅=+∞, then every perturbed residualized datum Ry+e with e _2<Γ/2 selects q_* uniquely by nearest-ray dist…
Exact mathematical passage
\Gamma=\inf_{q\ne q_*}\operatorname{dist}(Ry,C_q)>0,
JUse in source · 14 passages
Search aliases: J
Use in source · Signal_Reconstruction_Traditional.tex, line 186
P_bTP_a=∑_σ∈ F_b∑_ρ∈ F_aQ_σ TQ_ρ. The coordinate changes J= A_Q S_P and C= A_P S_Q satisfy CJ=I and JC=I on their packet coordinate spaces. Refinement is exact before any child packet is discarded.
Exact mathematical passage
J=\mathcal A_Q\mathcal S_P
\mathcal JUse in source · 5 passages
Search aliases: J mathcal
Use in source · Signal_Reconstruction_Traditional.tex, line 294
smallDomain and hypotheses. An arbitrary measurement model and linear packet classes. Let M:X→ Y be any map and let an edge assignment J:X→ J be specified. Separately, let S, J be linear subspaces of a common vector space. [SG-07 Measurement and decomposition obstructions]
Exact mathematical passage
J:X\to\mathcal J
jUse in source · 61 passages
Search aliases: j
Use in source · Signal_Reconstruction_Traditional.tex, line 256
smallDomain and hypotheses. Unit-modulus exponential nodes. Let z∈ C, |z|=1, and suppose u=z^N and w_j=z^r_j are known, with N≥1 and integer offsets r_j. Let d=gcd(N,r_1,…,r_k). [SG-06 The arithmetic of residual alias ambiguity]
Exact mathematical passage
w_j=z^{r_j}
KUse in source · 26 passages
Search aliases: K
Use in source · Signal_Reconstruction_Traditional.tex, line 359
smallDomain and hypotheses. Periodic integrable signals with one known edge shape. Let S_K be the trigonometric polynomials of degree at most K. Let m∈ L^1( T) and a finite nonempty set F⊂{n:|n|>K} satisfy D_F=∑_n∈ F|widehat m(n)|^2>0. [SG-09 Out-of-band matched identification]
Exact mathematical passage
\mathcal S_K
kUse in source · 29 passages
Search aliases: k
Use in source · Signal_Reconstruction_Traditional.tex, line 158
T=∑_a,bP_bTP_a, (ST)_ca=∑_bS_cbT_ba. The block representation is an algebra isomorphism to End(⊕_aX_a), with multiplication by contraction of intermediate packets. For every integer k≥1, iterating the product gives P_a_kT^kP_a_0 =∑_a_1,…,a_k-1 (P_a_kTP_a_k-1)⋯(P_a_1TP_a_0).
Exact mathematical passage
k\ge1
\kerUse in source · 1 passage
Search aliases: ker
Use in source · Signal_Reconstruction_Traditional.tex, line 525
Every vector α with α _0≤ s is uniquely determined by Aα if and only if ker A∩{h: h _0≤min(2s,d)}={0}. Equivalently every set of at most min(2s,d) columns is independent. With the convention spark(A)=d+1 when all d columns are independent, the equivalent condition is spark(A)>min(2s,d). For s=0 uniqueness is automatic.
Exact mathematical passage
\ker A\cap\{h:\|h\|_0\le\min(2s,d)\}=\{0\}.
LUse in source · 32 passages
Search aliases: L
Use in source · Signal_Reconstruction_Traditional.tex, line 59
ments determine an unknown finite jump count through Hankel rank, followed by an annihilating polynomial and Vandermonde recovery. Matched adjacent offsets recover the physical phase lost by decimation. Noisy order, root, and branch conclusions retain explicit conditioning and margin gates, with conditional asymptotic rates. A corrected periodic mean-zero jump estimate controls edge synthesis in L^1. Classical Gibbs ringing is derived and localized to the globally truncated edge block. Finite di…
Exact mathematical passage
L^1
L^1Use in source · 13 passages
Search aliases: L^1
Use in source · Signal_Reconstruction_Traditional.tex, line 59
ments determine an unknown finite jump count through Hankel rank, followed by an annihilating polynomial and Vandermonde recovery. Matched adjacent offsets recover the physical phase lost by decimation. Noisy order, root, and branch conclusions retain explicit conditioning and margin gates, with conditional asymptotic rates. A corrected periodic mean-zero jump estimate controls edge synthesis in L^1. Classical Gibbs ringing is derived and localized to the globally truncated edge block. Finite di…
Exact mathematical passage
L^1
L_{\max}Use in source · 9 passages
Search aliases: L_max max
Use in source · Signal_Reconstruction_Traditional.tex, line 1193
smallDomain and hypotheses. Bandlimited background with a bounded number of jumps. Let f∈ U_K,L_max, with L_max≥1. Supply N≥2, q_0≥1 with q_0N>K, and assume the active aliased nodes u_ℓ=z_ℓ^N are distinct. [SG-34 An explicit exact self-calibration budget]
Exact mathematical passage
f\in\mathcal U_{K,L_{\max}}
\limUse in source · 1 passage
Search aliases: lim
Use in source · Signal_Reconstruction_Traditional.tex, line 1639
For x∈ X choose x_n∈ D with x_n→ x. Continuity gives Ax=lim_nAx_n=lim_nBx_n=Bx. This proves uniqueness of bounded extensions. For an infinite packet sum one would need partial sums to converge to the declared operator, in a stated operator or strong topology. Neither density nor the finite identity by itself proves such a limit. In SG-42 density of the actual selected spans provides a specific strong convergence theorem; a capped dictionary store or an arbitrary greedy prefix does not inherit it…
Exact mathematical passage
Ax=\lim_nAx_n=\lim_nBx_n=Bx
\lim_nUse in source · 1 passage
Search aliases: lim_n lim
Use in source · Signal_Reconstruction_Traditional.tex, line 1639
For x∈ X choose x_n∈ D with x_n→ x. Continuity gives Ax=lim_nAx_n=lim_nBx_n=Bx. This proves uniqueness of bounded extensions. For an infinite packet sum one would need partial sums to converge to the declared operator, in a stated operator or strong topology. Neither density nor the finite identity by itself proves such a limit. In SG-42 density of the actual selected spans provides a specific strong convergence theorem; a capped dictionary store or an arbitrary greedy prefix does not inherit it…
Exact mathematical passage
Ax=\lim_nAx_n=\lim_nBx_n=Bx
\logUse in source · 3 passages
Search aliases: log
Use in source · Signal_Reconstruction_Traditional.tex, line 468
_Mt)/sin(t/2) yields S_Mj(x)=frac12π∫_-a^aD_M(x-t) dt, (S_Mj)'(x)=fracD_M(x+a)-D_M(x-a)2π. At x=a, the coefficient formula gives S_Mj(a)=frac aπ+frac1π∑_n=1^Mfracsin(2na)n ⟶frac12. For completeness, the sine-series limit follows by summation by parts from bounded partial sums of e^2ina and by taking radial limits in ∑_n≥1r^ne^2ina/n=-log(1-re^2ia); its imaginary part tends to (π-2a)/2 for 0<2a<2π. Now differentiate the scaled function: frac ddzS_Mj(x_M(z))= fracD_M(-z/λ_M)-D_M(2a-z/λ_M)2πλ_M. Th…
Exact mathematical passage
\sum_{n\ge1}r^ne^{2ina}/n=-\log(1-re^{2ia})
MUse in source · 50 passages
Search aliases: M
Use in source · Signal_Reconstruction_Traditional.tex, line 294
smallDomain and hypotheses. An arbitrary measurement model and linear packet classes. Let M:X→ Y be any map and let an edge assignment J:X→ J be specified. Separately, let S, J be linear subspaces of a common vector space. [SG-07 Measurement and decomposition obstructions]
Exact mathematical passage
M:X\to Y
mUse in source · 38 passages
Search aliases: m
Use in source · Signal_Reconstruction_Traditional.tex, line 124
smallDomain and hypotheses. Vector spaces over a field. Let P_1,…,P_m∈End(X) satisfy P_aP_b=δ_abP_a and ∑_aP_a=I_X. Set X_a=P_aX. [SG-01 Complete packet analysis and synthesis]
Exact mathematical passage
P_1,\ldots,P_m\in\operatorname{End}(X)
M^{\mathsf T}Use in source · 2 passages
Search aliases: M^ T mathsf
Use in source · Signal_Reconstruction_Traditional.tex, line 1550
The minimizer is unique and (M^ TM+λ I)c=M^ Ty, M^ Tr=λ c. Adding candidate columns cannot increase the minimized ridge objective, but exact orthogonality M^ Tr=0 is not the ridge stationarity law. Even a complete pixel basis need not give zero residual when λ>0.
Exact mathematical passage
(M^{\mathsf T}M+\lambda I)c=M^{\mathsf T}y, \qquad M^{\mathsf T}r=\lambda c.
M^{-p-1}Use in source · 2 passages
Search aliases: M^-p-1
Use in source · Signal_Reconstruction_Traditional.tex, line 1365
j-widehat j _1=O(N^-p-1), f-widehat f_M _1≤ b_M+O((1+C_M)N^-p-1). For Fourier truncation, C_M=O(1+log(M+1)). If also |widehat s(n)|≤ C_s(1+|n|)^-p-2, its smooth remainder satisfies b_M=O(M^-p-1). Taking M comparable to N yields O((1+log N)N^-p-1).
Exact mathematical passage
b_M=O(M^{-p-1})
M^TUse in source · 3 passages
Search aliases: M^T
Use in source · Signal_Reconstruction_Traditional.tex, line 1561
The Hessian is twice M^TM+λ I, which is positive definite because v^T(M^TM+λ I)v= Mv ^2+λ v ^2>0 for nonzero v. Setting the gradient to zero gives the unique minimizer and rearrangement gives M^Tr=λ c. When columns are added, extending the previous coefficient vector by zero is a feasible point with exactly the previous objective, so the new optimum is no larger. This concerns the ridge objective, not an automatic monotonicity theorem for the unpenalized residual alone. For the full pixel basis …
Exact mathematical passage
M^TM+\lambda I
m_{(q_0+q)N+r}Use in source · 1 passage
Search aliases: m_(q_0+q)N+r
Use in source · Signal_Reconstruction_Traditional.tex, line 976
m_(q_0+q)N+r=∑_ℓ b_ℓ,ru_ℓ^q+e_(q_0+q)N+r, u_ℓ=z_ℓ^N, b_ℓ,r=a_ℓ u_ℓ^q_0z_ℓ^r. For each matched edge b_ℓ,r+1=z_ℓ b_ℓ,r. If |widehat s(n)|≤ C_s(1+|n|)^-p-2 with p≥0, then for nonnegative q,r |e_(q_0+q)N+r|≤2π C_s(q_0N)^-p-1.
Exact mathematical passage
m_{(q_0+q)N+r}=\sum_\ell b_{\ell,r}u_\ell^q+e_{(q_0+q)N+r}, \quad u_\ell=z_\ell^N,\quad b_{\ell,r}=a_\ell u_\ell^{q_0}z_\ell^r.
m_{(q_0+q)N+v}Use in source · 1 passage
Search aliases: m_(q_0+q)N+v
Use in source · Signal_Reconstruction_Traditional.tex, line 1426
∏_ℓ=1^r(E-z_ℓ)^d+1m=0. For offset v≥0, writing a decimated channel as m_(q_0+q)N+v=∑_ℓ u_ℓ^q ∑_h=0^dγ_ℓ,v,hq^h, u_ℓ=z_ℓ^N, the highest-degree coefficients satisfy γ_ℓ,1,d/γ_ℓ,0,d=z_ℓ. This ratio applies to separated, matched confluent terms with nonzero leading coefficient. The displayed annihilator is not asserted to have minimal order when leading coefficients vanish.
Exact mathematical passage
m_{(q_0+q)N+v}=\sum_\ell u_\ell^q \sum_{h=0}^d\gamma_{\ell,v,h}q^h, \qquad u_\ell=z_\ell^N,
m_{4q}Use in source · 1 passage
Search aliases: m_4q
Use in source · Signal_Reconstruction_Traditional.tex, line 1678
Take a bandlimited background s(x)=frac14+frac25cos(2x)-frac13sin(3x)+frac15cos(5x) and a mean-zero edge with locations and jumps (τ_ℓ)=(0,π/2,π,3π/2), (a_ℓ)=(1,-2,-1,2). The physical nodes are (1,-i,-1,i). Their derivative moments are m_n=1-2(-i)^n-(-1)^n+2i^n, so in residues modulo four they are m_4q=0, m_4q+1=2+4i, m_4q+2=0, m_4q+3=2-4i. Since the smooth degree is five, the first clean consecutive moment is n_0=6. With an order bound four, coefficients at n=6,…,13 determine the edge. The Hank…
Exact mathematical passage
m_{4q}=0,\quad m_{4q+1}=2+4i,\quad m_{4q+2}=0,\quad m_{4q+3}=2-4i.
m_{4q+1}Use in source · 1 passage
Search aliases: m_4q+1
Use in source · Signal_Reconstruction_Traditional.tex, line 1678
Take a bandlimited background s(x)=frac14+frac25cos(2x)-frac13sin(3x)+frac15cos(5x) and a mean-zero edge with locations and jumps (τ_ℓ)=(0,π/2,π,3π/2), (a_ℓ)=(1,-2,-1,2). The physical nodes are (1,-i,-1,i). Their derivative moments are m_n=1-2(-i)^n-(-1)^n+2i^n, so in residues modulo four they are m_4q=0, m_4q+1=2+4i, m_4q+2=0, m_4q+3=2-4i. Since the smooth degree is five, the first clean consecutive moment is n_0=6. With an order bound four, coefficients at n=6,…,13 determine the edge. The Hank…
Exact mathematical passage
m_{4q}=0,\quad m_{4q+1}=2+4i,\quad m_{4q+2}=0,\quad m_{4q+3}=2-4i.
m_{4q+2}Use in source · 1 passage
Search aliases: m_4q+2
Use in source · Signal_Reconstruction_Traditional.tex, line 1678
Take a bandlimited background s(x)=frac14+frac25cos(2x)-frac13sin(3x)+frac15cos(5x) and a mean-zero edge with locations and jumps (τ_ℓ)=(0,π/2,π,3π/2), (a_ℓ)=(1,-2,-1,2). The physical nodes are (1,-i,-1,i). Their derivative moments are m_n=1-2(-i)^n-(-1)^n+2i^n, so in residues modulo four they are m_4q=0, m_4q+1=2+4i, m_4q+2=0, m_4q+3=2-4i. Since the smooth degree is five, the first clean consecutive moment is n_0=6. With an order bound four, coefficients at n=6,…,13 determine the edge. The Hank…
Exact mathematical passage
m_{4q}=0,\quad m_{4q+1}=2+4i,\quad m_{4q+2}=0,\quad m_{4q+3}=2-4i.
m_{4q+3}Use in source · 1 passage
Search aliases: m_4q+3
Use in source · Signal_Reconstruction_Traditional.tex, line 1678
Take a bandlimited background s(x)=frac14+frac25cos(2x)-frac13sin(3x)+frac15cos(5x) and a mean-zero edge with locations and jumps (τ_ℓ)=(0,π/2,π,3π/2), (a_ℓ)=(1,-2,-1,2). The physical nodes are (1,-i,-1,i). Their derivative moments are m_n=1-2(-i)^n-(-1)^n+2i^n, so in residues modulo four they are m_4q=0, m_4q+1=2+4i, m_4q+2=0, m_4q+3=2-4i. Since the smooth degree is five, the first clean consecutive moment is n_0=6. With an order bound four, coefficients at n=6,…,13 determine the edge. The Hank…
Exact mathematical passage
m_{4q}=0,\quad m_{4q+1}=2+4i,\quad m_{4q+2}=0,\quad m_{4q+3}=2-4i.
m_nUse in source · 6 passages
Search aliases: m_n
Use in source · Signal_Reconstruction_Traditional.tex, line 322
Their unordered location sets have permutation-matched bottleneck distance at least π-δ, while their measured derivative moments satisfy |m_n-widetilde m_n| =|(1-e^-inπ)(1-e^-inδ)| ≤2|n|δ (n∈ F). There is no Lipschitz inverse for locations with one constant on all distinct two-jump configurations, even at this fixed order and with known amplitudes.
Exact mathematical passage
|m_n-\widetilde m_n| =|(1-e^{-in\pi})(1-e^{-in\delta})| \le2|n|\delta\qquad(n\in F).
m_{n+4}Use in source · 1 passage
Search aliases: m_n+4
Use in source · Signal_Reconstruction_Traditional.tex, line 1686
-(-1)^n+2i^n, so in residues modulo four they are m_4q=0, m_4q+1=2+4i, m_4q+2=0, m_4q+3=2-4i. Since the smooth degree is five, the first clean consecutive moment is n_0=6. With an order bound four, coefficients at n=6,…,13 determine the edge. The Hankel factorization has four distinct nodes and four nonzero amplitudes, hence rank four. Its recurrence is m_n+4-m_n=0, with monic polynomial Q(z)=z^4-1. Recovering those roots and solving the Vandermonde system returns the displayed jumps. SG-21 then…
Exact mathematical passage
m_{n+4}-m_n=0
m_{n+k}Use in source · 1 passage
Search aliases: m_n+k
Use in source · Signal_Reconstruction_Traditional.tex, line 753
For every available recurrence index, ∑_k=0^rq_km_n+k =∑_ℓ c_ℓ z_ℓ^n-n_0Q(z_ℓ)=0, q_r=1. SG-19 makes H_r,r invertible. Thus the system has one solution and the true polynomial supplies it. Factoring the resulting polynomial recovers the distinct nodes up to a permutation. The amplitude matrix for consecutive moments is a square Vandermonde multiplied by an invertible diagonal node power matrix, so it too is invertible. The argument convention identifies locations modulo a period, as required. Po…
Exact mathematical passage
\sum_{k=0}^rq_km_{n+k} =\sum_\ell c_\ell z_\ell^{n-n_0}Q(z_\ell)=0, \qquad q_r=1.
m_n^{(c)}Use in source · 1 passage
Search aliases: m_n^(c)
Use in source · Signal_Reconstruction_Traditional.tex, line 1396
smallDomain and hypotheses. A finite collection of exponential moment channels. Let m_n^(c)=∑_ℓ=1^ra_cℓz_ℓ^n, with distinct common unit-modulus nodes. At each node at least one channel amplitude is nonzero. Form the stacked linear recurrence system for a monic polynomial of degree r and assume its coefficient matrix has column rank r. [SG-39 Shared-location multichannel recovery]
Exact mathematical passage
m_n^{(c)}=\sum_{\ell=1}^ra_{c\ell}z_\ell^n
m_{n_0}Use in source · 1 passage
Search aliases: m_n_0
Use in source · Signal_Reconstruction_Traditional.tex, line 733
smallDomain and hypotheses. The nonzero-order moment model of SG-19. Assume r≥1 and 2r consecutive moments m_n_0,…,m_n_0+2r-1 are known. [SG-20 Exact Prony annihilation and location recovery]
Exact mathematical passage
m_{n_0},\ldots,m_{n_0+2r-1}
m_{n_0+2r-1}Use in source · 1 passage
Search aliases: m_n_0+2r-1
Use in source · Signal_Reconstruction_Traditional.tex, line 733
smallDomain and hypotheses. The nonzero-order moment model of SG-19. Assume r≥1 and 2r consecutive moments m_n_0,…,m_n_0+2r-1 are known. [SG-20 Exact Prony annihilation and location recovery]
Exact mathematical passage
m_{n_0},\ldots,m_{n_0+2r-1}
m_{n_0+i+j}Use in source · 1 passage
Search aliases: m_n_0+i+j
Use in source · Signal_Reconstruction_Traditional.tex, line 706
smallDomain and hypotheses. Finite exponential moments over C. Let m_n_0+k=∑_ℓ=1^r c_ℓ z_ℓ^k, with distinct nonzero nodes and nonzero coefficients. Let H_h,w=(m_n_0+i+j)_0≤ i<h, 0≤ j<w, with h,w≥ r. [SG-19 Hankel rank detects the actual order]
Exact mathematical passage
H_{h,w}=(m_{n_0+i+j})_{0\le i<h,\ 0\le j<w}
m_{n_0+k}Use in source · 1 passage
Search aliases: m_n_0+k
Use in source · Signal_Reconstruction_Traditional.tex, line 706
smallDomain and hypotheses. Finite exponential moments over C. Let m_n_0+k=∑_ℓ=1^r c_ℓ z_ℓ^k, with distinct nonzero nodes and nonzero coefficients. Let H_h,w=(m_n_0+i+j)_0≤ i<h, 0≤ j<w, with h,w≥ r. [SG-19 Hankel rank detects the actual order]
Exact mathematical passage
m_{n_0+k}=\sum_{\ell=1}^r c_\ell z_\ell^k
m_{n_0+r+i}Use in source · 1 passage
Search aliases: m_n_0+r+i
Use in source · Signal_Reconstruction_Traditional.tex, line 741
The unique monic annihilator Q(z)=∏_ℓ=1^r(z-z_ℓ)=∑_k=0^r-1q_kz^k+z^r is recovered from H_r,rq=-v, v=(m_n_0+r+i)_i=0^r-1. Its roots are exactly the nodes. Once the nodes are known, r consecutive moments determine the coefficients uniquely by a Vandermonde solve. For unit-modulus nodes the corresponding locations are τ_ℓ=-arg z_ℓpmod2π.
Exact mathematical passage
H_{r,r}q=-v,\qquad v=(m_{n_0+r+i})_{i=0}^{r-1}.
\maxUse in source · 14 passages
Search aliases: max
Use in source · Signal_Reconstruction_Traditional.tex, line 626
ent, the distance to any nonempty closed set is 1-Lipschitz by the triangle inequality. The true ray has perturbed distance at most e _2, and every wrong ray has distance at least Γ- e _2> e _2. Thus the true ray wins strictly. A squared objective gap must be square-rooted before it is used in this norm-distance condition. For positive constraints the optimum on a closed ray uses max(0,widehat a(q)), rather than an unconditionally signed amplitude.
Exact mathematical passage
\max(0,\widehat a(q))
\max_{0\le i<h,\ 0\le j<w}Use in source · 1 passage
Search aliases: max_0≤ i<h, 0≤ j<w max le le
Use in source · Signal_Reconstruction_Traditional.tex, line 843
E _2≤ E _F ≤sqrthwmax_0≤ i<h, 0≤ j<w|ϵ_n_0+i+j|. In particular, an error bound |η_n|≤η implies E _2≤2πsqrthwηmax_i,j|n_0+i+j|.
Exact mathematical passage
\|E\|_2\le\|E\|_F \le\sqrt{hw}\max_{0\le i<h,\ 0\le j<w}|\epsilon_{n_0+i+j}|.
\max_{i,j}Use in source · 1 passage
Search aliases: max_i,j max
Use in source · Signal_Reconstruction_Traditional.tex, line 848
E _2≤ E _F ≤sqrthwmax_0≤ i<h, 0≤ j<w|ϵ_n_0+i+j|. In particular, an error bound |η_n|≤η implies E _2≤2πsqrthwηmax_i,j|n_0+i+j|.
Exact mathematical passage
\|E\|_2\le2\pi\sqrt{hw}\eta\max_{i,j}|n_0+i+j|.
\max_{x\in V,\,\|x\|=1}Use in source · 1 passage
Search aliases: max_x∈ V, x =1 max in
Use in source · Signal_Reconstruction_Traditional.tex, line 1015
The singular-value perturbation inequality [source reference] gives |σ_j(H)-σ_j(H_0)|≤ E _2 for 1≤ j≤ k. It holds equally over the complex field: in the min--max characterization σ_j(A)=min_dim V=w-j+1max_x∈ V, x =1 Ax . the triangle inequality bounds each restricted maximum for A+E by that for A plus E _2. Minimize and then interchange the two matrices. For j≤ r this yields σ_j(H)≥σ_r(H_0)-ϵ>t. For j>r, when such indices exist, it gives σ_j(H)≤ϵ<t. For rank zero the same upper bound applies to …
Exact mathematical passage
\sigma_j(A)=\min_{\dim V=w-j+1}\max_{x\in V,\,\|x\|=1}\|Ax\|.
\minUse in source · 15 passages
Search aliases: min
Use in source · Signal_Reconstruction_Traditional.tex, line 525
Every vector α with α _0≤ s is uniquely determined by Aα if and only if ker A∩{h: h _0≤min(2s,d)}={0}. Equivalently every set of at most min(2s,d) columns is independent. With the convention spark(A)=d+1 when all d columns are independent, the equivalent condition is spark(A)>min(2s,d). For s=0 uniqueness is automatic.
Exact mathematical passage
\ker A\cap\{h:\|h\|_0\le\min(2s,d)\}=\{0\}.
\min_{\dim V=w-j+1}Use in source · 1 passage
Search aliases: min_dim V=w-j+1 min dim
Use in source · Signal_Reconstruction_Traditional.tex, line 1015
The singular-value perturbation inequality [source reference] gives |σ_j(H)-σ_j(H_0)|≤ E _2 for 1≤ j≤ k. It holds equally over the complex field: in the min--max characterization σ_j(A)=min_dim V=w-j+1max_x∈ V, x =1 Ax . the triangle inequality bounds each restricted maximum for A+E by that for A plus E _2. Minimize and then interchange the two matrices. For j≤ r this yields σ_j(H)≥σ_r(H_0)-ϵ>t. For j>r, when such indices exist, it gives σ_j(H)≤ϵ<t. For rank zero the same upper bound applies to …
Exact mathematical passage
\sigma_j(A)=\min_{\dim V=w-j+1}\max_{x\in V,\,\|x\|=1}\|Ax\|.
\min_{\substack{\|h\|_2=1\\\|h\|_0\le\min(2s,d)}}Use in source · 1 passage
Search aliases: min_substack h _2=1 h _0≤min(2s,d) min substack le min
Use in source · Signal_Reconstruction_Traditional.tex, line 560
Define γ=min_substack h _2=1 h _0≤min(2s,d) Ah _2>0. Then widehatα-α_0 _2≤2ϵ/γ, D(widehatα-α_0) _2 ≤2 D _2ϵ/γ. If every nonzero true coefficient has magnitude greater than 4ϵ/γ, thresholding at 2ϵ/γ with a strict greater-than rule recovers the true support.
Exact mathematical passage
\gamma=\min_{\substack{\|h\|_2=1\\\|h\|_0\le\min(2s,d)}}\|Ah\|_2>0.
\min_{|z-z_\ell|=\rho}Use in source · 1 passage
Search aliases: min_|z-z_ℓ|=ρ min ell rho
Use in source · Signal_Reconstruction_Traditional.tex, line 897
If for every ℓ sup_|z-z_ℓ|=ρ|E_Q(z)| <min_|z-z_ℓ|=ρ|Q(z)|, then widetilde Q has exactly one root, counted with multiplicity, in each disk |z-z_ℓ|<ρ. All its roots are thereby matched with error less than ρ. A sufficient explicit bound on each circle is ∑_k=0^r-1|widetilde q_k-q_k|(|z_ℓ|+ρ)^k <ρ∏_j≠ℓ(|z_ℓ-z_j|-ρ).
Exact mathematical passage
\sup_{|z-z_\ell|=\rho}|E_Q(z)| <\min_{|z-z_\ell|=\rho}|Q(z)|,
\min_{\ell\ne k}Use in source · 1 passage
Search aliases: min_ℓ≠ k min ell ne
Use in source · Signal_Reconstruction_Traditional.tex, line 893
smallDomain and hypotheses. Monic complex polynomials of the same degree. Let r≥1 and Q(z)=∏_ℓ=1^r(z-z_ℓ) have distinct roots and widetilde Q=Q+E_Q. Choose 0<ρ<tfrac12min_ℓ≠ k|z_ℓ-z_k| when r≥2, or any ρ>0 when r=1. [SG-25 Conditional simple-root disk localization]
Exact mathematical passage
0<\rho<\tfrac12\min_{\ell\ne k}|z_\ell-z_k|
NUse in source · 61 passages
Search aliases: N
Use in source · Signal_Reconstruction_Traditional.tex, line 227
smallDomain and hypotheses. Absolutely summable sequences on Z. Fix an integer N≥1. Let h,x∈ℓ^1( Z) and y=h*x. Define x_a[q]=x[Nq+a] and y_b[q]=y[Nq+b], for 0≤ a,b<N. [SG-05 Polyphase analysis and convolution kernels]
Exact mathematical passage
N\ge1
nUse in source · 66 passages
Search aliases: n
Use in source · Signal_Reconstruction_Traditional.tex, line 85
The default periodic domain is T= R/(2π Z), with widehat f(n)=frac12π∫_0^2πf(x)e^-inx dx. Signal identities inferred from Fourier coefficients are identities of L^1 classes. For a piecewise-constant edge we select the right-continuous representative. This convention fixes the values at jumps, which Fourier integrals alone cannot determine. When an L^1 norm occurs it is the unnormalized integral over a period. On the unit circle we use chord distance |z-w| and circular angular distance d_ T(τ,σ).
Exact mathematical passage
\widehat f(n)=\frac1{2\pi}\int_0^{2\pi}f(x)e^{-inx}\,dx.
N^{-1}Use in source · 1 passage
Search aliases: N^-1
Use in source · Signal_Reconstruction_Traditional.tex, line 1251
verse bounds transfer the moment error to aliased node and matched coefficient errors of order N^-p-1. The supplied rank gap and threshold select the correct order by SG-28. Nonvanishing base coefficients allow SG-30 to give the same order for the coarse phase error. The strict branch margin selects the associated root by SG-32. Normalize the aliased estimate and use SG-33: its root error is N^-1 times an error of order N^-p-1, giving N^-p-2. On the unit circle, d_ T=2arcsin(|widehat z-z|/2), wh…
Exact mathematical passage
N^{-1}
N^{-p-1}Use in source · 7 passages
Search aliases: N^-p-1
Use in source · Signal_Reconstruction_Traditional.tex, line 1231
cted sequence of decimation scales. Let p≥0, fix q_0 and the finite number of channel samples and offsets, and take a sequence N→∞. Assume distinct aliased nodes, uniform rank-threshold gaps/local inverse bounds for the aligned Prony and coefficient fits, |b_ℓ,0|≥ b_min>0, uniformly bounded amplitudes, and the strict branch margin of SG-32. Suppose the moment defect is O(N^-p-1). [SG-35 Conditional asymptotic rates after calibrated lifting]
Exact mathematical passage
O(N^{-p-1})
N^{-p-2}Use in source · 3 passages
Search aliases: N^-p-2
Use in source · Signal_Reconstruction_Traditional.tex, line 1236
The threshold-selected order is eventually correct. The aligned lifted nodes and locations satisfy |widehat z_ℓ-z_ℓ|=O(N^-p-2), d_ T(widehatτ_ℓ,τ_ℓ)=O(N^-p-2). The recovered amplitudes satisfy |widehat a_ℓ-a_ℓ|=O(N^-p-1).
Exact mathematical passage
|\widehat z_\ell-z_\ell|=O(N^{-p-2}),\qquad d_{\mathbb T}(\widehat\tau_\ell,\tau_\ell)=O(N^{-p-2}).
N^dUse in source · 2 passages
Search aliases: N^d
Use in source · Signal_Reconstruction_Traditional.tex, line 1445
For one term, (E-z)(z^nP(n))=z^n+1(P(n+1)-P(n)). A finite difference lowers a nonconstant polynomial's degree by one, and annihilates a constant. After d+1 differences the term vanishes. The shift factors commute, so the product annihilates every term in the sum. If P_ℓ(n) has leading coefficient c_ℓ≠0 at degree d, substitution of (q_0+q)N+v has leading q coefficient c_ℓ N^d. The remaining exponential prefactor is z_ℓ^q_0N+v; hence γ_ℓ,v,d=c_ℓ N^dz_ℓ^q_0N+v and the ratio is z_ℓ. Repeated roots r…
Exact mathematical passage
c_\ell N^d
n_0Use in source · 11 passages
Search aliases: n_0
Use in source · Signal_Reconstruction_Traditional.tex, line 706
smallDomain and hypotheses. Finite exponential moments over C. Let m_n_0+k=∑_ℓ=1^r c_ℓ z_ℓ^k, with distinct nonzero nodes and nonzero coefficients. Let H_h,w=(m_n_0+i+j)_0≤ i<h, 0≤ j<w, with h,w≥ r. [SG-19 Hankel rank detects the actual order]
Exact mathematical passage
m_{n_0+k}=\sum_{\ell=1}^r c_\ell z_\ell^k
OUse in source · 8 passages
Search aliases: O
Use in source · Signal_Reconstruction_Traditional.tex, line 1231
cted sequence of decimation scales. Let p≥0, fix q_0 and the finite number of channel samples and offsets, and take a sequence N→∞. Assume distinct aliased nodes, uniform rank-threshold gaps/local inverse bounds for the aligned Prony and coefficient fits, |b_ℓ,0|≥ b_min>0, uniformly bounded amplitudes, and the strict branch margin of SG-32. Suppose the moment defect is O(N^-p-1). [SG-35 Conditional asymptotic rates after calibrated lifting]
Exact mathematical passage
O(N^{-p-1})
\overline uUse in source · 8 passages
Search aliases: overline u overline
Use in source · Signal_Reconstruction_Traditional.tex, line 1130
smallDomain and hypotheses. Nth roots of a unit-modulus estimate. Let N≥2, |z|=1, u=z^N, and overline u=widehat u/|widehat u| with widehat u≠0. Let ρ be a root of overline u continuously associated with z, and widetilde z a coarse phase estimate. [SG-32 Unique branch selection under a strict margin]
Exact mathematical passage
\overline u=\widehat u/|\widehat u|
\overline u^{-q_0}Use in source · 1 passage
Search aliases: overline u^-q_0 overline
Use in source · Signal_Reconstruction_Traditional.tex, line 1255
se coefficients allow SG-30 to give the same order for the coarse phase error. The strict branch margin selects the associated root by SG-32. Normalize the aliased estimate and use SG-33: its root error is N^-1 times an error of order N^-p-1, giving N^-p-2. On the unit circle, d_ T=2arcsin(|widehat z-z|/2), which has the same order for small errors. For the amplitude use widehat a=widehat b_0 overline u^-q_0 and a=b_0u^-q_0. Since q_0 is fixed and both powers have unit modulus, SG-26's telescopi…
Exact mathematical passage
\widehat a=\widehat b_0\,\overline u^{-q_0}
\overline{bz^r}Use in source · 2 passages
Search aliases: overlinebz^r overline
Use in source · Signal_Reconstruction_Traditional.tex, line 1060
Each summand is overlinebz^r bz^r+1=|b|^2z because |z|=1. There are R summands and R≥1, so normalization returns z. For two channels direct division cancels b. No statistical independence is assumed, so one cannot improve a worst-case noise order merely by increasing R; the next bound shows the cancellation of the number of products.
Exact mathematical passage
\overline{bz^r}\,bz^{r+1}=|b|^2z
\overline{c_r}Use in source · 2 passages
Search aliases: overlinec_r overline
Use in source · Signal_Reconstruction_Traditional.tex, line 1047
smallDomain and hypotheses. One matched unit-circle coefficient progression. Let R≥1 be an integer counting adjacent products, and let c_r=bz^r for 0≤ r≤ R, with b≠0 and |z|=1. Define S(c)=∑_r=0^R-1overlinec_rc_r+1. [SG-29 Exact phase from adjacent offset products]
Exact mathematical passage
S(c)=\sum_{r=0}^{R-1}\overline{c_r}c_{r+1}
\overline{e_r}Use in source · 1 passage
Search aliases: overlinee_r overline
Use in source · Signal_Reconstruction_Traditional.tex, line 1085
Expand each product: overlinec_rc_r+1=|b|^2z +overlinebz^re_r+1+overlinee_rbz^r+1 +overlinee_re_r+1. The three error terms have total magnitude at most 2|b|ϵ+ϵ^2. Summing gives S(c)=Az+E with |E|≤Δ. Therefore |S(c)|≥ A-Δ>0. For any nonzero v and unit z, |frac v|v|-z| ≤|frac v|v|-frac vA|+|frac vA-z| =frac|A-|v||A+frac|v-Az|A ≤frac2|v-Az|A. Apply this inequality to v=S(c) and substitute the definitions. The number of products cancels, as appropriate for coherent adversarial errors.
Exact mathematical passage
\overline{c_r}c_{r+1}=|b|^2z +\overline{bz^r}e_{r+1}+\overline{e_r}bz^{r+1} +\overline{e_r}e_{r+1}.
\overline{\widehat m(n)}Use in source · 1 passage
Search aliases: overlinewidehat m(n) overline widehat
Use in source · Signal_Reconstruction_Traditional.tex, line 363
On X= S_K+span{m}, the functional ℓ_F(g)=D_F^-1∑_n∈ Foverlinewidehat m(n)widehat g(n) satisfies ℓ_F(s+am)=a. The decomposition is direct. If the measured coefficients have errors η_n, then |widetilde a-a|≤ D_F^-1/2(∑_n∈ F|η_n|^2)^1/2.
Exact mathematical passage
\ell_F(g)=D_F^{-1}\sum_{n\in F}\overline{\widehat m(n)}\widehat g(n)
PUse in source · 44 passages
Search aliases: P
Use in source · Signal_Reconstruction_Traditional.tex, line 124
smallDomain and hypotheses. Vector spaces over a field. Let P_1,…,P_m∈End(X) satisfy P_aP_b=δ_abP_a and ∑_aP_a=I_X. Set X_a=P_aX. [SG-01 Complete packet analysis and synthesis]
Exact mathematical passage
P_1,\ldots,P_m\in\operatorname{End}(X)
pUse in source · 18 passages
Search aliases: p
Use in source · Signal_Reconstruction_Traditional.tex, line 981
m_(q_0+q)N+r=∑_ℓ b_ℓ,ru_ℓ^q+e_(q_0+q)N+r, u_ℓ=z_ℓ^N, b_ℓ,r=a_ℓ u_ℓ^q_0z_ℓ^r. For each matched edge b_ℓ,r+1=z_ℓ b_ℓ,r. If |widehat s(n)|≤ C_s(1+|n|)^-p-2 with p≥0, then for nonnegative q,r |e_(q_0+q)N+r|≤2π C_s(q_0N)^-p-1.
Exact mathematical passage
|\widehat s(n)|\le C_s(1+|n|)^{-p-2}
P_1Use in source · 1 passage
Search aliases: P_1
Use in source · Signal_Reconstruction_Traditional.tex, line 124
smallDomain and hypotheses. Vector spaces over a field. Let P_1,…,P_m∈End(X) satisfy P_aP_b=δ_abP_a and ∑_aP_a=I_X. Set X_a=P_aX. [SG-01 Complete packet analysis and synthesis]
Exact mathematical passage
P_1,\ldots,P_m\in\operatorname{End}(X)
P_aUse in source · 17 passages
Search aliases: P_a
Use in source · Signal_Reconstruction_Traditional.tex, line 124
smallDomain and hypotheses. Vector spaces over a field. Let P_1,…,P_m∈End(X) satisfy P_aP_b=δ_abP_a and ∑_aP_a=I_X. Set X_a=P_aX. [SG-01 Complete packet analysis and synthesis]
Exact mathematical passage
P_aP_b=\delta_{ab}P_a
P_a^*Use in source · 1 passage
Search aliases: P_a^*
Use in source · Signal_Reconstruction_Traditional.tex, line 218
Compute P_aP_b=V^-1E_aE_bV=δ_abP_a and ∑_aP_a=V^-1IV=I. In the unitary case the pulled-back projections are orthogonal: P_a^*=P_a. Their images are mutually orthogonal, and SG-01 writes x as their finite orthogonal sum. The Pythagorean identity gives the energy formula. Without unitarity and orthogonality the algebraic identities remain valid while that norm identity need not hold.
Exact mathematical passage
P_a^*=P_a
P_{a_0}Use in source · 1 passage
Search aliases: P_a_0
Use in source · Signal_Reconstruction_Traditional.tex, line 159
T=∑_a,bP_bTP_a, (ST)_ca=∑_bS_cbT_ba. The block representation is an algebra isomorphism to End(⊕_aX_a), with multiplication by contraction of intermediate packets. For every integer k≥1, iterating the product gives P_a_kT^kP_a_0 =∑_a_1,…,a_k-1 (P_a_kTP_a_k-1)⋯(P_a_1TP_a_0).
Exact mathematical passage
P_{a_k}T^kP_{a_0} =\sum_{a_1,\ldots,a_{k-1}} (P_{a_k}TP_{a_{k-1}})\cdots(P_{a_1}TP_{a_0}).
P_{a_1}Use in source · 1 passage
Search aliases: P_a_1
Use in source · Signal_Reconstruction_Traditional.tex, line 159
T=∑_a,bP_bTP_a, (ST)_ca=∑_bS_cbT_ba. The block representation is an algebra isomorphism to End(⊕_aX_a), with multiplication by contraction of intermediate packets. For every integer k≥1, iterating the product gives P_a_kT^kP_a_0 =∑_a_1,…,a_k-1 (P_a_kTP_a_k-1)⋯(P_a_1TP_a_0).
Exact mathematical passage
P_{a_k}T^kP_{a_0} =\sum_{a_1,\ldots,a_{k-1}} (P_{a_k}TP_{a_{k-1}})\cdots(P_{a_1}TP_{a_0}).
P_{a_k}Use in source · 1 passage
Search aliases: P_a_k
Use in source · Signal_Reconstruction_Traditional.tex, line 159
T=∑_a,bP_bTP_a, (ST)_ca=∑_bS_cbT_ba. The block representation is an algebra isomorphism to End(⊕_aX_a), with multiplication by contraction of intermediate packets. For every integer k≥1, iterating the product gives P_a_kT^kP_a_0 =∑_a_1,…,a_k-1 (P_a_kTP_a_k-1)⋯(P_a_1TP_a_0).
Exact mathematical passage
P_{a_k}T^kP_{a_0} =\sum_{a_1,\ldots,a_{k-1}} (P_{a_k}TP_{a_{k-1}})\cdots(P_{a_1}TP_{a_0}).
P_{a_{k-1}}Use in source · 1 passage
Search aliases: P_a_k-1
Use in source · Signal_Reconstruction_Traditional.tex, line 159
T=∑_a,bP_bTP_a, (ST)_ca=∑_bS_cbT_ba. The block representation is an algebra isomorphism to End(⊕_aX_a), with multiplication by contraction of intermediate packets. For every integer k≥1, iterating the product gives P_a_kT^kP_a_0 =∑_a_1,…,a_k-1 (P_a_kTP_a_k-1)⋯(P_a_1TP_a_0).
Exact mathematical passage
P_{a_k}T^kP_{a_0} =\sum_{a_1,\ldots,a_{k-1}} (P_{a_k}TP_{a_{k-1}})\cdots(P_{a_1}TP_{a_0}).
P_bUse in source · 8 passages
Search aliases: P_b
Use in source · Signal_Reconstruction_Traditional.tex, line 124
smallDomain and hypotheses. Vector spaces over a field. Let P_1,…,P_m∈End(X) satisfy P_aP_b=δ_abP_a and ∑_aP_a=I_X. Set X_a=P_aX. [SG-01 Complete packet analysis and synthesis]
Exact mathematical passage
P_aP_b=\delta_{ab}P_a
P_JUse in source · 5 passages
Search aliases: P_J
Use in source · Signal_Reconstruction_Traditional.tex, line 387
smallDomain and hypotheses. The model space of SG-09. Let P_Jg=ℓ_F(g)m, P_S=I-P_J, and let S_M be Fourier truncation of degree M≥ K. [SG-10 The singularity-exact block operator]
Exact mathematical passage
P_Jg=\ell_F(g)m
P_J^2Use in source · 1 passage
Search aliases: P_J^2
Use in source · Signal_Reconstruction_Traditional.tex, line 390
P_J^2=P_J, P_S^2=P_S, P_SP_J=P_JP_S=0, P_S+P_J=I_X, and the resolved reconstruction obeys R_M=S_MP_S+P_J=I_X.
Exact mathematical passage
P_J^2=P_J,\quad P_S^2=P_S,\quad P_SP_J=P_JP_S=0, \quad P_S+P_J=I_X,
P_kUse in source · 4 passages
Search aliases: P_k
Use in source · Signal_Reconstruction_Traditional.tex, line 1519
smallDomain and hypotheses. A Hilbert space with finite selected packet spans. Let V_1⊆ V_2⊆⋯ be finite-dimensional subspaces. Let P_k be the orthogonal projector onto V_k. [SG-42 Ideal nested-span refitting decreases residual]
Exact mathematical passage
P_k
P_{k+1}Use in source · 1 passage
Search aliases: P_k+1
Use in source · Signal_Reconstruction_Traditional.tex, line 1522
f-P_k+1f ≤ f-P_kf , f-P_kf⊥ V_k. If ⋃_kV_k is dense, then P_kf→ f in norm.
Exact mathematical passage
\|f-P_{k+1}f\|\le\|f-P_kf\|, \qquad f-P_kf\perp V_k.
P_mUse in source · 1 passage
Search aliases: P_m
Use in source · Signal_Reconstruction_Traditional.tex, line 124
smallDomain and hypotheses. Vector spaces over a field. Let P_1,…,P_m∈End(X) satisfy P_aP_b=δ_abP_a and ∑_aP_a=I_X. Set X_a=P_aX. [SG-01 Complete packet analysis and synthesis]
Exact mathematical passage
P_1,\ldots,P_m\in\operatorname{End}(X)
P_SUse in source · 4 passages
Search aliases: P_S
Use in source · Signal_Reconstruction_Traditional.tex, line 387
smallDomain and hypotheses. The model space of SG-09. Let P_Jg=ℓ_F(g)m, P_S=I-P_J, and let S_M be Fourier truncation of degree M≥ K. [SG-10 The singularity-exact block operator]
Exact mathematical passage
P_S=I-P_J
P_S^2Use in source · 1 passage
Search aliases: P_S^2
Use in source · Signal_Reconstruction_Traditional.tex, line 390
P_J^2=P_J, P_S^2=P_S, P_SP_J=P_JP_S=0, P_S+P_J=I_X, and the resolved reconstruction obeys R_M=S_MP_S+P_J=I_X.
Exact mathematical passage
P_J^2=P_J,\quad P_S^2=P_S,\quad P_SP_J=P_JP_S=0, \quad P_S+P_J=I_X,
P_\ellUse in source · 4 passages
Search aliases: P_ℓ ell
Use in source · Signal_Reconstruction_Traditional.tex, line 1419
smallDomain and hypotheses. Polynomial-exponential sequences. Let m_n=∑_ℓ=1^rz_ℓ^nP_ℓ(n) with distinct nonzero nodes and deg P_ℓ≤ d. Let E be the forward shift. For an offset ratio assume N≥1 and the leading coefficient of P_ℓ at degree d is nonzero. [SG-40 Confluent annihilation and the leading offset phase]
Exact mathematical passage
m_n=\sum_{\ell=1}^rz_\ell^nP_\ell(n)
\pmodUse in source · 1 passage
Search aliases: pmod
Use in source · Signal_Reconstruction_Traditional.tex, line 747
The unique monic annihilator Q(z)=∏_ℓ=1^r(z-z_ℓ)=∑_k=0^r-1q_kz^k+z^r is recovered from H_r,rq=-v, v=(m_n_0+r+i)_i=0^r-1. Its roots are exactly the nodes. Once the nodes are known, r consecutive moments determine the coefficients uniquely by a Vandermonde solve. For unit-modulus nodes the corresponding locations are τ_ℓ=-arg z_ℓpmod2π.
Exact mathematical passage
\tau_\ell=-\arg z_\ell\pmod{2\pi}
QUse in source · 16 passages
Search aliases: Q
Use in source · Signal_Reconstruction_Traditional.tex, line 180
smallDomain and hypotheses. Finite complete packet systems. Let (Q_ρ) refine (P_a), with a partition of fine labels into F_a and P_a=∑_ρ∈ F_aQ_ρ. [SG-03 Refinement preserves represented operators]
Exact mathematical passage
(Q_\rho)
\mathcal QUse in source · 2 passages
Search aliases: Q mathcal
Use in source · Signal_Reconstruction_Traditional.tex, line 594
smallDomain and hypotheses. Finite real dictionaries with positive amplitudes. Let B have full column rank, P be the orthogonal projector onto its range, R=I-P, and let Q be a finite mask library with Rq≠0 for every q. Suppose y=Bθ_*+a_*q_*, with a_*>0 and q_*∈ Q. [SG-16 Profiled one-packet discovery and its finite gap]
Exact mathematical passage
\mathcal Q
qUse in source · 57 passages
Search aliases: q
Use in source · Signal_Reconstruction_Traditional.tex, line 227
smallDomain and hypotheses. Absolutely summable sequences on Z. Fix an integer N≥1. Let h,x∈ℓ^1( Z) and y=h*x. Define x_a[q]=x[Nq+a] and y_b[q]=y[Nq+b], for 0≤ a,b<N. [SG-05 Polyphase analysis and convolution kernels]
Exact mathematical passage
x_a[q]=x[Nq+a]
q^hUse in source · 1 passage
Search aliases: q^h
Use in source · Signal_Reconstruction_Traditional.tex, line 1426
∏_ℓ=1^r(E-z_ℓ)^d+1m=0. For offset v≥0, writing a decimated channel as m_(q_0+q)N+v=∑_ℓ u_ℓ^q ∑_h=0^dγ_ℓ,v,hq^h, u_ℓ=z_ℓ^N, the highest-degree coefficients satisfy γ_ℓ,1,d/γ_ℓ,0,d=z_ℓ. This ratio applies to separated, matched confluent terms with nonzero leading coefficient. The displayed annihilator is not asserted to have minimal order when leading coefficients vanish.
Exact mathematical passage
m_{(q_0+q)N+v}=\sum_\ell u_\ell^q \sum_{h=0}^d\gamma_{\ell,v,h}q^h, \qquad u_\ell=z_\ell^N,
q_*Use in source · 7 passages
Search aliases: q_*
Use in source · Signal_Reconstruction_Traditional.tex, line 594
smallDomain and hypotheses. Finite real dictionaries with positive amplitudes. Let B have full column rank, P be the orthogonal projector onto its range, R=I-P, and let Q be a finite mask library with Rq≠0 for every q. Suppose y=Bθ_*+a_*q_*, with a_*>0 and q_*∈ Q. [SG-16 Profiled one-packet discovery and its finite gap]
Exact mathematical passage
y=B\theta_*+a_*q_*
q_0Use in source · 20 passages
Search aliases: q_0
Use in source · Signal_Reconstruction_Traditional.tex, line 973
smallDomain and hypotheses. Derivative moments with an additive defect. Fix integers N≥2, q_0≥1, and an offset r≥0. Suppose m_n=∑_ℓ a_ℓ z_ℓ^n+e_n, with |z_ℓ|=1. [SG-27 Shared-node offset channels]
Exact mathematical passage
q_0\ge1
q_kUse in source · 3 passages
Search aliases: q_k
Use in source · Signal_Reconstruction_Traditional.tex, line 737
The unique monic annihilator Q(z)=∏_ℓ=1^r(z-z_ℓ)=∑_k=0^r-1q_kz^k+z^r is recovered from H_r,rq=-v, v=(m_n_0+r+i)_i=0^r-1. Its roots are exactly the nodes. Once the nodes are known, r consecutive moments determine the coefficients uniquely by a Vandermonde solve. For unit-modulus nodes the corresponding locations are τ_ℓ=-arg z_ℓpmod2π.
Exact mathematical passage
Q(z)=\prod_{\ell=1}^r(z-z_\ell)=\sum_{k=0}^{r-1}q_kz^k+z^r
q_rUse in source · 1 passage
Search aliases: q_r
Use in source · Signal_Reconstruction_Traditional.tex, line 753
For every available recurrence index, ∑_k=0^rq_km_n+k =∑_ℓ c_ℓ z_ℓ^n-n_0Q(z_ℓ)=0, q_r=1. SG-19 makes H_r,r invertible. Thus the system has one solution and the true polynomial supplies it. Factoring the resulting polynomial recovers the distinct nodes up to a permutation. The amplitude matrix for consecutive moments is a square Vandermonde multiplied by an invertible diagonal node power matrix, so it too is invertible. The argument convention identifies locations modulo a period, as required. Po…
Exact mathematical passage
\sum_{k=0}^rq_km_{n+k} =\sum_\ell c_\ell z_\ell^{n-n_0}Q(z_\ell)=0, \qquad q_r=1.
Q_\rhoUse in source · 3 passages
Search aliases: Q_ρ rho
Use in source · Signal_Reconstruction_Traditional.tex, line 180
smallDomain and hypotheses. Finite complete packet systems. Let (Q_ρ) refine (P_a), with a partition of fine labels into F_a and P_a=∑_ρ∈ F_aQ_ρ. [SG-03 Refinement preserves represented operators]
Exact mathematical passage
(Q_\rho)
Q_\sigmaUse in source · 1 passage
Search aliases: Q_σ sigma
Use in source · Signal_Reconstruction_Traditional.tex, line 183
P_bTP_a=∑_σ∈ F_b∑_ρ∈ F_aQ_σ TQ_ρ. The coordinate changes J= A_Q S_P and C= A_P S_Q satisfy CJ=I and JC=I on their packet coordinate spaces. Refinement is exact before any child packet is discarded.
Exact mathematical passage
P_bTP_a=\sum_{\sigma\in F_b}\sum_{\rho\in F_a}Q_\sigma TQ_\rho.
RUse in source · 33 passages
Search aliases: R
Use in source · Signal_Reconstruction_Traditional.tex, line 297
If Mf_0=Mf_1 but J(f_0)≠ J(f_1), no map R:Y→ J recovers the correct edge for both. If S∩ J≠{0}, the sum decomposition f=s+j is not unique. In particular, finitely many point samples on a periodic interval cannot recover every arbitrary interval packet.
Exact mathematical passage
R:Y\to\mathcal J
\mathbb RUse in source · 6 passages
Search aliases: R mathbb
Use in source · Signal_Reconstruction_Traditional.tex, line 84
The default periodic domain is T= R/(2π Z), with widehat f(n)=frac12π∫_0^2πf(x)e^-inx dx. Signal identities inferred from Fourier coefficients are identities of L^1 classes. For a piecewise-constant edge we select the right-continuous representative. This convention fixes the values at jumps, which Fourier integrals alone cannot determine. When an L^1 norm occurs it is the unnormalized integral over a period. On the unit circle we use chord distance |z-w| and circular angular distance d_ T(τ,σ).
Exact mathematical passage
\mathbb T=\mathbb R/(2\pi\mathbb Z)
\mathcal RUse in source · 5 passages
Search aliases: R mathcal
Use in source · Signal_Reconstruction_Traditional.tex, line 395
P_J^2=P_J, P_S^2=P_S, P_SP_J=P_JP_S=0, P_S+P_J=I_X, and the resolved reconstruction obeys R_M=S_MP_S+P_J=I_X.
Exact mathematical passage
\mathcal R_M=S_MP_S+P_J=I_X.
rUse in source · 83 passages
Search aliases: r
Use in source · Signal_Reconstruction_Traditional.tex, line 256
smallDomain and hypotheses. Unit-modulus exponential nodes. Let z∈ C, |z|=1, and suppose u=z^N and w_j=z^r_j are known, with N≥1 and integer offsets r_j. Let d=gcd(N,r_1,…,r_k). [SG-06 The arithmetic of residual alias ambiguity]
Exact mathematical passage
w_j=z^{r_j}
\mathbb R^mUse in source · 2 passages
Search aliases: R^m mathbb
Use in source · Signal_Reconstruction_Traditional.tex, line 1546
smallDomain and hypotheses. Finite-dimensional real packet designs. Let M∈ R^m× d, y∈ R^m, and λ>0. Let c minimize y-Mc _2^2+λ c _2^2 and set r=y-Mc. [SG-43 Ridge refitting has a different stationarity law]
Exact mathematical passage
y\in\mathbb R^m
\mathbb R^{m\times d}Use in source · 1 passage
Search aliases: R^m× d mathbb times
Use in source · Signal_Reconstruction_Traditional.tex, line 1546
smallDomain and hypotheses. Finite-dimensional real packet designs. Let M∈ R^m× d, y∈ R^m, and λ>0. Let c minimize y-Mc _2^2+λ c _2^2 and set r=y-Mc. [SG-43 Ridge refitting has a different stationarity law]
Exact mathematical passage
M\in\mathbb R^{m\times d}
r^nUse in source · 1 passage
Search aliases: r^n
Use in source · Signal_Reconstruction_Traditional.tex, line 468
_Mt)/sin(t/2) yields S_Mj(x)=frac12π∫_-a^aD_M(x-t) dt, (S_Mj)'(x)=fracD_M(x+a)-D_M(x-a)2π. At x=a, the coefficient formula gives S_Mj(a)=frac aπ+frac1π∑_n=1^Mfracsin(2na)n ⟶frac12. For completeness, the sine-series limit follows by summation by parts from bounded partial sums of e^2ina and by taking radial limits in ∑_n≥1r^ne^2ina/n=-log(1-re^2ia); its imaginary part tends to (π-2a)/2 for 0<2a<2π. Now differentiate the scaled function: frac ddzS_Mj(x_M(z))= fracD_M(-z/λ_M)-D_M(2a-z/λ_M)2πλ_M. Th…
Exact mathematical passage
\sum_{n\ge1}r^ne^{2ina}/n=-\log(1-re^{2ia})
r_1Use in source · 2 passages
Search aliases: r_1
Use in source · Signal_Reconstruction_Traditional.tex, line 256
smallDomain and hypotheses. Unit-modulus exponential nodes. Let z∈ C, |z|=1, and suppose u=z^N and w_j=z^r_j are known, with N≥1 and integer offsets r_j. Let d=gcd(N,r_1,…,r_k). [SG-06 The arithmetic of residual alias ambiguity]
Exact mathematical passage
d=\gcd(N,r_1,\ldots,r_k)
r_jUse in source · 3 passages
Search aliases: r_j
Use in source · Signal_Reconstruction_Traditional.tex, line 256
smallDomain and hypotheses. Unit-modulus exponential nodes. Let z∈ C, |z|=1, and suppose u=z^N and w_j=z^r_j are known, with N≥1 and integer offsets r_j. Let d=gcd(N,r_1,…,r_k). [SG-06 The arithmetic of residual alias ambiguity]
Exact mathematical passage
w_j=z^{r_j}
r_kUse in source · 2 passages
Search aliases: r_k
Use in source · Signal_Reconstruction_Traditional.tex, line 256
smallDomain and hypotheses. Unit-modulus exponential nodes. Let z∈ C, |z|=1, and suppose u=z^N and w_j=z^r_j are known, with N≥1 and integer offsets r_j. Let d=gcd(N,r_1,…,r_k). [SG-06 The arithmetic of residual alias ambiguity]
Exact mathematical passage
d=\gcd(N,r_1,\ldots,r_k)
\mathcal R_MUse in source · 5 passages
Search aliases: R_M mathcal
Use in source · Signal_Reconstruction_Traditional.tex, line 395
P_J^2=P_J, P_S^2=P_S, P_SP_J=P_JP_S=0, P_S+P_J=I_X, and the resolved reconstruction obeys R_M=S_MP_S+P_J=I_X.
Exact mathematical passage
\mathcal R_M=S_MP_S+P_J=I_X.
\operatorname{rank}Use in source · 1 passage
Search aliases: rank operatorname
Use in source · Signal_Reconstruction_Traditional.tex, line 710
Writing V_h=(z_ℓ^i)_i,ℓ and V_w=(z_ℓ^j)_j,ℓ, H_h,w=V_hdiag(c_ℓ)V_w^ T, rankH_h,w=r. For r=0 the Hankel matrix is zero. The transpose is algebraic and is not a conjugate transpose.
Exact mathematical passage
H_{h,w}=V_h\operatorname{diag}(c_\ell)V_w^{\mathsf T}, \qquad\operatorname{rank}H_{h,w}=r.
SUse in source · 21 passages
Search aliases: S
Use in source · Signal_Reconstruction_Traditional.tex, line 149
smallDomain and hypotheses. The packet space of SG-01. Let S,T∈End(X). Set T_ba=P_bTP_a, viewed as a map X_a→ X_b. [SG-02 Transfer algebra and packet paths]
Exact mathematical passage
S,T\in\operatorname{End}(X)
\mathcal SUse in source · 20 passages
Search aliases: S mathcal
Use in source · Signal_Reconstruction_Traditional.tex, line 128
The maps A x=(P_ax)_a, S(x_a)_a=∑_ax_a are inverse isomorphisms between X and ⊕_aX_a. In particular, x=∑_aP_ax uniquely.
Exact mathematical passage
\mathcal A x=(P_ax)_a,\qquad \mathcal S(x_a)_a=\sum_ax_a
sUse in source · 61 passages
Search aliases: s
Use in source · Signal_Reconstruction_Traditional.tex, line 299
If Mf_0=Mf_1 but J(f_0)≠ J(f_1), no map R:Y→ J recovers the correct edge for both. If S∩ J≠{0}, the sum decomposition f=s+j is not unique. In particular, finitely many point samples on a periodic interval cannot recover every arbitrary interval packet.
Exact mathematical passage
f=s+j
s_0Use in source · 6 passages
Search aliases: s_0
Use in source · Signal_Reconstruction_Traditional.tex, line 547
-h on the second set. Then h=α-β, and Aα=Aβ contradicts uniqueness. Column dependence is exactly the existence of a nonzero kernel vector supported on the selected columns. The spark convention then gives the stated capped inequality, including a fully independent matrix when 2s>d. When s=0, the only admissible vector is zero and the kernel test concerns only zero. If y=s_0+Dα with s_0∈ S, then Ry=Aα, so the same theorem applies after background elimination.
Exact mathematical passage
y=s_0+D\alpha
S_{cb}Use in source · 1 passage
Search aliases: S_cb
Use in source · Signal_Reconstruction_Traditional.tex, line 152
T=∑_a,bP_bTP_a, (ST)_ca=∑_bS_cbT_ba. The block representation is an algebra isomorphism to End(⊕_aX_a), with multiplication by contraction of intermediate packets. For every integer k≥1, iterating the product gives P_a_kT^kP_a_0 =∑_a_1,…,a_k-1 (P_a_kTP_a_k-1)⋯(P_a_1TP_a_0).
Exact mathematical passage
T=\sum_{a,b}P_bTP_a,\qquad (ST)_{ca}=\sum_bS_{cb}T_{ba}.
S_kUse in source · 1 passage
Search aliases: S_k
Use in source · Signal_Reconstruction_Traditional.tex, line 488
smallDomain and hypotheses. Periodic bounded real signals. Let σ_M=(M+1)^-1∑_k=0^MS_k be the Fejer mean. Suppose 0≤ j≤1 almost everywhere. [SG-13 Positive Fourier averaging preserves range]
Exact mathematical passage
\sigma_M=(M+1)^{-1}\sum_{k=0}^MS_k
\mathcal S_KUse in source · 5 passages
Search aliases: S_K mathcal
Use in source · Signal_Reconstruction_Traditional.tex, line 359
smallDomain and hypotheses. Periodic integrable signals with one known edge shape. Let S_K be the trigonometric polynomials of degree at most K. Let m∈ L^1( T) and a finite nonempty set F⊂{n:|n|>K} satisfy D_F=∑_n∈ F|widehat m(n)|^2>0. [SG-09 Out-of-band matched identification]
Exact mathematical passage
\mathcal S_K
S_MUse in source · 9 passages
Search aliases: S_M
Use in source · Signal_Reconstruction_Traditional.tex, line 387
smallDomain and hypotheses. The model space of SG-09. Let P_Jg=ℓ_F(g)m, P_S=I-P_J, and let S_M be Fourier truncation of degree M≥ K. [SG-10 The singularity-exact block operator]
Exact mathematical passage
S_M
\mathcal S_PUse in source · 2 passages
Search aliases: S_P mathcal
Use in source · Signal_Reconstruction_Traditional.tex, line 186
P_bTP_a=∑_σ∈ F_b∑_ρ∈ F_aQ_σ TQ_ρ. The coordinate changes J= A_Q S_P and C= A_P S_Q satisfy CJ=I and JC=I on their packet coordinate spaces. Refinement is exact before any child packet is discarded.
Exact mathematical passage
J=\mathcal A_Q\mathcal S_P
\mathcal S_QUse in source · 2 passages
Search aliases: S_Q mathcal
Use in source · Signal_Reconstruction_Traditional.tex, line 187
P_bTP_a=∑_σ∈ F_b∑_ρ∈ F_aQ_σ TQ_ρ. The coordinate changes J= A_Q S_P and C= A_P S_Q satisfy CJ=I and JC=I on their packet coordinate spaces. Refinement is exact before any child packet is discarded.
Exact mathematical passage
C=\mathcal A_P\mathcal S_Q
\operatorname{Si}Use in source · 2 passages
Search aliases: Si operatorname
Use in source · Signal_Reconstruction_Traditional.tex, line 441
Uniformly for z in compact real intervals, S_Mj(x_M(z))⟶frac12+frac1πSi(z), Si(z)=∫_0^zfracsin tt dt. The first positive maximum of the limiting profile is 1+G, where G=fracSi(π)π-frac12 =0.089489872236….
Exact mathematical passage
S_Mj(x_M(z))\longrightarrow\frac12+\frac1\pi\operatorname{Si}(z), \qquad \operatorname{Si}(z)=\int_0^z\frac{\sin t}{t}\,dt.
\sinUse in source · 15 passages
Search aliases: sin
Use in source · Signal_Reconstruction_Traditional.tex, line 441
Uniformly for z in compact real intervals, S_Mj(x_M(z))⟶frac12+frac1πSi(z), Si(z)=∫_0^zfracsin tt dt. The first positive maximum of the limiting profile is 1+G, where G=fracSi(π)π-frac12 =0.089489872236….
Exact mathematical passage
S_Mj(x_M(z))\longrightarrow\frac12+\frac1\pi\operatorname{Si}(z), \qquad \operatorname{Si}(z)=\int_0^z\frac{\sin t}{t}\,dt.
\operatorname{span}Use in source · 1 passage
Search aliases: span operatorname
Use in source · Signal_Reconstruction_Traditional.tex, line 362
On X= S_K+span{m}, the functional ℓ_F(g)=D_F^-1∑_n∈ Foverlinewidehat m(n)widehat g(n) satisfies ℓ_F(s+am)=a. The decomposition is direct. If the measured coefficients have errors η_n, then |widetilde a-a|≤ D_F^-1/2(∑_n∈ F|η_n|^2)^1/2.
Exact mathematical passage
X=\mathcal S_K+\operatorname{span}\{m\}
\operatorname{spark}Use in source · 2 passages
Search aliases: spark operatorname
Use in source · Signal_Reconstruction_Traditional.tex, line 529
Every vector α with α _0≤ s is uniquely determined by Aα if and only if ker A∩{h: h _0≤min(2s,d)}={0}. Equivalently every set of at most min(2s,d) columns is independent. With the convention spark(A)=d+1 when all d columns are independent, the equivalent condition is spark(A)>min(2s,d). For s=0 uniqueness is automatic.
Exact mathematical passage
\operatorname{spark}(A)=d+1
\supUse in source · 1 passage
Search aliases: sup
Use in source · Signal_Reconstruction_Traditional.tex, line 897
If for every ℓ sup_|z-z_ℓ|=ρ|E_Q(z)| <min_|z-z_ℓ|=ρ|Q(z)|, then widetilde Q has exactly one root, counted with multiplicity, in each disk |z-z_ℓ|<ρ. All its roots are thereby matched with error less than ρ. A sufficient explicit bound on each circle is ∑_k=0^r-1|widetilde q_k-q_k|(|z_ℓ|+ρ)^k <ρ∏_j≠ℓ(|z_ℓ-z_j|-ρ).
Exact mathematical passage
\sup_{|z-z_\ell|=\rho}|E_Q(z)| <\min_{|z-z_\ell|=\rho}|Q(z)|,
\sup_{|z-z_\ell|=\rho}Use in source · 1 passage
Search aliases: sup_|z-z_ℓ|=ρ sup ell rho
Use in source · Signal_Reconstruction_Traditional.tex, line 897
If for every ℓ sup_|z-z_ℓ|=ρ|E_Q(z)| <min_|z-z_ℓ|=ρ|Q(z)|, then widetilde Q has exactly one root, counted with multiplicity, in each disk |z-z_ℓ|<ρ. All its roots are thereby matched with error less than ρ. A sufficient explicit bound on each circle is ∑_k=0^r-1|widetilde q_k-q_k|(|z_ℓ|+ρ)^k <ρ∏_j≠ℓ(|z_ℓ-z_j|-ρ).
Exact mathematical passage
\sup_{|z-z_\ell|=\rho}|E_Q(z)| <\min_{|z-z_\ell|=\rho}|Q(z)|,
\operatorname{supp}Use in source · 1 passage
Search aliases: supp operatorname
Use in source · Signal_Reconstruction_Traditional.tex, line 636
aq-widetilde awidetilde q _2^2 · =|a-widetilde a|^2|suppq∩suppwidetilde q| · +|a|^2|suppq∖suppwidetilde q| +|widetilde a|^2|suppwidetilde q∖suppq|. For a=widetilde a=1 the squared norm is the cardinality of the symmetric support difference.
Exact mathematical passage
\|aq-\widetilde a\widetilde q\|_2^2 &=|a-\widetilde a|^2|\operatorname{supp}q\cap\operatorname{supp}\widetilde q|\\ &\quad+|a|^2|\operatorname{supp}q\setminus\operatorname{supp}\widetilde q| +|\widetilde a|^2|\operatorname{supp}\widetilde q\setminus\operatorname{supp}q|.
TUse in source · 36 passages
Search aliases: T
Use in source · Signal_Reconstruction_Traditional.tex, line 149
smallDomain and hypotheses. The packet space of SG-01. Let S,T∈End(X). Set T_ba=P_bTP_a, viewed as a map X_a→ X_b. [SG-02 Transfer algebra and packet paths]
Exact mathematical passage
S,T\in\operatorname{End}(X)
\mathbb TUse in source · 7 passages
Search aliases: T mathbb
Use in source · Signal_Reconstruction_Traditional.tex, line 84
The default periodic domain is T= R/(2π Z), with widehat f(n)=frac12π∫_0^2πf(x)e^-inx dx. Signal identities inferred from Fourier coefficients are identities of L^1 classes. For a piecewise-constant edge we select the right-continuous representative. This convention fixes the values at jumps, which Fourier integrals alone cannot determine. When an L^1 norm occurs it is the unnormalized integral over a period. On the unit circle we use chord distance |z-w| and circular angular distance d_ T(τ,σ).
Exact mathematical passage
\mathbb T=\mathbb R/(2\pi\mathbb Z)
\mathsf TUse in source · 3 passages
Search aliases: T mathsf
Use in source · Signal_Reconstruction_Traditional.tex, line 710
Writing V_h=(z_ℓ^i)_i,ℓ and V_w=(z_ℓ^j)_j,ℓ, H_h,w=V_hdiag(c_ℓ)V_w^ T, rankH_h,w=r. For r=0 the Hankel matrix is zero. The transpose is algebraic and is not a conjugate transpose.
Exact mathematical passage
H_{h,w}=V_h\operatorname{diag}(c_\ell)V_w^{\mathsf T}, \qquad\operatorname{rank}H_{h,w}=r.
tUse in source · 19 passages
Search aliases: t
Use in source · Signal_Reconstruction_Traditional.tex, line 235
Interleaving the N channels recovers x exactly. With h_ba[k]=h[Nk+b-a], the output channels satisfy y_b[q]=∑_a=0^N-1∑_t∈ Zh_ba[q-t]x_a[t]. Thus convolution becomes an N× N transfer-kernel system.
Exact mathematical passage
y_b[q]=\sum_{a=0}^{N-1}\sum_{t\in\mathbb Z}h_{ba}[q-t]x_a[t].
T^{-1}Use in source · 2 passages
Search aliases: T^-1
Use in source · Signal_Reconstruction_Traditional.tex, line 783
The sawtooth derivative is DG_T=δ_0-T^-1dx: its slope is -1/T away from the endpoint and its endpoint jump is one. Summing translated copies gives the prescribed Dirac masses and a constant-density term -T^-1∑ a_ℓ, which vanishes. Consequently j is constant between the jump points. Each translated sawtooth has mean zero, so j does too. Any other packet with the same jumps has distributional derivative difference zero. Since both are piecewise constant, their difference is the same constant on ev…
Exact mathematical passage
DG_T=\delta_0-T^{-1}dx
T^kUse in source · 1 passage
Search aliases: T^k
Use in source · Signal_Reconstruction_Traditional.tex, line 159
T=∑_a,bP_bTP_a, (ST)_ca=∑_bS_cbT_ba. The block representation is an algebra isomorphism to End(⊕_aX_a), with multiplication by contraction of intermediate packets. For every integer k≥1, iterating the product gives P_a_kT^kP_a_0 =∑_a_1,…,a_k-1 (P_a_kTP_a_k-1)⋯(P_a_1TP_a_0).
Exact mathematical passage
P_{a_k}T^kP_{a_0} =\sum_{a_1,\ldots,a_{k-1}} (P_{a_k}TP_{a_{k-1}})\cdots(P_{a_1}TP_{a_0}).
T_{ba}Use in source · 2 passages
Search aliases: T_ba
Use in source · Signal_Reconstruction_Traditional.tex, line 149
smallDomain and hypotheses. The packet space of SG-01. Let S,T∈End(X). Set T_ba=P_bTP_a, viewed as a map X_a→ X_b. [SG-02 Transfer algebra and packet paths]
Exact mathematical passage
T_{ba}=P_bTP_a
\mathcal UUse in source · 4 passages
Search aliases: U mathcal
Use in source · Signal_Reconstruction_Traditional.tex, line 660
Let U_K,L consist of real periodic signals f=s+j, where s∈ S_K and j is right-continuous, piecewise constant, mean zero, and has at most L distinct active jumps. An active jump has nonzero amplitude. For an actual jump count r, write Dj=∑_ℓ=1^r a_ℓδ_τ_ℓ, z_ℓ=e^-iτ_ℓ, ∑_ℓ a_ℓ=0. The sum-zero condition is required by periodicity. A single nonzero active jump cannot occur, so the nonconstant real class starts at r=2. The case r=0 is allowed and will be handled explicitly. The smooth-edge representa…
Exact mathematical passage
\mathcal U_{K,L}
uUse in source · 20 passages
Search aliases: u
Use in source · Signal_Reconstruction_Traditional.tex, line 256
smallDomain and hypotheses. Unit-modulus exponential nodes. Let z∈ C, |z|=1, and suppose u=z^N and w_j=z^r_j are known, with N≥1 and integer offsets r_j. Let d=gcd(N,r_1,…,r_k). [SG-06 The arithmetic of residual alias ambiguity]
Exact mathematical passage
u=z^N
u^{-q_0}Use in source · 1 passage
Search aliases: u^-q_0
Use in source · Signal_Reconstruction_Traditional.tex, line 1255
for the coarse phase error. The strict branch margin selects the associated root by SG-32. Normalize the aliased estimate and use SG-33: its root error is N^-1 times an error of order N^-p-1, giving N^-p-2. On the unit circle, d_ T=2arcsin(|widehat z-z|/2), which has the same order for small errors. For the amplitude use widehat a=widehat b_0 overline u^-q_0 and a=b_0u^-q_0. Since q_0 is fixed and both powers have unit modulus, SG-26's telescoping power estimate bounds the power error by q_0|ove…
Exact mathematical passage
a=b_0u^{-q_0}
u^cUse in source · 1 passage
Search aliases: u^c
Use in source · Signal_Reconstruction_Traditional.tex, line 262
Exactly d unit-modulus nodes satisfy the same data. Hence the physical node is unique precisely when d=1. If integers c,b_1,…,b_k obey Nc+∑_jr_jb_j=d, then u^c∏_jw_j^b_j=z^d. For d=1 this is an explicit exact de-aliasing formula.
Exact mathematical passage
u^c\prod_jw_j^{b_j}=z^d.
\mathcal U_{K,L}Use in source · 3 passages
Search aliases: U_K,L mathcal
Use in source · Signal_Reconstruction_Traditional.tex, line 660
Let U_K,L consist of real periodic signals f=s+j, where s∈ S_K and j is right-continuous, piecewise constant, mean zero, and has at most L distinct active jumps. An active jump has nonzero amplitude. For an actual jump count r, write Dj=∑_ℓ=1^r a_ℓδ_τ_ℓ, z_ℓ=e^-iτ_ℓ, ∑_ℓ a_ℓ=0. The sum-zero condition is required by periodicity. A single nonzero active jump cannot occur, so the nonconstant real class starts at r=2. The case r=0 is allowed and will be handled explicitly. The smooth-edge representa…
Exact mathematical passage
\mathcal U_{K,L}
\mathcal U_{K,L_{\max}}Use in source · 1 passage
Search aliases: U_K,L_max mathcal max
Use in source · Signal_Reconstruction_Traditional.tex, line 1193
smallDomain and hypotheses. Bandlimited background with a bounded number of jumps. Let f∈ U_K,L_max, with L_max≥1. Supply N≥2, q_0≥1 with q_0N>K, and assume the active aliased nodes u_ℓ=z_ℓ^N are distinct. [SG-34 An explicit exact self-calibration budget]
Exact mathematical passage
f\in\mathcal U_{K,L_{\max}}
u_\ellUse in source · 5 passages
Search aliases: u_ℓ ell
Use in source · Signal_Reconstruction_Traditional.tex, line 976
m_(q_0+q)N+r=∑_ℓ b_ℓ,ru_ℓ^q+e_(q_0+q)N+r, u_ℓ=z_ℓ^N, b_ℓ,r=a_ℓ u_ℓ^q_0z_ℓ^r. For each matched edge b_ℓ,r+1=z_ℓ b_ℓ,r. If |widehat s(n)|≤ C_s(1+|n|)^-p-2 with p≥0, then for nonnegative q,r |e_(q_0+q)N+r|≤2π C_s(q_0N)^-p-1.
Exact mathematical passage
m_{(q_0+q)N+r}=\sum_\ell b_{\ell,r}u_\ell^q+e_{(q_0+q)N+r}, \quad u_\ell=z_\ell^N,\quad b_{\ell,r}=a_\ell u_\ell^{q_0}z_\ell^r.
u_\ell^{-q_0}Use in source · 1 passage
Search aliases: u_ℓ^-q_0 ell
Use in source · Signal_Reconstruction_Traditional.tex, line 1219
-19 gives its rank r. If r=0, the edge is zero and the low modes finish the recovery. Otherwise SG-20 recovers the u_ℓ and then the base coefficients. The adjacent channel supplies at least r moments of ∑_ℓ b_ℓ,0z_ℓ u_ℓ^q. Using the same recovered nodes, its Vandermonde system recovers b_ℓ,1=b_ℓ,0z_ℓ. Division gives z_ℓ=b_ℓ,1/b_ℓ,0, and a_ℓ=b_ℓ,0u_ℓ^-q_0. SG-21 constructs j, and subtraction from the low modes constructs s. The two index sets are disjoint because N≥2 and their residues are zero a…
Exact mathematical passage
a_\ell=b_{\ell,0}u_\ell^{-q_0}
u_\ell^qUse in source · 3 passages
Search aliases: u_ℓ^q ell
Use in source · Signal_Reconstruction_Traditional.tex, line 976
m_(q_0+q)N+r=∑_ℓ b_ℓ,ru_ℓ^q+e_(q_0+q)N+r, u_ℓ=z_ℓ^N, b_ℓ,r=a_ℓ u_ℓ^q_0z_ℓ^r. For each matched edge b_ℓ,r+1=z_ℓ b_ℓ,r. If |widehat s(n)|≤ C_s(1+|n|)^-p-2 with p≥0, then for nonnegative q,r |e_(q_0+q)N+r|≤2π C_s(q_0N)^-p-1.
Exact mathematical passage
m_{(q_0+q)N+r}=\sum_\ell b_{\ell,r}u_\ell^q+e_{(q_0+q)N+r}, \quad u_\ell=z_\ell^N,\quad b_{\ell,r}=a_\ell u_\ell^{q_0}z_\ell^r.
u_\ell^{q_0}Use in source · 2 passages
Search aliases: u_ℓ^q_0 ell
Use in source · Signal_Reconstruction_Traditional.tex, line 976
m_(q_0+q)N+r=∑_ℓ b_ℓ,ru_ℓ^q+e_(q_0+q)N+r, u_ℓ=z_ℓ^N, b_ℓ,r=a_ℓ u_ℓ^q_0z_ℓ^r. For each matched edge b_ℓ,r+1=z_ℓ b_ℓ,r. If |widehat s(n)|≤ C_s(1+|n|)^-p-2 with p≥0, then for nonnegative q,r |e_(q_0+q)N+r|≤2π C_s(q_0N)^-p-1.
Exact mathematical passage
m_{(q_0+q)N+r}=\sum_\ell b_{\ell,r}u_\ell^q+e_{(q_0+q)N+r}, \quad u_\ell=z_\ell^N,\quad b_{\ell,r}=a_\ell u_\ell^{q_0}z_\ell^r.
VUse in source · 26 passages
Search aliases: V
Use in source · Signal_Reconstruction_Traditional.tex, line 204
smallDomain and hypotheses. Vector spaces with an invertible transform. Let V:X→ Y be an isomorphism and (E_a) a finite complete packet family on Y. [SG-04 Pulled-back transform packets]
Exact mathematical passage
V:X\to Y
vUse in source · 18 passages
Search aliases: v
Use in source · Signal_Reconstruction_Traditional.tex, line 270
If z' gives the same data, the quotient v=z'/z satisfies v^N=v^r_1=⋯=v^r_k=1. Bezout's identity gives v^d=1. Conversely d divides every supplied exponent, so every dth root of unity has all those powers equal to one. Therefore the fiber consists exactly of z multiplied by the d distinct dth roots of unity. Applying the Bezout identity directly to the measured powers gives the displayed formula. Negative integer powers are legitimate because the data are nonzero.
Exact mathematical passage
v=z'/z
V^{-1}Use in source · 3 passages
Search aliases: V^-1
Use in source · Signal_Reconstruction_Traditional.tex, line 207
The operators P_a=V^-1E_aV are a complete packet family on X. For a unitary V and orthogonal coordinate selectors E_a in a Hilbert space, x ^2=∑_a P_ax ^2.
Exact mathematical passage
P_a=V^{-1}E_aV
v^dUse in source · 1 passage
Search aliases: v^d
Use in source · Signal_Reconstruction_Traditional.tex, line 271
If z' gives the same data, the quotient v=z'/z satisfies v^N=v^r_1=⋯=v^r_k=1. Bezout's identity gives v^d=1. Conversely d divides every supplied exponent, so every dth root of unity has all those powers equal to one. Therefore the fiber consists exactly of z multiplied by the d distinct dth roots of unity. Applying the Bezout identity directly to the measured powers gives the displayed formula. Negative integer powers are legitimate because the data are nonzero.
Exact mathematical passage
v^d=1
v^NUse in source · 1 passage
Search aliases: v^N
Use in source · Signal_Reconstruction_Traditional.tex, line 271
If z' gives the same data, the quotient v=z'/z satisfies v^N=v^r_1=⋯=v^r_k=1. Bezout's identity gives v^d=1. Conversely d divides every supplied exponent, so every dth root of unity has all those powers equal to one. Therefore the fiber consists exactly of z multiplied by the d distinct dth roots of unity. Applying the Bezout identity directly to the measured powers gives the displayed formula. Negative integer powers are legitimate because the data are nonzero.
Exact mathematical passage
v^N=v^{r_1}=\cdots=v^{r_k}=1
v^{r_1}Use in source · 1 passage
Search aliases: v^r_1
Use in source · Signal_Reconstruction_Traditional.tex, line 271
If z' gives the same data, the quotient v=z'/z satisfies v^N=v^r_1=⋯=v^r_k=1. Bezout's identity gives v^d=1. Conversely d divides every supplied exponent, so every dth root of unity has all those powers equal to one. Therefore the fiber consists exactly of z multiplied by the d distinct dth roots of unity. Applying the Bezout identity directly to the measured powers gives the displayed formula. Negative integer powers are legitimate because the data are nonzero.
Exact mathematical passage
v^N=v^{r_1}=\cdots=v^{r_k}=1
v^{r_k}Use in source · 1 passage
Search aliases: v^r_k
Use in source · Signal_Reconstruction_Traditional.tex, line 271
If z' gives the same data, the quotient v=z'/z satisfies v^N=v^r_1=⋯=v^r_k=1. Bezout's identity gives v^d=1. Conversely d divides every supplied exponent, so every dth root of unity has all those powers equal to one. Therefore the fiber consists exactly of z multiplied by the d distinct dth roots of unity. Applying the Bezout identity directly to the measured powers gives the displayed formula. Negative integer powers are legitimate because the data are nonzero.
Exact mathematical passage
v^N=v^{r_1}=\cdots=v^{r_k}=1
v^TUse in source · 1 passage
Search aliases: v^T
Use in source · Signal_Reconstruction_Traditional.tex, line 1562
The Hessian is twice M^TM+λ I, which is positive definite because v^T(M^TM+λ I)v= Mv ^2+λ v ^2>0 for nonzero v. Setting the gradient to zero gives the unique minimizer and rearrangement gives M^Tr=λ c. When columns are added, extending the previous coefficient vector by zero is a feasible point with exactly the previous objective, so the new optimum is no larger. This concerns the ridge objective, not an automatic monotonicity theorem for the unpenalized residual alone. For the full pixel basis …
Exact mathematical passage
v^T(M^TM+\lambda I)v=\|Mv\|^2+\lambda\|v\|^2>0
V_1Use in source · 1 passage
Search aliases: V_1
Use in source · Signal_Reconstruction_Traditional.tex, line 1519
smallDomain and hypotheses. A Hilbert space with finite selected packet spans. Let V_1⊆ V_2⊆⋯ be finite-dimensional subspaces. Let P_k be the orthogonal projector onto V_k. [SG-42 Ideal nested-span refitting decreases residual]
Exact mathematical passage
V_1\subseteq V_2\subseteq\cdots
V_2Use in source · 1 passage
Search aliases: V_2
Use in source · Signal_Reconstruction_Traditional.tex, line 1519
smallDomain and hypotheses. A Hilbert space with finite selected packet spans. Let V_1⊆ V_2⊆⋯ be finite-dimensional subspaces. Let P_k be the orthogonal projector onto V_k. [SG-42 Ideal nested-span refitting decreases residual]
Exact mathematical passage
V_1\subseteq V_2\subseteq\cdots
V_hUse in source · 3 passages
Search aliases: V_h
Use in source · Signal_Reconstruction_Traditional.tex, line 709
Writing V_h=(z_ℓ^i)_i,ℓ and V_w=(z_ℓ^j)_j,ℓ, H_h,w=V_hdiag(c_ℓ)V_w^ T, rankH_h,w=r. For r=0 the Hankel matrix is zero. The transpose is algebraic and is not a conjugate transpose.
Exact mathematical passage
V_h=(z_\ell^i)_{i,\ell}
V_KUse in source · 1 passage
Search aliases: V_K
Use in source · Signal_Reconstruction_Traditional.tex, line 1535
Orthogonal projection minimizes distance to its subspace. Since P_kf belongs to the larger subspace V_k+1, its residual is a feasible upper bound on the next minimum. Orthogonality follows from the defining projection identity, or by varying the quadratic norm along every vector in V_k. For convergence, fix ϵ>0 and choose v∈ V_K within ϵ of f by density. For all k≥ K, minimum distance is at most f-v <ϵ. Thus the residual converges to zero. This theorem assumes dense actual selected spans. A coun…
Exact mathematical passage
v\in V_K
V_kUse in source · 4 passages
Search aliases: V_k
Use in source · Signal_Reconstruction_Traditional.tex, line 1519
smallDomain and hypotheses. A Hilbert space with finite selected packet spans. Let V_1⊆ V_2⊆⋯ be finite-dimensional subspaces. Let P_k be the orthogonal projector onto V_k. [SG-42 Ideal nested-span refitting decreases residual]
Exact mathematical passage
V_k
V_{k+1}Use in source · 1 passage
Search aliases: V_k+1
Use in source · Signal_Reconstruction_Traditional.tex, line 1532
Orthogonal projection minimizes distance to its subspace. Since P_kf belongs to the larger subspace V_k+1, its residual is a feasible upper bound on the next minimum. Orthogonality follows from the defining projection identity, or by varying the quadratic norm along every vector in V_k. For convergence, fix ϵ>0 and choose v∈ V_K within ϵ of f by density. For all k≥ K, minimum distance is at most f-v <ϵ. Thus the residual converges to zero. This theorem assumes dense actual selected spans. A coun…
Exact mathematical passage
V_{k+1}
V_wUse in source · 1 passage
Search aliases: V_w
Use in source · Signal_Reconstruction_Traditional.tex, line 709
Writing V_h=(z_ℓ^i)_i,ℓ and V_w=(z_ℓ^j)_j,ℓ, H_h,w=V_hdiag(c_ℓ)V_w^ T, rankH_h,w=r. For r=0 the Hankel matrix is zero. The transpose is algebraic and is not a conjugate transpose.
Exact mathematical passage
V_w=(z_\ell^j)_{j,\ell}
V_w^{\mathsf T}Use in source · 1 passage
Search aliases: V_w^ T mathsf
Use in source · Signal_Reconstruction_Traditional.tex, line 710
Writing V_h=(z_ℓ^i)_i,ℓ and V_w=(z_ℓ^j)_j,ℓ, H_h,w=V_hdiag(c_ℓ)V_w^ T, rankH_h,w=r. For r=0 the Hankel matrix is zero. The transpose is algebraic and is not a conjugate transpose.
Exact mathematical passage
H_{h,w}=V_h\operatorname{diag}(c_\ell)V_w^{\mathsf T}, \qquad\operatorname{rank}H_{h,w}=r.
V_w^TUse in source · 1 passage
Search aliases: V_w^T
Use in source · Signal_Reconstruction_Traditional.tex, line 724
Expand a matrix entry to obtain the factorization. The leading r× r Vandermonde submatrix has determinant ∏_ℓ<k(z_k-z_ℓ), which is nonzero. Thus both rectangular Vandermonde factors have full column rank, and their transposes have full row rank. The diagonal factor is invertible. Its product with V_w^T is surjective onto C^r, and subsequent multiplication by the injective V_h has image dimension r. For an empty exponential sum every entry is zero and the rank is zero.
Exact mathematical passage
V_w^T
wUse in source · 14 passages
Search aliases: w
Use in source · Signal_Reconstruction_Traditional.tex, line 93
t_0^2πf(x)e^-inx dx. Signal identities inferred from Fourier coefficients are identities of L^1 classes. For a piecewise-constant edge we select the right-continuous representative. This convention fixes the values at jumps, which Fourier integrals alone cannot determine. When an L^1 norm occurs it is the unnormalized integral over a period. On the unit circle we use chord distance |z-w| and circular angular distance d_ T(τ,σ).
Exact mathematical passage
|z-w|
w_jUse in source · 1 passage
Search aliases: w_j
Use in source · Signal_Reconstruction_Traditional.tex, line 256
smallDomain and hypotheses. Unit-modulus exponential nodes. Let z∈ C, |z|=1, and suppose u=z^N and w_j=z^r_j are known, with N≥1 and integer offsets r_j. Let d=gcd(N,r_1,…,r_k). [SG-06 The arithmetic of residual alias ambiguity]
Exact mathematical passage
w_j=z^{r_j}
w_j^{b_j}Use in source · 1 passage
Search aliases: w_j^b_j
Use in source · Signal_Reconstruction_Traditional.tex, line 262
Exactly d unit-modulus nodes satisfy the same data. Hence the physical node is unique precisely when d=1. If integers c,b_1,…,b_k obey Nc+∑_jr_jb_j=d, then u^c∏_jw_j^b_j=z^d. For d=1 this is an explicit exact de-aliasing formula.
Exact mathematical passage
u^c\prod_jw_j^{b_j}=z^d.
\widehat aUse in source · 12 passages
Search aliases: widehat a widehat
Use in source · Signal_Reconstruction_Traditional.tex, line 598
For an unrestricted scalar amplitude the least-squares profile is widehat a(q)=frac⟨ Rq,Ry⟩ Rq _2^2, J(q)= Ry-widehat a(q)Rq _2^2. The true mask has widehat a(q_*)=a_* and J(q_*)=0. It is the unique zero-residual positive-amplitude mask exactly when no other residualized mask lies on its positive ray. Define the closed rays C_q={aRq:a≥0}. If Γ=inf_q≠ q_*dist(Ry,C_q)>0, with the convention inf∅=+∞, then every perturbed residualized datum Ry+e with e _2<Γ/2 selects q_* uniquely by nearest-ray dist…
Exact mathematical passage
\widehat a(q)=\frac{\langle Rq,Ry\rangle}{\|Rq\|_2^2}, \qquad J(q)=\|Ry-\widehat a(q)Rq\|_2^2.
\widehat a_\ellUse in source · 5 passages
Search aliases: widehat a_ℓ widehat ell
Use in source · Signal_Reconstruction_Traditional.tex, line 1241
The threshold-selected order is eventually correct. The aligned lifted nodes and locations satisfy |widehat z_ℓ-z_ℓ|=O(N^-p-2), d_ T(widehatτ_ℓ,τ_ℓ)=O(N^-p-2). The recovered amplitudes satisfy |widehat a_ℓ-a_ℓ|=O(N^-p-1).
Exact mathematical passage
|\widehat a_\ell-a_\ell|=O(N^{-p-1})
\widehat bUse in source · 1 passage
Search aliases: widehat b widehat
Use in source · Signal_Reconstruction_Traditional.tex, line 1255
se coefficients allow SG-30 to give the same order for the coarse phase error. The strict branch margin selects the associated root by SG-32. Normalize the aliased estimate and use SG-33: its root error is N^-1 times an error of order N^-p-1, giving N^-p-2. On the unit circle, d_ T=2arcsin(|widehat z-z|/2), which has the same order for small errors. For the amplitude use widehat a=widehat b_0 overline u^-q_0 and a=b_0u^-q_0. Since q_0 is fixed and both powers have unit modulus, SG-26's telescopi…
Exact mathematical passage
\widehat a=\widehat b_0\,\overline u^{-q_0}
\widehat b_0Use in source · 1 passage
Search aliases: widehat b_0 widehat
Use in source · Signal_Reconstruction_Traditional.tex, line 1255
se coefficients allow SG-30 to give the same order for the coarse phase error. The strict branch margin selects the associated root by SG-32. Normalize the aliased estimate and use SG-33: its root error is N^-1 times an error of order N^-p-1, giving N^-p-2. On the unit circle, d_ T=2arcsin(|widehat z-z|/2), which has the same order for small errors. For the amplitude use widehat a=widehat b_0 overline u^-q_0 and a=b_0u^-q_0. Since q_0 is fixed and both powers have unit modulus, SG-26's telescopi…
Exact mathematical passage
\widehat a=\widehat b_0\,\overline u^{-q_0}
\widehat fUse in source · 7 passages
Search aliases: widehat f widehat
Use in source · Signal_Reconstruction_Traditional.tex, line 85
The default periodic domain is T= R/(2π Z), with widehat f(n)=frac12π∫_0^2πf(x)e^-inx dx. Signal identities inferred from Fourier coefficients are identities of L^1 classes. For a piecewise-constant edge we select the right-continuous representative. This convention fixes the values at jumps, which Fourier integrals alone cannot determine. When an L^1 norm occurs it is the unnormalized integral over a period. On the unit circle we use chord distance |z-w| and circular angular distance d_ T(τ,σ).
Exact mathematical passage
\widehat f(n)=\frac1{2\pi}\int_0^{2\pi}f(x)e^{-inx}\,dx.
\widehat f_MUse in source · 4 passages
Search aliases: widehat f_M widehat
Use in source · Signal_Reconstruction_Traditional.tex, line 1282
smallDomain and hypotheses. A normed signal space. Let f=s+j, let widehat j belong to the same space, and let A_M be a bounded linear operator with norm at most C_M. Define widehat f_M=A_M(f-widehat j)+widehat j. [SG-36 Resolved synthesis splits the two error sources]
Exact mathematical passage
\widehat f_M=A_M(f-\widehat j)+\widehat j
\widehat gUse in source · 1 passage
Search aliases: widehat g widehat
Use in source · Signal_Reconstruction_Traditional.tex, line 363
On X= S_K+span{m}, the functional ℓ_F(g)=D_F^-1∑_n∈ Foverlinewidehat m(n)widehat g(n) satisfies ℓ_F(s+am)=a. The decomposition is direct. If the measured coefficients have errors η_n, then |widetilde a-a|≤ D_F^-1/2(∑_n∈ F|η_n|^2)^1/2.
Exact mathematical passage
\ell_F(g)=D_F^{-1}\sum_{n\in F}\overline{\widehat m(n)}\widehat g(n)
\widehat jUse in source · 15 passages
Search aliases: widehat j widehat
Use in source · Signal_Reconstruction_Traditional.tex, line 454
Direct integration gives widehat j(0)=a/π and widehat j(n)=sin(na)/(π n) for n≠0. The Dirichlet kernel D_M(t)=sin(λ_Mt)/sin(t/2) yields S_Mj(x)=frac12π∫_-a^aD_M(x-t) dt, (S_Mj)'(x)=fracD_M(x+a)-D_M(x-a)2π. At x=a, the coefficient formula gives S_Mj(a)=frac aπ+frac1π∑_n=1^Mfracsin(2na)n ⟶frac12. For completeness, the sine-series limit follows by summation by parts from bounded partial sums of e^2ina and by taking radial limits in ∑_n≥1r^ne^2ina/n=-log(1-re^2ia); its imaginary part tends to (π-2a)…
Exact mathematical passage
\widehat j(0)=a/\pi
\widehat mUse in source · 1 passage
Search aliases: widehat m widehat
Use in source · Signal_Reconstruction_Traditional.tex, line 359
smallDomain and hypotheses. Periodic integrable signals with one known edge shape. Let S_K be the trigonometric polynomials of degree at most K. Let m∈ L^1( T) and a finite nonempty set F⊂{n:|n|>K} satisfy D_F=∑_n∈ F|widehat m(n)|^2>0. [SG-09 Out-of-band matched identification]
Exact mathematical passage
D_F=\sum_{n\in F}|\widehat m(n)|^2>0
\widehat sUse in source · 6 passages
Search aliases: widehat s widehat
Use in source · Signal_Reconstruction_Traditional.tex, line 414
smallDomain and hypotheses. Periodic signals with a supplied exact edge packet. Let f=s+j with |widehat s(n)|≤ Ce^-ρ|n|, where C≥0 and ρ>0. Use the canonical uniformly convergent Fourier-series representative of s. Set R_Mf=S_Ms+j. [SG-11 Resolved error for an analytic background]
Exact mathematical passage
|\widehat s(n)|\le Ce^{-\rho|n|}
\widehat uUse in source · 6 passages
Search aliases: widehat u widehat
Use in source · Signal_Reconstruction_Traditional.tex, line 1130
smallDomain and hypotheses. Nth roots of a unit-modulus estimate. Let N≥2, |z|=1, u=z^N, and overline u=widehat u/|widehat u| with widehat u≠0. Let ρ be a root of overline u continuously associated with z, and widetilde z a coarse phase estimate. [SG-32 Unique branch selection under a strict margin]
Exact mathematical passage
\overline u=\widehat u/|\widehat u|
\widehat zUse in source · 5 passages
Search aliases: widehat z widehat
Use in source · Signal_Reconstruction_Traditional.tex, line 940
widehat a-a _2 ≤frac e _2+ Δ V _2 a _2 σ_min(V)- Δ V _2. For unit nodes in rows with nonnegative powers n, matched chord error |widehat z_ℓ-z_ℓ|≤δ gives the entry bound |widehat z_ℓ^n-z_ℓ^n|≤ nδ.
Exact mathematical passage
|\widehat z_\ell-z_\ell|\le\delta
\widehat z^nUse in source · 1 passage
Search aliases: widehat z^n widehat
Use in source · Signal_Reconstruction_Traditional.tex, line 954
The minimum singular value of widetilde V is at least σ_min(V)- Δ V _2>0. Thus its least-squares inverse exists. Since widetilde V a=m+Δ V a, widehat a-a=widetilde V^†(e-Δ V a). Apply widetilde V^† _2=1/σ_min(widetilde V). The power bound follows by factoring widehat z^n-z^n=(widehat z-z)∑_k=0^n-1widehat z^n-1-kz^k; each summand has modulus one. The statement assumes aligned nodes and unit normalization. Arbitrary unnormalized roots require extra modulus factors.
Exact mathematical passage
\widehat z^n-z^n=(\widehat z-z)\sum_{k=0}^{n-1}\widehat z^{n-1-k}z^k
\widehat z^{n-1-k}Use in source · 1 passage
Search aliases: widehat z^n-1-k widehat
Use in source · Signal_Reconstruction_Traditional.tex, line 954
The minimum singular value of widetilde V is at least σ_min(V)- Δ V _2>0. Thus its least-squares inverse exists. Since widetilde V a=m+Δ V a, widehat a-a=widetilde V^†(e-Δ V a). Apply widetilde V^† _2=1/σ_min(widetilde V). The power bound follows by factoring widehat z^n-z^n=(widehat z-z)∑_k=0^n-1widehat z^n-1-kz^k; each summand has modulus one. The statement assumes aligned nodes and unit normalization. Arbitrary unnormalized roots require extra modulus factors.
Exact mathematical passage
\widehat z^n-z^n=(\widehat z-z)\sum_{k=0}^{n-1}\widehat z^{n-1-k}z^k
\widehat z_\ellUse in source · 2 passages
Search aliases: widehat z_ℓ widehat ell
Use in source · Signal_Reconstruction_Traditional.tex, line 940
widehat a-a _2 ≤frac e _2+ Δ V _2 a _2 σ_min(V)- Δ V _2. For unit nodes in rows with nonnegative powers n, matched chord error |widehat z_ℓ-z_ℓ|≤δ gives the entry bound |widehat z_ℓ^n-z_ℓ^n|≤ nδ.
Exact mathematical passage
|\widehat z_\ell-z_\ell|\le\delta
\widehat z_\ell^nUse in source · 1 passage
Search aliases: widehat z_ℓ^n widehat ell
Use in source · Signal_Reconstruction_Traditional.tex, line 941
widehat a-a _2 ≤frac e _2+ Δ V _2 a _2 σ_min(V)- Δ V _2. For unit nodes in rows with nonnegative powers n, matched chord error |widehat z_ℓ-z_ℓ|≤δ gives the entry bound |widehat z_ℓ^n-z_ℓ^n|≤ nδ.
Exact mathematical passage
|\widehat z_\ell^n-z_\ell^n|\le n\delta
\widehat\alphaUse in source · 9 passages
Search aliases: widehatα widehat alpha
Use in source · Signal_Reconstruction_Traditional.tex, line 556
smallDomain and hypotheses. The Euclidean model of SG-14. Assume s≥1, d≥1, and the injectivity condition of SG-14. Let y=s_0+Dα_0+η, s_0∈ S, α_0 _0≤ s, and Rη _2≤ϵ. Let widehatα be s-sparse with Awidehatα-Ry _2≤ϵ. [SG-15 Sparse noise stability and support thresholding]
Exact mathematical passage
\widehat\alpha
\widehat\tauUse in source · 4 passages
Search aliases: widehatτ widehat tau
Use in source · Signal_Reconstruction_Traditional.tex, line 1236
The threshold-selected order is eventually correct. The aligned lifted nodes and locations satisfy |widehat z_ℓ-z_ℓ|=O(N^-p-2), d_ T(widehatτ_ℓ,τ_ℓ)=O(N^-p-2). The recovered amplitudes satisfy |widehat a_ℓ-a_ℓ|=O(N^-p-1).
Exact mathematical passage
|\widehat z_\ell-z_\ell|=O(N^{-p-2}),\qquad d_{\mathbb T}(\widehat\tau_\ell,\tau_\ell)=O(N^{-p-2}).
\widehat\tau_\ellUse in source · 4 passages
Search aliases: widehatτ_ℓ widehat tau ell
Use in source · Signal_Reconstruction_Traditional.tex, line 1236
The threshold-selected order is eventually correct. The aligned lifted nodes and locations satisfy |widehat z_ℓ-z_ℓ|=O(N^-p-2), d_ T(widehatτ_ℓ,τ_ℓ)=O(N^-p-2). The recovered amplitudes satisfy |widehat a_ℓ-a_ℓ|=O(N^-p-1).
Exact mathematical passage
|\widehat z_\ell-z_\ell|=O(N^{-p-2}),\qquad d_{\mathbb T}(\widehat\tau_\ell,\tau_\ell)=O(N^{-p-2}).
\widetilde aUse in source · 6 passages
Search aliases: widetilde a widetilde
Use in source · Signal_Reconstruction_Traditional.tex, line 368
On X= S_K+span{m}, the functional ℓ_F(g)=D_F^-1∑_n∈ Foverlinewidehat m(n)widehat g(n) satisfies ℓ_F(s+am)=a. The decomposition is direct. If the measured coefficients have errors η_n, then |widetilde a-a|≤ D_F^-1/2(∑_n∈ F|η_n|^2)^1/2.
Exact mathematical passage
|\widetilde a-a|\le D_F^{-1/2}\Bigl(\sum_{n\in F}|\eta_n|^2\Bigr)^{1/2}.
\widetilde HUse in source · 4 passages
Search aliases: widetilde H widetilde
Use in source · Signal_Reconstruction_Traditional.tex, line 867
smallDomain and hypotheses. The invertible square Prony system of SG-20. Write Hq=-v, and suppose widetilde H=H+E, widetilde v=v+e. Assume H^-1 _2 E _2<1. [SG-24 Annihilator coefficient perturbation]
Exact mathematical passage
\widetilde H=H+E
\widetilde H^{-1}Use in source · 1 passage
Search aliases: widetilde H^-1 widetilde
Use in source · Signal_Reconstruction_Traditional.tex, line 882
Factor widetilde H=H(I+H^-1E). The Neumann series converges in operator norm under the supplied inequality, so widetilde H^-1 _2≤ H^-1 _2/(1- H^-1 _2 E _2). Subtract the equations to obtain widetilde H(widetilde q-q)=-e-Eq. Apply the inverse bound and the triangle inequality. The estimate assumes that the same true order is used; it does not itself choose the order from noisy data.
Exact mathematical passage
\|\widetilde H^{-1}\|_2\le \|H^{-1}\|_2/(1-\|H^{-1}\|_2\|E\|_2)
\widetilde mUse in source · 3 passages
Search aliases: widetilde m widetilde
Use in source · Signal_Reconstruction_Traditional.tex, line 322
Their unordered location sets have permutation-matched bottleneck distance at least π-δ, while their measured derivative moments satisfy |m_n-widetilde m_n| =|(1-e^-inπ)(1-e^-inδ)| ≤2|n|δ (n∈ F). There is no Lipschitz inverse for locations with one constant on all distinct two-jump configurations, even at this fixed order and with known amplitudes.
Exact mathematical passage
|m_n-\widetilde m_n| =|(1-e^{-in\pi})(1-e^{-in\delta})| \le2|n|\delta\qquad(n\in F).
\widetilde m_nUse in source · 1 passage
Search aliases: widetilde m_n widetilde
Use in source · Signal_Reconstruction_Traditional.tex, line 322
Their unordered location sets have permutation-matched bottleneck distance at least π-δ, while their measured derivative moments satisfy |m_n-widetilde m_n| =|(1-e^-inπ)(1-e^-inδ)| ≤2|n|δ (n∈ F). There is no Lipschitz inverse for locations with one constant on all distinct two-jump configurations, even at this fixed order and with known amplitudes.
Exact mathematical passage
|m_n-\widetilde m_n| =|(1-e^{-in\pi})(1-e^{-in\delta})| \le2|n|\delta\qquad(n\in F).
\widetilde QUse in source · 2 passages
Search aliases: widetilde Q widetilde
Use in source · Signal_Reconstruction_Traditional.tex, line 893
smallDomain and hypotheses. Monic complex polynomials of the same degree. Let r≥1 and Q(z)=∏_ℓ=1^r(z-z_ℓ) have distinct roots and widetilde Q=Q+E_Q. Choose 0<ρ<tfrac12min_ℓ≠ k|z_ℓ-z_k| when r≥2, or any ρ>0 when r=1. [SG-25 Conditional simple-root disk localization]
Exact mathematical passage
\widetilde Q=Q+E_Q
\widetilde qUse in source · 5 passages
Search aliases: widetilde q widetilde
Use in source · Signal_Reconstruction_Traditional.tex, line 633
smallDomain and hypotheses. Finite binary masks. Let q,widetilde q∈{0,1}^m and a,widetilde a∈ R. [SG-17 Mask error is a support geometry calculation]
Exact mathematical passage
q,\widetilde q\in\{0,1\}^m
\widetilde q_kUse in source · 1 passage
Search aliases: widetilde q_k widetilde
Use in source · Signal_Reconstruction_Traditional.tex, line 904
If for every ℓ sup_|z-z_ℓ|=ρ|E_Q(z)| <min_|z-z_ℓ|=ρ|Q(z)|, then widetilde Q has exactly one root, counted with multiplicity, in each disk |z-z_ℓ|<ρ. All its roots are thereby matched with error less than ρ. A sufficient explicit bound on each circle is ∑_k=0^r-1|widetilde q_k-q_k|(|z_ℓ|+ρ)^k <ρ∏_j≠ℓ(|z_ℓ-z_j|-ρ).
Exact mathematical passage
\sum_{k=0}^{r-1}|\widetilde q_k-q_k|(|z_\ell|+\rho)^k <\rho\prod_{j\ne\ell}(|z_\ell-z_j|-\rho).
\widetilde VUse in source · 6 passages
Search aliases: widetilde V widetilde
Use in source · Signal_Reconstruction_Traditional.tex, line 931
smallDomain and hypotheses. A square or overdetermined Vandermonde amplitude fit. Let V have full column rank, m=Va, widetilde V=V+Δ V, and widetilde m=m+e. Assume σ_min(V)> Δ V _2. Let widehat a minimize widetilde Vb-widetilde m _2. [SG-26 Amplitude stability with perturbed nodes]
Exact mathematical passage
\widetilde V=V+\Delta V
\widetilde vUse in source · 1 passage
Search aliases: widetilde v widetilde
Use in source · Signal_Reconstruction_Traditional.tex, line 867
smallDomain and hypotheses. The invertible square Prony system of SG-20. Write Hq=-v, and suppose widetilde H=H+E, widetilde v=v+e. Assume H^-1 _2 E _2<1. [SG-24 Annihilator coefficient perturbation]
Exact mathematical passage
\widetilde v=v+e
\widetilde V^\daggerUse in source · 2 passages
Search aliases: widetilde V^† widetilde dagger
Use in source · Signal_Reconstruction_Traditional.tex, line 949
The minimum singular value of widetilde V is at least σ_min(V)- Δ V _2>0. Thus its least-squares inverse exists. Since widetilde V a=m+Δ V a, widehat a-a=widetilde V^†(e-Δ V a). Apply widetilde V^† _2=1/σ_min(widetilde V). The power bound follows by factoring widehat z^n-z^n=(widehat z-z)∑_k=0^n-1widehat z^n-1-kz^k; each summand has modulus one. The statement assumes aligned nodes and unit normalization. Arbitrary unnormalized roots require extra modulus factors.
Exact mathematical passage
\widehat a-a=\widetilde V^\dagger(e-\Delta V a).
\widetilde zUse in source · 5 passages
Search aliases: widetilde z widetilde
Use in source · Signal_Reconstruction_Traditional.tex, line 1130
smallDomain and hypotheses. Nth roots of a unit-modulus estimate. Let N≥2, |z|=1, u=z^N, and overline u=widehat u/|widehat u| with widehat u≠0. Let ρ be a root of overline u continuously associated with z, and widetilde z a coarse phase estimate. [SG-32 Unique branch selection under a strict margin]
Exact mathematical passage
\widetilde z
XUse in source · 20 passages
Search aliases: X
Use in source · Signal_Reconstruction_Traditional.tex, line 124
smallDomain and hypotheses. Vector spaces over a field. Let P_1,…,P_m∈End(X) satisfy P_aP_b=δ_abP_a and ∑_aP_a=I_X. Set X_a=P_aX. [SG-01 Complete packet analysis and synthesis]
Exact mathematical passage
P_1,\ldots,P_m\in\operatorname{End}(X)
xUse in source · 39 passages
Search aliases: x
Use in source · Signal_Reconstruction_Traditional.tex, line 85
The default periodic domain is T= R/(2π Z), with widehat f(n)=frac12π∫_0^2πf(x)e^-inx dx. Signal identities inferred from Fourier coefficients are identities of L^1 classes. For a piecewise-constant edge we select the right-continuous representative. This convention fixes the values at jumps, which Fourier integrals alone cannot determine. When an L^1 norm occurs it is the unnormalized integral over a period. On the unit circle we use chord distance |z-w| and circular angular distance d_ T(τ,σ).
Exact mathematical passage
\widehat f(n)=\frac1{2\pi}\int_0^{2\pi}f(x)e^{-inx}\,dx.
X_aUse in source · 5 passages
Search aliases: X_a
Use in source · Signal_Reconstruction_Traditional.tex, line 124
smallDomain and hypotheses. Vector spaces over a field. Let P_1,…,P_m∈End(X) satisfy P_aP_b=δ_abP_a and ∑_aP_a=I_X. Set X_a=P_aX. [SG-01 Complete packet analysis and synthesis]
Exact mathematical passage
X_a=P_aX
x_aUse in source · 8 passages
Search aliases: x_a
Use in source · Signal_Reconstruction_Traditional.tex, line 128
The maps A x=(P_ax)_a, S(x_a)_a=∑_ax_a are inverse isomorphisms between X and ⊕_aX_a. In particular, x=∑_aP_ax uniquely.
Exact mathematical passage
\mathcal A x=(P_ax)_a,\qquad \mathcal S(x_a)_a=\sum_ax_a
X_bUse in source · 1 passage
Search aliases: X_b
Use in source · Signal_Reconstruction_Traditional.tex, line 149
smallDomain and hypotheses. The packet space of SG-01. Let S,T∈End(X). Set T_ba=P_bTP_a, viewed as a map X_a→ X_b. [SG-02 Transfer algebra and packet paths]
Exact mathematical passage
X_a\to X_b
x_bUse in source · 1 passage
Search aliases: x_b
Use in source · Signal_Reconstruction_Traditional.tex, line 140
Completeness gives S A x=x. If x_a∈ X_a, then P_ax_a=x_a and P_bx_a=0 for b≠ a. Thus the bth component of A S(x_a)_a is P_b∑_ax_a=x_b. Both inverse identities follow. For uniqueness, apply P_b to any proposed sum of components. This proof uses pairwise annihilation, not orthogonality; the packet projections can be oblique.
Exact mathematical passage
P_b\sum_ax_a=x_b
x_MUse in source · 3 passages
Search aliases: x_M
Use in source · Signal_Reconstruction_Traditional.tex, line 437
smallDomain and hypotheses. The interval packet j=1_[-a,a) on T. Assume 0<a<π, let λ_M=M+1/2, and use the right edge coordinate x_M(z)=a-z/λ_M. [SG-12 The scaled interval-edge Gibbs profile]
Exact mathematical passage
x_M(z)=a-z/\lambda_M
x_nUse in source · 3 passages
Search aliases: x_n
Use in source · Signal_Reconstruction_Traditional.tex, line 1638
For x∈ X choose x_n∈ D with x_n→ x. Continuity gives Ax=lim_nAx_n=lim_nBx_n=Bx. This proves uniqueness of bounded extensions. For an infinite packet sum one would need partial sums to converge to the declared operator, in a stated operator or strong topology. Neither density nor the finite identity by itself proves such a limit. In SG-42 density of the actual selected spans provides a specific strong convergence theorem; a capped dictionary store or an arbitrary greedy prefix does not inherit it…
Exact mathematical passage
x_n\in D
YUse in source · 5 passages
Search aliases: Y
Use in source · Signal_Reconstruction_Traditional.tex, line 204
smallDomain and hypotheses. Vector spaces with an invertible transform. Let V:X→ Y be an isomorphism and (E_a) a finite complete packet family on Y. [SG-04 Pulled-back transform packets]
Exact mathematical passage
V:X\to Y
yUse in source · 28 passages
Search aliases: y
Use in source · Signal_Reconstruction_Traditional.tex, line 227
smallDomain and hypotheses. Absolutely summable sequences on Z. Fix an integer N≥1. Let h,x∈ℓ^1( Z) and y=h*x. Define x_a[q]=x[Nq+a] and y_b[q]=y[Nq+b], for 0≤ a,b<N. [SG-05 Polyphase analysis and convolution kernels]
Exact mathematical passage
y=h*x
y_bUse in source · 2 passages
Search aliases: y_b
Use in source · Signal_Reconstruction_Traditional.tex, line 227
smallDomain and hypotheses. Absolutely summable sequences on Z. Fix an integer N≥1. Let h,x∈ℓ^1( Z) and y=h*x. Define x_a[q]=x[Nq+a] and y_b[q]=y[Nq+b], for 0≤ a,b<N. [SG-05 Polyphase analysis and convolution kernels]
Exact mathematical passage
y_b[q]=y[Nq+b]
\mathbb ZUse in source · 6 passages
Search aliases: Z mathbb
Use in source · Signal_Reconstruction_Traditional.tex, line 84
The default periodic domain is T= R/(2π Z), with widehat f(n)=frac12π∫_0^2πf(x)e^-inx dx. Signal identities inferred from Fourier coefficients are identities of L^1 classes. For a piecewise-constant edge we select the right-continuous representative. This convention fixes the values at jumps, which Fourier integrals alone cannot determine. When an L^1 norm occurs it is the unnormalized integral over a period. On the unit circle we use chord distance |z-w| and circular angular distance d_ T(τ,σ).
Exact mathematical passage
\mathbb T=\mathbb R/(2\pi\mathbb Z)
zUse in source · 89 passages
Search aliases: z
Use in source · Signal_Reconstruction_Traditional.tex, line 93
t_0^2πf(x)e^-inx dx. Signal identities inferred from Fourier coefficients are identities of L^1 classes. For a piecewise-constant edge we select the right-continuous representative. This convention fixes the values at jumps, which Fourier integrals alone cannot determine. When an L^1 norm occurs it is the unnormalized integral over a period. On the unit circle we use chord distance |z-w| and circular angular distance d_ T(τ,σ).
Exact mathematical passage
|z-w|
z'Use in source · 2 passages
Search aliases: z'
Use in source · Signal_Reconstruction_Traditional.tex, line 270
If z' gives the same data, the quotient v=z'/z satisfies v^N=v^r_1=⋯=v^r_k=1. Bezout's identity gives v^d=1. Conversely d divides every supplied exponent, so every dth root of unity has all those powers equal to one. Therefore the fiber consists exactly of z multiplied by the d distinct dth roots of unity. Applying the Bezout identity directly to the measured powers gives the displayed formula. Negative integer powers are legitimate because the data are nonzero.
Exact mathematical passage
z'
z^4Use in source · 1 passage
Search aliases: z^4
Use in source · Signal_Reconstruction_Traditional.tex, line 1686
four they are m_4q=0, m_4q+1=2+4i, m_4q+2=0, m_4q+3=2-4i. Since the smooth degree is five, the first clean consecutive moment is n_0=6. With an order bound four, coefficients at n=6,…,13 determine the edge. The Hankel factorization has four distinct nodes and four nonzero amplitudes, hence rank four. Its recurrence is m_n+4-m_n=0, with monic polynomial Q(z)=z^4-1. Recovering those roots and solving the Vandermonde system returns the displayed jumps. SG-21 then constructs the edge, and coefficien…
Exact mathematical passage
Q(z)=z^4-1
z^dUse in source · 1 passage
Search aliases: z^d
Use in source · Signal_Reconstruction_Traditional.tex, line 262
Exactly d unit-modulus nodes satisfy the same data. Hence the physical node is unique precisely when d=1. If integers c,b_1,…,b_k obey Nc+∑_jr_jb_j=d, then u^c∏_jw_j^b_j=z^d. For d=1 this is an explicit exact de-aliasing formula.
Exact mathematical passage
u^c\prod_jw_j^{b_j}=z^d.
z^kUse in source · 2 passages
Search aliases: z^k
Use in source · Signal_Reconstruction_Traditional.tex, line 737
The unique monic annihilator Q(z)=∏_ℓ=1^r(z-z_ℓ)=∑_k=0^r-1q_kz^k+z^r is recovered from H_r,rq=-v, v=(m_n_0+r+i)_i=0^r-1. Its roots are exactly the nodes. Once the nodes are known, r consecutive moments determine the coefficients uniquely by a Vandermonde solve. For unit-modulus nodes the corresponding locations are τ_ℓ=-arg z_ℓpmod2π.
Exact mathematical passage
Q(z)=\prod_{\ell=1}^r(z-z_\ell)=\sum_{k=0}^{r-1}q_kz^k+z^r
z^NUse in source · 4 passages
Search aliases: z^N
Use in source · Signal_Reconstruction_Traditional.tex, line 256
smallDomain and hypotheses. Unit-modulus exponential nodes. Let z∈ C, |z|=1, and suppose u=z^N and w_j=z^r_j are known, with N≥1 and integer offsets r_j. Let d=gcd(N,r_1,…,r_k). [SG-06 The arithmetic of residual alias ambiguity]
Exact mathematical passage
u=z^N
z^nUse in source · 2 passages
Search aliases: z^n
Use in source · Signal_Reconstruction_Traditional.tex, line 954
The minimum singular value of widetilde V is at least σ_min(V)- Δ V _2>0. Thus its least-squares inverse exists. Since widetilde V a=m+Δ V a, widehat a-a=widetilde V^†(e-Δ V a). Apply widetilde V^† _2=1/σ_min(widetilde V). The power bound follows by factoring widehat z^n-z^n=(widehat z-z)∑_k=0^n-1widehat z^n-1-kz^k; each summand has modulus one. The statement assumes aligned nodes and unit normalization. Arbitrary unnormalized roots require extra modulus factors.
Exact mathematical passage
\widehat z^n-z^n=(\widehat z-z)\sum_{k=0}^{n-1}\widehat z^{n-1-k}z^k
z^{n+1}Use in source · 1 passage
Search aliases: z^n+1
Use in source · Signal_Reconstruction_Traditional.tex, line 1440
For one term, (E-z)(z^nP(n))=z^n+1(P(n+1)-P(n)). A finite difference lowers a nonconstant polynomial's degree by one, and annihilates a constant. After d+1 differences the term vanishes. The shift factors commute, so the product annihilates every term in the sum. If P_ℓ(n) has leading coefficient c_ℓ≠0 at degree d, substitution of (q_0+q)N+v has leading q coefficient c_ℓ N^d. The remaining exponential prefactor is z_ℓ^q_0N+v; hence γ_ℓ,v,d=c_ℓ N^dz_ℓ^q_0N+v and the ratio is z_ℓ. Repeated roots r…
Exact mathematical passage
(E-z)(z^nP(n))=z^{n+1}(P(n+1)-P(n))
z^rUse in source · 3 passages
Search aliases: z^r
Use in source · Signal_Reconstruction_Traditional.tex, line 737
The unique monic annihilator Q(z)=∏_ℓ=1^r(z-z_ℓ)=∑_k=0^r-1q_kz^k+z^r is recovered from H_r,rq=-v, v=(m_n_0+r+i)_i=0^r-1. Its roots are exactly the nodes. Once the nodes are known, r consecutive moments determine the coefficients uniquely by a Vandermonde solve. For unit-modulus nodes the corresponding locations are τ_ℓ=-arg z_ℓpmod2π.
Exact mathematical passage
Q(z)=\prod_{\ell=1}^r(z-z_\ell)=\sum_{k=0}^{r-1}q_kz^k+z^r
z^{r+1}Use in source · 2 passages
Search aliases: z^r+1
Use in source · Signal_Reconstruction_Traditional.tex, line 1060
Each summand is overlinebz^r bz^r+1=|b|^2z because |z|=1. There are R summands and R≥1, so normalization returns z. For two channels direct division cancels b. No statistical independence is assumed, so one cannot improve a worst-case noise order merely by increasing R; the next bound shows the cancellation of the number of products.
Exact mathematical passage
\overline{bz^r}\,bz^{r+1}=|b|^2z
z^{r_j}Use in source · 1 passage
Search aliases: z^r_j
Use in source · Signal_Reconstruction_Traditional.tex, line 256
smallDomain and hypotheses. Unit-modulus exponential nodes. Let z∈ C, |z|=1, and suppose u=z^N and w_j=z^r_j are known, with N≥1 and integer offsets r_j. Let d=gcd(N,r_1,…,r_k). [SG-06 The arithmetic of residual alias ambiguity]
Exact mathematical passage
w_j=z^{r_j}
z_jUse in source · 2 passages
Search aliases: z_j
Use in source · Signal_Reconstruction_Traditional.tex, line 904
If for every ℓ sup_|z-z_ℓ|=ρ|E_Q(z)| <min_|z-z_ℓ|=ρ|Q(z)|, then widetilde Q has exactly one root, counted with multiplicity, in each disk |z-z_ℓ|<ρ. All its roots are thereby matched with error less than ρ. A sufficient explicit bound on each circle is ∑_k=0^r-1|widetilde q_k-q_k|(|z_ℓ|+ρ)^k <ρ∏_j≠ℓ(|z_ℓ-z_j|-ρ).
Exact mathematical passage
\sum_{k=0}^{r-1}|\widetilde q_k-q_k|(|z_\ell|+\rho)^k <\rho\prod_{j\ne\ell}(|z_\ell-z_j|-\rho).
z_kUse in source · 2 passages
Search aliases: z_k
Use in source · Signal_Reconstruction_Traditional.tex, line 722
Expand a matrix entry to obtain the factorization. The leading r× r Vandermonde submatrix has determinant ∏_ℓ<k(z_k-z_ℓ), which is nonzero. Thus both rectangular Vandermonde factors have full column rank, and their transposes have full row rank. The diagonal factor is invertible. Its product with V_w^T is surjective onto C^r, and subsequent multiplication by the injective V_h has image dimension r. For an empty exponential sum every entry is zero and the rank is zero.
Exact mathematical passage
\prod_{\ell<k}(z_k-z_\ell)
z_\ellUse in source · 27 passages
Search aliases: z_ℓ ell
Use in source · Signal_Reconstruction_Traditional.tex, line 664
Let U_K,L consist of real periodic signals f=s+j, where s∈ S_K and j is right-continuous, piecewise constant, mean zero, and has at most L distinct active jumps. An active jump has nonzero amplitude. For an actual jump count r, write Dj=∑_ℓ=1^r a_ℓδ_τ_ℓ, z_ℓ=e^-iτ_ℓ, ∑_ℓ a_ℓ=0. The sum-zero condition is required by periodicity. A single nonzero active jump cannot occur, so the nonconstant real class starts at r=2. The case r=0 is allowed and will be handled explicitly. The smooth-edge representa…
Exact mathematical passage
Dj=\sum_{\ell=1}^r a_\ell\delta_{\tau_\ell},\quad z_\ell=e^{-i\tau_\ell},\quad\sum_\ell a_\ell=0.
z_\ell^6Use in source · 1 passage
Search aliases: z_ℓ^6 ell
Use in source · Signal_Reconstruction_Traditional.tex, line 1699
The same object demonstrates internal phase recovery. Choose N=3 and q_0=2. The nodes are permuted by cubing, so the aliased nodes remain distinct. The sufficient self-calibration sets for L_max=4 are I_0={6,9,12,15,18,21,24,27,30}, I_1={7,10,13,16}. For each matched term the base coefficient is b_ℓ,0=a_ℓ z_ℓ^6, and the adjacent coefficient is b_ℓ,1=b_ℓ,0z_ℓ. Their ratio is the original physical node, not the principal cube root. The total high budget is thirteen, agreeing with 3L_max+1. The un-…
Exact mathematical passage
b_{\ell,0}=a_\ell z_\ell^6
z_\ell^iUse in source · 1 passage
Search aliases: z_ℓ^i ell
Use in source · Signal_Reconstruction_Traditional.tex, line 709
Writing V_h=(z_ℓ^i)_i,ℓ and V_w=(z_ℓ^j)_j,ℓ, H_h,w=V_hdiag(c_ℓ)V_w^ T, rankH_h,w=r. For r=0 the Hankel matrix is zero. The transpose is algebraic and is not a conjugate transpose.
Exact mathematical passage
V_h=(z_\ell^i)_{i,\ell}
z_\ell^jUse in source · 1 passage
Search aliases: z_ℓ^j ell
Use in source · Signal_Reconstruction_Traditional.tex, line 709
Writing V_h=(z_ℓ^i)_i,ℓ and V_w=(z_ℓ^j)_j,ℓ, H_h,w=V_hdiag(c_ℓ)V_w^ T, rankH_h,w=r. For r=0 the Hankel matrix is zero. The transpose is algebraic and is not a conjugate transpose.
Exact mathematical passage
V_w=(z_\ell^j)_{j,\ell}
z_\ell^kUse in source · 1 passage
Search aliases: z_ℓ^k ell
Use in source · Signal_Reconstruction_Traditional.tex, line 706
smallDomain and hypotheses. Finite exponential moments over C. Let m_n_0+k=∑_ℓ=1^r c_ℓ z_ℓ^k, with distinct nonzero nodes and nonzero coefficients. Let H_h,w=(m_n_0+i+j)_0≤ i<h, 0≤ j<w, with h,w≥ r. [SG-19 Hankel rank detects the actual order]
Exact mathematical passage
m_{n_0+k}=\sum_{\ell=1}^r c_\ell z_\ell^k
z_\ell^NUse in source · 3 passages
Search aliases: z_ℓ^N ell
Use in source · Signal_Reconstruction_Traditional.tex, line 976
m_(q_0+q)N+r=∑_ℓ b_ℓ,ru_ℓ^q+e_(q_0+q)N+r, u_ℓ=z_ℓ^N, b_ℓ,r=a_ℓ u_ℓ^q_0z_ℓ^r. For each matched edge b_ℓ,r+1=z_ℓ b_ℓ,r. If |widehat s(n)|≤ C_s(1+|n|)^-p-2 with p≥0, then for nonnegative q,r |e_(q_0+q)N+r|≤2π C_s(q_0N)^-p-1.
Exact mathematical passage
m_{(q_0+q)N+r}=\sum_\ell b_{\ell,r}u_\ell^q+e_{(q_0+q)N+r}, \quad u_\ell=z_\ell^N,\quad b_{\ell,r}=a_\ell u_\ell^{q_0}z_\ell^r.
z_\ell^nUse in source · 5 passages
Search aliases: z_ℓ^n ell
Use in source · Signal_Reconstruction_Traditional.tex, line 678
Dj=∑_ℓ a_ℓδ_τ_ℓ, m_n:=2π inwidehat f(n)=∑_ℓ=1^ra_ℓ z_ℓ^n. For a non-bandlimited integrable background the same formula holds with an additive defect e_n=2π inwidehat s(n).
Exact mathematical passage
Dj=\sum_\ell a_\ell\delta_{\tau_\ell},\qquad m_n:=2\pi in\widehat f(n)=\sum_{\ell=1}^ra_\ell z_\ell^n.
z_\ell^{n-n_0}Use in source · 1 passage
Search aliases: z_ℓ^n-n_0 ell
Use in source · Signal_Reconstruction_Traditional.tex, line 753
For every available recurrence index, ∑_k=0^rq_km_n+k =∑_ℓ c_ℓ z_ℓ^n-n_0Q(z_ℓ)=0, q_r=1. SG-19 makes H_r,r invertible. Thus the system has one solution and the true polynomial supplies it. Factoring the resulting polynomial recovers the distinct nodes up to a permutation. The amplitude matrix for consecutive moments is a square Vandermonde multiplied by an invertible diagonal node power matrix, so it too is invertible. The argument convention identifies locations modulo a period, as required. Po…
Exact mathematical passage
\sum_{k=0}^rq_km_{n+k} =\sum_\ell c_\ell z_\ell^{n-n_0}Q(z_\ell)=0, \qquad q_r=1.
z_\ell^{q_0N+v}Use in source · 2 passages
Search aliases: z_ℓ^q_0N+v ell
Use in source · Signal_Reconstruction_Traditional.tex, line 1446
)). A finite difference lowers a nonconstant polynomial's degree by one, and annihilates a constant. After d+1 differences the term vanishes. The shift factors commute, so the product annihilates every term in the sum. If P_ℓ(n) has leading coefficient c_ℓ≠0 at degree d, substitution of (q_0+q)N+v has leading q coefficient c_ℓ N^d. The remaining exponential prefactor is z_ℓ^q_0N+v; hence γ_ℓ,v,d=c_ℓ N^dz_ℓ^q_0N+v and the ratio is z_ℓ. Repeated roots require confluent recovery and stability argum…
Exact mathematical passage
z_\ell^{q_0N+v}
z_\ell^rUse in source · 2 passages
Search aliases: z_ℓ^r ell
Use in source · Signal_Reconstruction_Traditional.tex, line 976
m_(q_0+q)N+r=∑_ℓ b_ℓ,ru_ℓ^q+e_(q_0+q)N+r, u_ℓ=z_ℓ^N, b_ℓ,r=a_ℓ u_ℓ^q_0z_ℓ^r. For each matched edge b_ℓ,r+1=z_ℓ b_ℓ,r. If |widehat s(n)|≤ C_s(1+|n|)^-p-2 with p≥0, then for nonnegative q,r |e_(q_0+q)N+r|≤2π C_s(q_0N)^-p-1.
Exact mathematical passage
m_{(q_0+q)N+r}=\sum_\ell b_{\ell,r}u_\ell^q+e_{(q_0+q)N+r}, \quad u_\ell=z_\ell^N,\quad b_{\ell,r}=a_\ell u_\ell^{q_0}z_\ell^r.
|Use in source · 71 passages
Search aliases: |
Use in source · Signal_Reconstruction_Traditional.tex, line 93
t_0^2πf(x)e^-inx dx. Signal identities inferred from Fourier coefficients are identities of L^1 classes. For a piecewise-constant edge we select the right-continuous representative. This convention fixes the values at jumps, which Fourier integrals alone cannot determine. When an L^1 norm occurs it is the unnormalized integral over a period. On the unit circle we use chord distance |z-w| and circular angular distance d_ T(τ,σ).
Exact mathematical passage
|z-w|
|^2Use in source · 10 passages
Search aliases: |^2
Use in source · Signal_Reconstruction_Traditional.tex, line 359
smallDomain and hypotheses. Periodic integrable signals with one known edge shape. Let S_K be the trigonometric polynomials of degree at most K. Let m∈ L^1( T) and a finite nonempty set F⊂{n:|n|>K} satisfy D_F=∑_n∈ F|widehat m(n)|^2>0. [SG-09 Out-of-band matched identification]
Exact mathematical passage
D_F=\sum_{n\in F}|\widehat m(n)|^2>0
\cdotUse in source · 1 passage
Search aliases: · cdot
Use in source · Signal_Reconstruction_Traditional.tex, line 1325
SG-21 writes both packets as translated sawtooth sums. Subtract in the form j-widehat j= ∑_ℓ(a_ℓ-widehat a_ℓ)G_T(·-τ_ℓ) +∑_ℓwidehat a_ℓ [G_T(·-τ_ℓ)-G_T(·-widehatτ_ℓ)]. Direct integration gives G_T _1=T/4. For a shift by d∈[0,T/2], on an interval of length T-d the difference has magnitude d/T, and on an interval of length d it has magnitude 1-d/T. Hence its L^1 norm is (T-d)d/T+d(1-d/T)=2d(1-d/T). Apply the triangle inequality term by term. Periodic translation invariance makes the calculation de…
Exact mathematical passage
j-\widehat j= \sum_\ell(a_\ell-\widehat a_\ell)G_T(\cdot-\tau_\ell) +\sum_\ell\widehat a_\ell [G_T(\cdot-\tau_\ell)-G_T(\cdot-\widehat\tau_\ell)].
\timesUse in source · 4 passages
Search aliases: × times
Use in source · Signal_Reconstruction_Traditional.tex, line 238
Interleaving the N channels recovers x exactly. With h_ba[k]=h[Nk+b-a], the output channels satisfy y_b[q]=∑_a=0^N-1∑_t∈ Zh_ba[q-t]x_a[t]. Thus convolution becomes an N× N transfer-kernel system.
Exact mathematical passage
N\times N
\alphaUse in source · 17 passages
Search aliases: α alpha
Use in source · Signal_Reconstruction_Traditional.tex, line 523
Every vector α with α _0≤ s is uniquely determined by Aα if and only if ker A∩{h: h _0≤min(2s,d)}={0}. Equivalently every set of at most min(2s,d) columns is independent. With the convention spark(A)=d+1 when all d columns are independent, the equivalent condition is spark(A)>min(2s,d). For s=0 uniqueness is automatic.
Exact mathematical passage
\alpha
\alpha_0Use in source · 9 passages
Search aliases: α_0 alpha
Use in source · Signal_Reconstruction_Traditional.tex, line 556
smallDomain and hypotheses. The Euclidean model of SG-14. Assume s≥1, d≥1, and the injectivity condition of SG-14. Let y=s_0+Dα_0+η, s_0∈ S, α_0 _0≤ s, and Rη _2≤ϵ. Let widehatα be s-sparse with Awidehatα-Ry _2≤ϵ. [SG-15 Sparse noise stability and support thresholding]
Exact mathematical passage
y=s_0+D\alpha_0+\eta
\betaUse in source · 3 passages
Search aliases: β beta
Use in source · Signal_Reconstruction_Traditional.tex, line 541
The difference of two s-sparse vectors has support of size at most 2s and at most d. Thus the kernel condition implies uniqueness. Conversely, if a nonzero kernel vector h has support size at most 2s, partition that support into two sets of at most s indices. Let α equal h on the first set and zero elsewhere; let β equal -h on the second set. Then h=α-β, and Aα=Aβ contradicts uniqueness. Column dependence is exactly the existence of a nonzero kernel vector supported on the selected columns. The …
Exact mathematical passage
\beta
\GammaUse in source · 3 passages
Search aliases: Γ Gamma
Use in source · Signal_Reconstruction_Traditional.tex, line 606
For an unrestricted scalar amplitude the least-squares profile is widehat a(q)=frac⟨ Rq,Ry⟩ Rq _2^2, J(q)= Ry-widehat a(q)Rq _2^2. The true mask has widehat a(q_*)=a_* and J(q_*)=0. It is the unique zero-residual positive-amplitude mask exactly when no other residualized mask lies on its positive ray. Define the closed rays C_q={aRq:a≥0}. If Γ=inf_q≠ q_*dist(Ry,C_q)>0, with the convention inf∅=+∞, then every perturbed residualized datum Ry+e with e _2<Γ/2 selects q_* uniquely by nearest-ray dist…
Exact mathematical passage
\Gamma=\inf_{q\ne q_*}\operatorname{dist}(Ry,C_q)>0,
\gammaUse in source · 13 passages
Search aliases: γ gamma
Use in source · Signal_Reconstruction_Traditional.tex, line 560
Define γ=min_substack h _2=1 h _0≤min(2s,d) Ah _2>0. Then widehatα-α_0 _2≤2ϵ/γ, D(widehatα-α_0) _2 ≤2 D _2ϵ/γ. If every nonzero true coefficient has magnitude greater than 4ϵ/γ, thresholding at 2ϵ/γ with a strict greater-than rule recovers the true support.
Exact mathematical passage
\gamma=\min_{\substack{\|h\|_2=1\\\|h\|_0\le\min(2s,d)}}\|Ah\|_2>0.
\gamma_{\ell,0,d}Use in source · 1 passage
Search aliases: γ_ℓ,0,d gamma ell
Use in source · Signal_Reconstruction_Traditional.tex, line 1432
∏_ℓ=1^r(E-z_ℓ)^d+1m=0. For offset v≥0, writing a decimated channel as m_(q_0+q)N+v=∑_ℓ u_ℓ^q ∑_h=0^dγ_ℓ,v,hq^h, u_ℓ=z_ℓ^N, the highest-degree coefficients satisfy γ_ℓ,1,d/γ_ℓ,0,d=z_ℓ. This ratio applies to separated, matched confluent terms with nonzero leading coefficient. The displayed annihilator is not asserted to have minimal order when leading coefficients vanish.
Exact mathematical passage
\gamma_{\ell,1,d}/\gamma_{\ell,0,d}=z_\ell
\gamma_{\ell,1,d}Use in source · 1 passage
Search aliases: γ_ℓ,1,d gamma ell
Use in source · Signal_Reconstruction_Traditional.tex, line 1432
∏_ℓ=1^r(E-z_ℓ)^d+1m=0. For offset v≥0, writing a decimated channel as m_(q_0+q)N+v=∑_ℓ u_ℓ^q ∑_h=0^dγ_ℓ,v,hq^h, u_ℓ=z_ℓ^N, the highest-degree coefficients satisfy γ_ℓ,1,d/γ_ℓ,0,d=z_ℓ. This ratio applies to separated, matched confluent terms with nonzero leading coefficient. The displayed annihilator is not asserted to have minimal order when leading coefficients vanish.
Exact mathematical passage
\gamma_{\ell,1,d}/\gamma_{\ell,0,d}=z_\ell
\gamma_{\ell,v,d}Use in source · 1 passage
Search aliases: γ_ℓ,v,d gamma ell
Use in source · Signal_Reconstruction_Traditional.tex, line 1447
lowers a nonconstant polynomial's degree by one, and annihilates a constant. After d+1 differences the term vanishes. The shift factors commute, so the product annihilates every term in the sum. If P_ℓ(n) has leading coefficient c_ℓ≠0 at degree d, substitution of (q_0+q)N+v has leading q coefficient c_ℓ N^d. The remaining exponential prefactor is z_ℓ^q_0N+v; hence γ_ℓ,v,d=c_ℓ N^dz_ℓ^q_0N+v and the ratio is z_ℓ. Repeated roots require confluent recovery and stability arguments; the simple-root di…
Exact mathematical passage
\gamma_{\ell,v,d}=c_\ell N^dz_\ell^{q_0N+v}
\gamma_{\ell,v,h}Use in source · 1 passage
Search aliases: γ_ℓ,v,h gamma ell
Use in source · Signal_Reconstruction_Traditional.tex, line 1426
∏_ℓ=1^r(E-z_ℓ)^d+1m=0. For offset v≥0, writing a decimated channel as m_(q_0+q)N+v=∑_ℓ u_ℓ^q ∑_h=0^dγ_ℓ,v,hq^h, u_ℓ=z_ℓ^N, the highest-degree coefficients satisfy γ_ℓ,1,d/γ_ℓ,0,d=z_ℓ. This ratio applies to separated, matched confluent terms with nonzero leading coefficient. The displayed annihilator is not asserted to have minimal order when leading coefficients vanish.
Exact mathematical passage
m_{(q_0+q)N+v}=\sum_\ell u_\ell^q \sum_{h=0}^d\gamma_{\ell,v,h}q^h, \qquad u_\ell=z_\ell^N,
\DeltaUse in source · 11 passages
Search aliases: Δ Delta
Use in source · Signal_Reconstruction_Traditional.tex, line 931
smallDomain and hypotheses. A square or overdetermined Vandermonde amplitude fit. Let V have full column rank, m=Va, widetilde V=V+Δ V, and widetilde m=m+e. Assume σ_min(V)> Δ V _2. Let widehat a minimize widetilde Vb-widetilde m _2. [SG-26 Amplitude stability with perturbed nodes]
Exact mathematical passage
\widetilde V=V+\Delta V
\deltaUse in source · 20 passages
Search aliases: δ delta
Use in source · Signal_Reconstruction_Traditional.tex, line 124
smallDomain and hypotheses. Vector spaces over a field. Let P_1,…,P_m∈End(X) satisfy P_aP_b=δ_abP_a and ∑_aP_a=I_X. Set X_a=P_aX. [SG-01 Complete packet analysis and synthesis]
Exact mathematical passage
P_aP_b=\delta_{ab}P_a
\delta_0Use in source · 1 passage
Search aliases: δ_0 delta
Use in source · Signal_Reconstruction_Traditional.tex, line 783
The sawtooth derivative is DG_T=δ_0-T^-1dx: its slope is -1/T away from the endpoint and its endpoint jump is one. Summing translated copies gives the prescribed Dirac masses and a constant-density term -T^-1∑ a_ℓ, which vanishes. Consequently j is constant between the jump points. Each translated sawtooth has mean zero, so j does too. Any other packet with the same jumps has distributional derivative difference zero. Since both are piecewise constant, their difference is the same constant on ev…
Exact mathematical passage
DG_T=\delta_0-T^{-1}dx
\delta_{ab}Use in source · 2 passages
Search aliases: δ_ab delta
Use in source · Signal_Reconstruction_Traditional.tex, line 124
smallDomain and hypotheses. Vector spaces over a field. Let P_1,…,P_m∈End(X) satisfy P_aP_b=δ_abP_a and ∑_aP_a=I_X. Set X_a=P_aX. [SG-01 Complete packet analysis and synthesis]
Exact mathematical passage
P_aP_b=\delta_{ab}P_a
\delta_{\tau_\ell}Use in source · 3 passages
Search aliases: δ_τ_ℓ delta tau ell
Use in source · Signal_Reconstruction_Traditional.tex, line 664
Let U_K,L consist of real periodic signals f=s+j, where s∈ S_K and j is right-continuous, piecewise constant, mean zero, and has at most L distinct active jumps. An active jump has nonzero amplitude. For an actual jump count r, write Dj=∑_ℓ=1^r a_ℓδ_τ_ℓ, z_ℓ=e^-iτ_ℓ, ∑_ℓ a_ℓ=0. The sum-zero condition is required by periodicity. A single nonzero active jump cannot occur, so the nonconstant real class starts at r=2. The case r=0 is allowed and will be handled explicitly. The smooth-edge representa…
Exact mathematical passage
Dj=\sum_{\ell=1}^r a_\ell\delta_{\tau_\ell},\quad z_\ell=e^{-i\tau_\ell},\quad\sum_\ell a_\ell=0.
\epsilonUse in source · 29 passages
Search aliases: ϵ epsilon
Use in source · Signal_Reconstruction_Traditional.tex, line 556
smallDomain and hypotheses. The Euclidean model of SG-14. Assume s≥1, d≥1, and the injectivity condition of SG-14. Let y=s_0+Dα_0+η, s_0∈ S, α_0 _0≤ s, and Rη _2≤ϵ. Let widehatα be s-sparse with Awidehatα-Ry _2≤ϵ. [SG-15 Sparse noise stability and support thresholding]
Exact mathematical passage
\|R\eta\|_2\le\epsilon
\epsilon^2Use in source · 2 passages
Search aliases: ϵ^2 epsilon
Use in source · Signal_Reconstruction_Traditional.tex, line 1072
smallDomain and hypotheses. A noisy progression as in SG-29. Let c_r=bz^r+e_r for 0≤ r≤ R, with R≥1, b≠0, |z|=1, and |e_r|≤ϵ. Put A=R|b|^2 and Δ=R(2|b|ϵ+ϵ^2). Assume Δ<A. [SG-30 Deterministic offset-phase perturbation]
Exact mathematical passage
\Delta=R(2|b|\epsilon+\epsilon^2)
\epsilon_nUse in source · 3 passages
Search aliases: ϵ_n epsilon
Use in source · Signal_Reconstruction_Traditional.tex, line 840
smallDomain and hypotheses. A rectangular Hankel matrix of derivative moments. Let measured Fourier coefficients have errors η_n and moment errors ϵ_n=2π inη_n. Let E=(ϵ_n_0+i+j) have h≥1 rows and w≥1 columns. [SG-23 Hankel perturbation from Fourier noise]
Exact mathematical passage
\epsilon_n=2\pi in\eta_n
\epsilon_{n_0+i+j}Use in source · 2 passages
Search aliases: ϵ_n_0+i+j epsilon
Use in source · Signal_Reconstruction_Traditional.tex, line 840
smallDomain and hypotheses. A rectangular Hankel matrix of derivative moments. Let measured Fourier coefficients have errors η_n and moment errors ϵ_n=2π inη_n. Let E=(ϵ_n_0+i+j) have h≥1 rows and w≥1 columns. [SG-23 Hankel perturbation from Fourier noise]
Exact mathematical passage
E=(\epsilon_{n_0+i+j})
\zetaUse in source · 5 passages
Search aliases: ζ zeta
Use in source · Signal_Reconstruction_Traditional.tex, line 1107
smallDomain and hypotheses. Finite complex offset data. Let R≥1, c_0≠0, and |ζ|=1. Define ν(c,ζ)=(∑_r=0^R-1|c_r+1-ζ c_r|^2)^1/2/(∑_r=0^R-1|c_r|^2)^1/2. [SG-31 A zero offset residual tests a progression]
Exact mathematical passage
|\zeta|=1
\zeta^rUse in source · 1 passage
Search aliases: ζ^r zeta
Use in source · Signal_Reconstruction_Traditional.tex, line 1110
Then ν(c,ζ)=0 if and only if c_r=c_0ζ^r for 0≤ r≤ R.
Exact mathematical passage
c_r=c_0\zeta^r
\etaUse in source · 17 passages
Search aliases: η eta
Use in source · Signal_Reconstruction_Traditional.tex, line 367
On X= S_K+span{m}, the functional ℓ_F(g)=D_F^-1∑_n∈ Foverlinewidehat m(n)widehat g(n) satisfies ℓ_F(s+am)=a. The decomposition is direct. If the measured coefficients have errors η_n, then |widetilde a-a|≤ D_F^-1/2(∑_n∈ F|η_n|^2)^1/2.
Exact mathematical passage
\eta_n
\eta_nUse in source · 6 passages
Search aliases: η_n eta
Use in source · Signal_Reconstruction_Traditional.tex, line 367
On X= S_K+span{m}, the functional ℓ_F(g)=D_F^-1∑_n∈ Foverlinewidehat m(n)widehat g(n) satisfies ℓ_F(s+am)=a. The decomposition is direct. If the measured coefficients have errors η_n, then |widetilde a-a|≤ D_F^-1/2(∑_n∈ F|η_n|^2)^1/2.
Exact mathematical passage
\eta_n
\thetaUse in source · 5 passages
Search aliases: θ theta
Use in source · Signal_Reconstruction_Traditional.tex, line 594
smallDomain and hypotheses. Finite real dictionaries with positive amplitudes. Let B have full column rank, P be the orthogonal projector onto its range, R=I-P, and let Q be a finite mask library with Rq≠0 for every q. Suppose y=Bθ_*+a_*q_*, with a_*>0 and q_*∈ Q. [SG-16 Profiled one-packet discovery and its finite gap]
Exact mathematical passage
y=B\theta_*+a_*q_*
\theta_*Use in source · 1 passage
Search aliases: θ_* theta
Use in source · Signal_Reconstruction_Traditional.tex, line 594
smallDomain and hypotheses. Finite real dictionaries with positive amplitudes. Let B have full column rank, P be the orthogonal projector onto its range, R=I-P, and let Q be a finite mask library with Rq≠0 for every q. Suppose y=Bθ_*+a_*q_*, with a_*>0 and q_*∈ Q. [SG-16 Profiled one-packet discovery and its finite gap]
Exact mathematical passage
y=B\theta_*+a_*q_*
\lambdaUse in source · 15 passages
Search aliases: λ lambda
Use in source · Signal_Reconstruction_Traditional.tex, line 437
smallDomain and hypotheses. The interval packet j=1_[-a,a) on T. Assume 0<a<π, let λ_M=M+1/2, and use the right edge coordinate x_M(z)=a-z/λ_M. [SG-12 The scaled interval-edge Gibbs profile]
Exact mathematical passage
\lambda_M=M+1/2
\lambda_MUse in source · 5 passages
Search aliases: λ_M lambda
Use in source · Signal_Reconstruction_Traditional.tex, line 437
smallDomain and hypotheses. The interval packet j=1_[-a,a) on T. Assume 0<a<π, let λ_M=M+1/2, and use the right edge coordinate x_M(z)=a-z/λ_M. [SG-12 The scaled interval-edge Gibbs profile]
Exact mathematical passage
\lambda_M=M+1/2
\nuUse in source · 2 passages
Search aliases: ν nu
Use in source · Signal_Reconstruction_Traditional.tex, line 1107
smallDomain and hypotheses. Finite complex offset data. Let R≥1, c_0≠0, and |ζ|=1. Define ν(c,ζ)=(∑_r=0^R-1|c_r+1-ζ c_r|^2)^1/2/(∑_r=0^R-1|c_r|^2)^1/2. [SG-31 A zero offset residual tests a progression]
Exact mathematical passage
\nu(c,\zeta)=\bigl(\sum_{r=0}^{R-1}|c_{r+1}-\zeta c_r|^2\bigr)^{1/2}/\bigl(\sum_{r=0}^{R-1}|c_r|^2\bigr)^{1/2}
\piUse in source · 46 passages
Search aliases: π pi
Use in source · Signal_Reconstruction_Traditional.tex, line 84
The default periodic domain is T= R/(2π Z), with widehat f(n)=frac12π∫_0^2πf(x)e^-inx dx. Signal identities inferred from Fourier coefficients are identities of L^1 classes. For a piecewise-constant edge we select the right-continuous representative. This convention fixes the values at jumps, which Fourier integrals alone cannot determine. When an L^1 norm occurs it is the unnormalized integral over a period. On the unit circle we use chord distance |z-w| and circular angular distance d_ T(τ,σ).
Exact mathematical passage
\mathbb T=\mathbb R/(2\pi\mathbb Z)
\rhoUse in source · 27 passages
Search aliases: ρ rho
Use in source · Signal_Reconstruction_Traditional.tex, line 180
smallDomain and hypotheses. Finite complete packet systems. Let (Q_ρ) refine (P_a), with a partition of fine labels into F_a and P_a=∑_ρ∈ F_aQ_ρ. [SG-03 Refinement preserves represented operators]
Exact mathematical passage
(Q_\rho)
\rho'Use in source · 2 passages
Search aliases: ρ' rho
Use in source · Signal_Reconstruction_Traditional.tex, line 1147
The N candidates lie on the unit circle at angles spaced by 2π/N. The smallest distance between distinct candidates is 2sin(π/N). The triangle inequality gives |widetilde z-ρ|≤|widetilde z-z|+|z-ρ|<sin(π/N). For any other candidate ρ', |widetilde z-ρ'|≥|ρ'-ρ|-|ρ-widetilde z| >sin(π/N). Thus the nearest candidate is unique. The same spacing argument places every wrong root farther from the true node than the associated root under the given margin. The oracle and data-based choices therefore coinc…
Exact mathematical passage
\rho'
\sigmaUse in source · 15 passages
Search aliases: σ sigma
Use in source · Signal_Reconstruction_Traditional.tex, line 93
identities inferred from Fourier coefficients are identities of L^1 classes. For a piecewise-constant edge we select the right-continuous representative. This convention fixes the values at jumps, which Fourier integrals alone cannot determine. When an L^1 norm occurs it is the unnormalized integral over a period. On the unit circle we use chord distance |z-w| and circular angular distance d_ T(τ,σ).
Exact mathematical passage
d_{\mathbb T}(\tau,\sigma)
\sigma_jUse in source · 4 passages
Search aliases: σ_j sigma
Use in source · Signal_Reconstruction_Traditional.tex, line 1013
The singular-value perturbation inequality [source reference] gives |σ_j(H)-σ_j(H_0)|≤ E _2 for 1≤ j≤ k. It holds equally over the complex field: in the min--max characterization σ_j(A)=min_dim V=w-j+1max_x∈ V, x =1 Ax . the triangle inequality bounds each restricted maximum for A+E by that for A plus E _2. Minimize and then interchange the two matrices. For j≤ r this yields σ_j(H)≥σ_r(H_0)-ϵ>t. For j>r, when such indices exist, it gives σ_j(H)≤ϵ<t. For rank zero the same upper bound applies to …
Exact mathematical passage
|\sigma_j(H)-\sigma_j(H_0)|\le\|E\|_2
\sigma_{k+1}Use in source · 1 passage
Search aliases: σ_k+1 sigma
Use in source · Signal_Reconstruction_Traditional.tex, line 1007
Exactly r singular values of H exceed t. If r=0, a supplied threshold t> E _2 selects zero. If r=k, all k singular values exceed the threshold and no nonexistent σ_k+1 is used.
Exact mathematical passage
\sigma_{k+1}
\sigma_MUse in source · 2 passages
Search aliases: σ_M sigma
Use in source · Signal_Reconstruction_Traditional.tex, line 488
smallDomain and hypotheses. Periodic bounded real signals. Let σ_M=(M+1)^-1∑_k=0^MS_k be the Fejer mean. Suppose 0≤ j≤1 almost everywhere. [SG-13 Positive Fourier averaging preserves range]
Exact mathematical passage
\sigma_M=(M+1)^{-1}\sum_{k=0}^MS_k
\sigma_{\min}Use in source · 4 passages
Search aliases: σ_min sigma min
Use in source · Signal_Reconstruction_Traditional.tex, line 931
smallDomain and hypotheses. A square or overdetermined Vandermonde amplitude fit. Let V have full column rank, m=Va, widetilde V=V+Δ V, and widetilde m=m+e. Assume σ_min(V)> Δ V _2. Let widehat a minimize widetilde Vb-widetilde m _2. [SG-26 Amplitude stability with perturbed nodes]
Exact mathematical passage
\sigma_{\min}(V)>\|\Delta V\|_2
\sigma_rUse in source · 3 passages
Search aliases: σ_r sigma
Use in source · Signal_Reconstruction_Traditional.tex, line 1002
smallDomain and hypotheses. Finite rectangular complex matrices. Let H=H_0+E, with H_0 of rank r≥1, k=min(h,w)≥ r, and ϵ= E _2<σ_r(H_0)/2. Supply a threshold t with ϵ<t<σ_r(H_0)-ϵ. [SG-28 Order recovery with a supplied singular threshold]
Exact mathematical passage
\epsilon=\|E\|_2<\sigma_r(H_0)/2
\tauUse in source · 15 passages
Search aliases: τ tau
Use in source · Signal_Reconstruction_Traditional.tex, line 93
identities inferred from Fourier coefficients are identities of L^1 classes. For a piecewise-constant edge we select the right-continuous representative. This convention fixes the values at jumps, which Fourier integrals alone cannot determine. When an L^1 norm occurs it is the unnormalized integral over a period. On the unit circle we use chord distance |z-w| and circular angular distance d_ T(τ,σ).
Exact mathematical passage
d_{\mathbb T}(\tau,\sigma)
\tau_\ellUse in source · 14 passages
Search aliases: τ_ℓ tau ell
Use in source · Signal_Reconstruction_Traditional.tex, line 664
Let U_K,L consist of real periodic signals f=s+j, where s∈ S_K and j is right-continuous, piecewise constant, mean zero, and has at most L distinct active jumps. An active jump has nonzero amplitude. For an actual jump count r, write Dj=∑_ℓ=1^r a_ℓδ_τ_ℓ, z_ℓ=e^-iτ_ℓ, ∑_ℓ a_ℓ=0. The sum-zero condition is required by periodicity. A single nonzero active jump cannot occur, so the nonconstant real class starts at r=2. The case r=0 is allowed and will be handled explicitly. The smooth-edge representa…
Exact mathematical passage
Dj=\sum_{\ell=1}^r a_\ell\delta_{\tau_\ell},\quad z_\ell=e^{-i\tau_\ell},\quad\sum_\ell a_\ell=0.
\PhiUse in source · 2 passages
Search aliases: Φ Phi
Use in source · Signal_Reconstruction_Traditional.tex, line 1050
S(c)=R|b|^2z≠0, Φ(c):=S(c)/|S(c)|=z. Two channels already give z=c_1/c_0 exactly. More channels provide repeated adjacent relations, without an unconditional deterministic averaging gain.
Exact mathematical passage
S(c)=R|b|^2z\ne0,\qquad\Phi(c):=S(c)/|S(c)|=z.
\varphiUse in source · 2 passages
Search aliases: φ varphi
Use in source · Signal_Reconstruction_Traditional.tex, line 688
For a smooth periodic test function φ, integrate by parts over each constant segment of j. Boundary terms telescope at each jump to a_ℓφ(τ_ℓ), which is exactly the displayed distribution. Evaluate the distribution on e^-inx: ⟨ Dj,e^-inx⟩ =-∫ j(x)(-in)e^-inx dx =2π inwidehat j(n). The Dirac sum gives ∑_ℓ a_ℓ e^-inτ_ℓ. For n>K, the smooth polynomial contributes zero. Without bandlimiting it contributes the stated defect. At n=0 the derivative moment is zero, in agreement with ∑ a_ℓ=0; it cannot de…
Exact mathematical passage
\varphi
\daggerUse in source · 2 passages
Search aliases: † dagger
Use in source · Signal_Reconstruction_Traditional.tex, line 949
The minimum singular value of widetilde V is at least σ_min(V)- Δ V _2>0. Thus its least-squares inverse exists. Since widetilde V a=m+Δ V a, widehat a-a=widetilde V^†(e-Δ V a). Apply widetilde V^† _2=1/σ_min(widetilde V). The power bound follows by factoring widehat z^n-z^n=(widehat z-z)∑_k=0^n-1widehat z^n-1-kz^k; each summand has modulus one. The statement assumes aligned nodes and unit normalization. Arbitrary unnormalized roots require extra modulus factors.
Exact mathematical passage
\widehat a-a=\widetilde V^\dagger(e-\Delta V a).
\ldotsUse in source · 8 passages
Search aliases: … ldots
Use in source · Signal_Reconstruction_Traditional.tex, line 124
smallDomain and hypotheses. Vector spaces over a field. Let P_1,…,P_m∈End(X) satisfy P_aP_b=δ_abP_a and ∑_aP_a=I_X. Set X_a=P_aX. [SG-01 Complete packet analysis and synthesis]
Exact mathematical passage
P_1,\ldots,P_m\in\operatorname{End}(X)
\ellUse in source · 76 passages
Search aliases: ℓ ell
Use in source · Signal_Reconstruction_Traditional.tex, line 227
smallDomain and hypotheses. Absolutely summable sequences on Z. Fix an integer N≥1. Let h,x∈ℓ^1( Z) and y=h*x. Define x_a[q]=x[Nq+a] and y_b[q]=y[Nq+b], for 0≤ a,b<N. [SG-05 Polyphase analysis and convolution kernels]
Exact mathematical passage
h,x\in\ell^1(\mathbb Z)
\ell^1Use in source · 2 passages
Search aliases: ℓ^1 ell
Use in source · Signal_Reconstruction_Traditional.tex, line 227
smallDomain and hypotheses. Absolutely summable sequences on Z. Fix an integer N≥1. Let h,x∈ℓ^1( Z) and y=h*x. Define x_a[q]=x[Nq+a] and y_b[q]=y[Nq+b], for 0≤ a,b<N. [SG-05 Polyphase analysis and convolution kernels]
Exact mathematical passage
h,x\in\ell^1(\mathbb Z)
\ell_FUse in source · 3 passages
Search aliases: ℓ_F ell
Use in source · Signal_Reconstruction_Traditional.tex, line 363
On X= S_K+span{m}, the functional ℓ_F(g)=D_F^-1∑_n∈ Foverlinewidehat m(n)widehat g(n) satisfies ℓ_F(s+am)=a. The decomposition is direct. If the measured coefficients have errors η_n, then |widetilde a-a|≤ D_F^-1/2(∑_n∈ F|η_n|^2)^1/2.
Exact mathematical passage
\ell_F(g)=D_F^{-1}\sum_{n\in F}\overline{\widehat m(n)}\widehat g(n)
\toUse in source · 11 passages
Search aliases: → to
Use in source · Signal_Reconstruction_Traditional.tex, line 149
smallDomain and hypotheses. The packet space of SG-01. Let S,T∈End(X). Set T_ba=P_bTP_a, viewed as a map X_a→ X_b. [SG-02 Transfer algebra and packet paths]
Exact mathematical passage
X_a\to X_b
\mapstoUse in source · 2 passages
Search aliases: ↦ mapsto
Use in source · Signal_Reconstruction_Traditional.tex, line 170
Insert I=∑_aP_a on both sides of T to obtain reconstruction. Insert that identity between S and T to obtain the product formula. Equivalently, the block map is T↦ A T S; SG-01 shows that its inverse is B↦ S B A and that it preserves multiplication. The path formula follows by inserting the same identity between every successive pair of operators. These are finite sums, so no convergence hypothesis is hidden in the insertion.
Exact mathematical passage
T\mapsto\mathcal A T\mathcal S
\varnothingUse in source · 1 passage
Search aliases: ∅ varnothing
Use in source · Signal_Reconstruction_Traditional.tex, line 609
ehat a(q)=frac⟨ Rq,Ry⟩ Rq _2^2, J(q)= Ry-widehat a(q)Rq _2^2. The true mask has widehat a(q_*)=a_* and J(q_*)=0. It is the unique zero-residual positive-amplitude mask exactly when no other residualized mask lies on its positive ray. Define the closed rays C_q={aRq:a≥0}. If Γ=inf_q≠ q_*dist(Ry,C_q)>0, with the convention inf∅=+∞, then every perturbed residualized datum Ry+e with e _2<Γ/2 selects q_* uniquely by nearest-ray distance.
Exact mathematical passage
\inf\varnothing=+\infty
\inUse in source · 34 passages
Search aliases: ∈ in
Use in source · Signal_Reconstruction_Traditional.tex, line 124
smallDomain and hypotheses. Vector spaces over a field. Let P_1,…,P_m∈End(X) satisfy P_aP_b=δ_abP_a and ∑_aP_a=I_X. Set X_a=P_aX. [SG-01 Complete packet analysis and synthesis]
Exact mathematical passage
P_1,\ldots,P_m\in\operatorname{End}(X)
\prodUse in source · 7 passages
Search aliases: ∏ prod
Use in source · Signal_Reconstruction_Traditional.tex, line 262
Exactly d unit-modulus nodes satisfy the same data. Hence the physical node is unique precisely when d=1. If integers c,b_1,…,b_k obey Nc+∑_jr_jb_j=d, then u^c∏_jw_j^b_j=z^d. For d=1 this is an explicit exact de-aliasing formula.
Exact mathematical passage
u^c\prod_jw_j^{b_j}=z^d.
\prod_jUse in source · 1 passage
Search aliases: ∏_j prod
Use in source · Signal_Reconstruction_Traditional.tex, line 262
Exactly d unit-modulus nodes satisfy the same data. Hence the physical node is unique precisely when d=1. If integers c,b_1,…,b_k obey Nc+∑_jr_jb_j=d, then u^c∏_jw_j^b_j=z^d. For d=1 this is an explicit exact de-aliasing formula.
Exact mathematical passage
u^c\prod_jw_j^{b_j}=z^d.
\prod_{j\ne\ell}Use in source · 1 passage
Search aliases: ∏_j≠ℓ prod ne ell
Use in source · Signal_Reconstruction_Traditional.tex, line 904
If for every ℓ sup_|z-z_ℓ|=ρ|E_Q(z)| <min_|z-z_ℓ|=ρ|Q(z)|, then widetilde Q has exactly one root, counted with multiplicity, in each disk |z-z_ℓ|<ρ. All its roots are thereby matched with error less than ρ. A sufficient explicit bound on each circle is ∑_k=0^r-1|widetilde q_k-q_k|(|z_ℓ|+ρ)^k <ρ∏_j≠ℓ(|z_ℓ-z_j|-ρ).
Exact mathematical passage
\sum_{k=0}^{r-1}|\widetilde q_k-q_k|(|z_\ell|+\rho)^k <\rho\prod_{j\ne\ell}(|z_\ell-z_j|-\rho).
\prod_\ellUse in source · 1 passage
Search aliases: ∏_ℓ prod ell
Use in source · Signal_Reconstruction_Traditional.tex, line 1400
The stacked system uniquely recovers the common annihilator Q(z)=∏_ℓ(z-z_ℓ). Its roots recover the common locations. With r consecutive moments per channel, a channelwise Vandermonde solve recovers every amplitude, including amplitudes that vanish in that channel.
Exact mathematical passage
Q(z)=\prod_\ell(z-z_\ell)
\prod_{\ell<k}Use in source · 1 passage
Search aliases: ∏_ℓ<k prod ell
Use in source · Signal_Reconstruction_Traditional.tex, line 722
Expand a matrix entry to obtain the factorization. The leading r× r Vandermonde submatrix has determinant ∏_ℓ<k(z_k-z_ℓ), which is nonzero. Thus both rectangular Vandermonde factors have full column rank, and their transposes have full row rank. The diagonal factor is invertible. Its product with V_w^T is surjective onto C^r, and subsequent multiplication by the injective V_h has image dimension r. For an empty exponential sum every entry is zero and the rank is zero.
Exact mathematical passage
\prod_{\ell<k}(z_k-z_\ell)
\prod_{\ell=1}^rUse in source · 3 passages
Search aliases: ∏_ℓ=1^r prod ell
Use in source · Signal_Reconstruction_Traditional.tex, line 737
The unique monic annihilator Q(z)=∏_ℓ=1^r(z-z_ℓ)=∑_k=0^r-1q_kz^k+z^r is recovered from H_r,rq=-v, v=(m_n_0+r+i)_i=0^r-1. Its roots are exactly the nodes. Once the nodes are known, r consecutive moments determine the coefficients uniquely by a Vandermonde solve. For unit-modulus nodes the corresponding locations are τ_ℓ=-arg z_ℓpmod2π.
Exact mathematical passage
Q(z)=\prod_{\ell=1}^r(z-z_\ell)=\sum_{k=0}^{r-1}q_kz^k+z^r
\sumUse in source · 47 passages
Search aliases: ∑ sum
Use in source · Signal_Reconstruction_Traditional.tex, line 124
smallDomain and hypotheses. Vector spaces over a field. Let P_1,…,P_m∈End(X) satisfy P_aP_b=δ_abP_a and ∑_aP_a=I_X. Set X_a=P_aX. [SG-01 Complete packet analysis and synthesis]
Exact mathematical passage
\sum_aP_a=I_X
\sum_aUse in source · 7 passages
Search aliases: ∑_a sum
Use in source · Signal_Reconstruction_Traditional.tex, line 124
smallDomain and hypotheses. Vector spaces over a field. Let P_1,…,P_m∈End(X) satisfy P_aP_b=δ_abP_a and ∑_aP_a=I_X. Set X_a=P_aX. [SG-01 Complete packet analysis and synthesis]
Exact mathematical passage
\sum_aP_a=I_X
\sum_{a,b}Use in source · 1 passage
Search aliases: ∑_a,b sum
Use in source · Signal_Reconstruction_Traditional.tex, line 152
T=∑_a,bP_bTP_a, (ST)_ca=∑_bS_cbT_ba. The block representation is an algebra isomorphism to End(⊕_aX_a), with multiplication by contraction of intermediate packets. For every integer k≥1, iterating the product gives P_a_kT^kP_a_0 =∑_a_1,…,a_k-1 (P_a_kTP_a_k-1)⋯(P_a_1TP_a_0).
Exact mathematical passage
T=\sum_{a,b}P_bTP_a,\qquad (ST)_{ca}=\sum_bS_{cb}T_{ba}.
\sum_{a=0}^{N-1}Use in source · 1 passage
Search aliases: ∑_a=0^N-1 sum
Use in source · Signal_Reconstruction_Traditional.tex, line 235
Interleaving the N channels recovers x exactly. With h_ba[k]=h[Nk+b-a], the output channels satisfy y_b[q]=∑_a=0^N-1∑_t∈ Zh_ba[q-t]x_a[t]. Thus convolution becomes an N× N transfer-kernel system.
Exact mathematical passage
y_b[q]=\sum_{a=0}^{N-1}\sum_{t\in\mathbb Z}h_{ba}[q-t]x_a[t].
\sum_{a_1,\ldots,a_{k-1}}Use in source · 1 passage
Search aliases: ∑_a_1,…,a_k-1 sum ldots
Use in source · Signal_Reconstruction_Traditional.tex, line 159
T=∑_a,bP_bTP_a, (ST)_ca=∑_bS_cbT_ba. The block representation is an algebra isomorphism to End(⊕_aX_a), with multiplication by contraction of intermediate packets. For every integer k≥1, iterating the product gives P_a_kT^kP_a_0 =∑_a_1,…,a_k-1 (P_a_kTP_a_k-1)⋯(P_a_1TP_a_0).
Exact mathematical passage
P_{a_k}T^kP_{a_0} =\sum_{a_1,\ldots,a_{k-1}} (P_{a_k}TP_{a_{k-1}})\cdots(P_{a_1}TP_{a_0}).
\sum_bUse in source · 1 passage
Search aliases: ∑_b sum
Use in source · Signal_Reconstruction_Traditional.tex, line 152
T=∑_a,bP_bTP_a, (ST)_ca=∑_bS_cbT_ba. The block representation is an algebra isomorphism to End(⊕_aX_a), with multiplication by contraction of intermediate packets. For every integer k≥1, iterating the product gives P_a_kT^kP_a_0 =∑_a_1,…,a_k-1 (P_a_kTP_a_k-1)⋯(P_a_1TP_a_0).
Exact mathematical passage
T=\sum_{a,b}P_bTP_a,\qquad (ST)_{ca}=\sum_bS_{cb}T_{ba}.
\sum_{h=0}^dUse in source · 1 passage
Search aliases: ∑_h=0^d sum
Use in source · Signal_Reconstruction_Traditional.tex, line 1426
∏_ℓ=1^r(E-z_ℓ)^d+1m=0. For offset v≥0, writing a decimated channel as m_(q_0+q)N+v=∑_ℓ u_ℓ^q ∑_h=0^dγ_ℓ,v,hq^h, u_ℓ=z_ℓ^N, the highest-degree coefficients satisfy γ_ℓ,1,d/γ_ℓ,0,d=z_ℓ. This ratio applies to separated, matched confluent terms with nonzero leading coefficient. The displayed annihilator is not asserted to have minimal order when leading coefficients vanish.
Exact mathematical passage
m_{(q_0+q)N+v}=\sum_\ell u_\ell^q \sum_{h=0}^d\gamma_{\ell,v,h}q^h, \qquad u_\ell=z_\ell^N,
\sum_jUse in source · 1 passage
Search aliases: ∑_j sum
Use in source · Signal_Reconstruction_Traditional.tex, line 261
Exactly d unit-modulus nodes satisfy the same data. Hence the physical node is unique precisely when d=1. If integers c,b_1,…,b_k obey Nc+∑_jr_jb_j=d, then u^c∏_jw_j^b_j=z^d. For d=1 this is an explicit exact de-aliasing formula.
Exact mathematical passage
Nc+\sum_jr_jb_j=d
\sum_{j=1}^mUse in source · 1 passage
Search aliases: ∑_j=1^m sum
Use in source · Signal_Reconstruction_Traditional.tex, line 1617
The first statement is the defining subtraction identity. Retaining r can carry every component omitted from q, so its data cost is not bounded by a small number of packet coefficients. For an image f∈ R^m, the standard pixel vectors satisfy f=∑_j=1^mf_je_j and form a complete finite basis. Those coefficients are all the pixels themselves. A program using the truth residual thus certifies faithful storage of its available input, rather than uniqueness under a smaller measurement map.
Exact mathematical passage
f=\sum_{j=1}^mf_je_j
\sum_{k=0}^MUse in source · 2 passages
Search aliases: ∑_k=0^M sum
Use in source · Signal_Reconstruction_Traditional.tex, line 488
smallDomain and hypotheses. Periodic bounded real signals. Let σ_M=(M+1)^-1∑_k=0^MS_k be the Fejer mean. Suppose 0≤ j≤1 almost everywhere. [SG-13 Positive Fourier averaging preserves range]
Exact mathematical passage
\sigma_M=(M+1)^{-1}\sum_{k=0}^MS_k
\sum_{k=0}^{n-1}Use in source · 1 passage
Search aliases: ∑_k=0^n-1 sum
Use in source · Signal_Reconstruction_Traditional.tex, line 954
The minimum singular value of widetilde V is at least σ_min(V)- Δ V _2>0. Thus its least-squares inverse exists. Since widetilde V a=m+Δ V a, widehat a-a=widetilde V^†(e-Δ V a). Apply widetilde V^† _2=1/σ_min(widetilde V). The power bound follows by factoring widehat z^n-z^n=(widehat z-z)∑_k=0^n-1widehat z^n-1-kz^k; each summand has modulus one. The statement assumes aligned nodes and unit normalization. Arbitrary unnormalized roots require extra modulus factors.
Exact mathematical passage
\widehat z^n-z^n=(\widehat z-z)\sum_{k=0}^{n-1}\widehat z^{n-1-k}z^k
\sum_{k=0}^rUse in source · 1 passage
Search aliases: ∑_k=0^r sum
Use in source · Signal_Reconstruction_Traditional.tex, line 753
For every available recurrence index, ∑_k=0^rq_km_n+k =∑_ℓ c_ℓ z_ℓ^n-n_0Q(z_ℓ)=0, q_r=1. SG-19 makes H_r,r invertible. Thus the system has one solution and the true polynomial supplies it. Factoring the resulting polynomial recovers the distinct nodes up to a permutation. The amplitude matrix for consecutive moments is a square Vandermonde multiplied by an invertible diagonal node power matrix, so it too is invertible. The argument convention identifies locations modulo a period, as required. Po…
Exact mathematical passage
\sum_{k=0}^rq_km_{n+k} =\sum_\ell c_\ell z_\ell^{n-n_0}Q(z_\ell)=0, \qquad q_r=1.
\sum_{k=0}^{r-1}Use in source · 2 passages
Search aliases: ∑_k=0^r-1 sum
Use in source · Signal_Reconstruction_Traditional.tex, line 737
The unique monic annihilator Q(z)=∏_ℓ=1^r(z-z_ℓ)=∑_k=0^r-1q_kz^k+z^r is recovered from H_r,rq=-v, v=(m_n_0+r+i)_i=0^r-1. Its roots are exactly the nodes. Once the nodes are known, r consecutive moments determine the coefficients uniquely by a Vandermonde solve. For unit-modulus nodes the corresponding locations are τ_ℓ=-arg z_ℓpmod2π.
Exact mathematical passage
Q(z)=\prod_{\ell=1}^r(z-z_\ell)=\sum_{k=0}^{r-1}q_kz^k+z^r
\sum_{n=1}^MUse in source · 1 passage
Search aliases: ∑_n=1^M sum
Use in source · Signal_Reconstruction_Traditional.tex, line 462
Direct integration gives widehat j(0)=a/π and widehat j(n)=sin(na)/(π n) for n≠0. The Dirichlet kernel D_M(t)=sin(λ_Mt)/sin(t/2) yields S_Mj(x)=frac12π∫_-a^aD_M(x-t) dt, (S_Mj)'(x)=fracD_M(x+a)-D_M(x-a)2π. At x=a, the coefficient formula gives S_Mj(a)=frac aπ+frac1π∑_n=1^Mfracsin(2na)n ⟶frac12. For completeness, the sine-series limit follows by summation by parts from bounded partial sums of e^2ina and by taking radial limits in ∑_n≥1r^ne^2ina/n=-log(1-re^2ia); its imaginary part tends to (π-2a)…
Exact mathematical passage
S_Mj(a)=\frac a\pi+\frac1\pi\sum_{n=1}^M\frac{\sin(2na)}n \longrightarrow\frac12.
\sum_{n>M}Use in source · 2 passages
Search aliases: ∑_n>M sum
Use in source · Signal_Reconstruction_Traditional.tex, line 430
Cancel j from the defining identity. Absolute summability of the Fourier coefficients gives an everywhere uniformly convergent series for s. The omitted positive and negative frequencies are bounded by 2C∑_n>Me^-ρ n, which is the displayed geometric series. No limiting trigonometric approximation of j occurs in this estimate.
Exact mathematical passage
2C\sum_{n>M}e^{-\rho n}
\sum_{n\in F}Use in source · 3 passages
Search aliases: ∑_n∈ F sum in
Use in source · Signal_Reconstruction_Traditional.tex, line 359
smallDomain and hypotheses. Periodic integrable signals with one known edge shape. Let S_K be the trigonometric polynomials of degree at most K. Let m∈ L^1( T) and a finite nonempty set F⊂{n:|n|>K} satisfy D_F=∑_n∈ F|widehat m(n)|^2>0. [SG-09 Out-of-band matched identification]
Exact mathematical passage
D_F=\sum_{n\in F}|\widehat m(n)|^2>0
\sum_{n\ge1}Use in source · 1 passage
Search aliases: ∑_n≥1 sum ge
Use in source · Signal_Reconstruction_Traditional.tex, line 468
_Mt)/sin(t/2) yields S_Mj(x)=frac12π∫_-a^aD_M(x-t) dt, (S_Mj)'(x)=fracD_M(x+a)-D_M(x-a)2π. At x=a, the coefficient formula gives S_Mj(a)=frac aπ+frac1π∑_n=1^Mfracsin(2na)n ⟶frac12. For completeness, the sine-series limit follows by summation by parts from bounded partial sums of e^2ina and by taking radial limits in ∑_n≥1r^ne^2ina/n=-log(1-re^2ia); its imaginary part tends to (π-2a)/2 for 0<2a<2π. Now differentiate the scaled function: frac ddzS_Mj(x_M(z))= fracD_M(-z/λ_M)-D_M(2a-z/λ_M)2πλ_M. Th…
Exact mathematical passage
\sum_{n\ge1}r^ne^{2ina}/n=-\log(1-re^{2ia})
\sum_{r=0}^{R-1}Use in source · 2 passages
Search aliases: ∑_r=0^R-1 sum
Use in source · Signal_Reconstruction_Traditional.tex, line 1047
smallDomain and hypotheses. One matched unit-circle coefficient progression. Let R≥1 be an integer counting adjacent products, and let c_r=bz^r for 0≤ r≤ R, with b≠0 and |z|=1. Define S(c)=∑_r=0^R-1overlinec_rc_r+1. [SG-29 Exact phase from adjacent offset products]
Exact mathematical passage
S(c)=\sum_{r=0}^{R-1}\overline{c_r}c_{r+1}
\sum_{t\in\mathbb Z}Use in source · 1 passage
Search aliases: ∑_t∈ Z sum in mathbb
Use in source · Signal_Reconstruction_Traditional.tex, line 235
Interleaving the N channels recovers x exactly. With h_ba[k]=h[Nk+b-a], the output channels satisfy y_b[q]=∑_a=0^N-1∑_t∈ Zh_ba[q-t]x_a[t]. Thus convolution becomes an N× N transfer-kernel system.
Exact mathematical passage
y_b[q]=\sum_{a=0}^{N-1}\sum_{t\in\mathbb Z}h_{ba}[q-t]x_a[t].
\sum_{\rho\in F_a}Use in source · 2 passages
Search aliases: ∑_ρ∈ F_a sum rho in
Use in source · Signal_Reconstruction_Traditional.tex, line 180
smallDomain and hypotheses. Finite complete packet systems. Let (Q_ρ) refine (P_a), with a partition of fine labels into F_a and P_a=∑_ρ∈ F_aQ_ρ. [SG-03 Refinement preserves represented operators]
Exact mathematical passage
P_a=\sum_{\rho\in F_a}Q_\rho
\sum_{\sigma\in F_b}Use in source · 1 passage
Search aliases: ∑_σ∈ F_b sum sigma in
Use in source · Signal_Reconstruction_Traditional.tex, line 183
P_bTP_a=∑_σ∈ F_b∑_ρ∈ F_aQ_σ TQ_ρ. The coordinate changes J= A_Q S_P and C= A_P S_Q satisfy CJ=I and JC=I on their packet coordinate spaces. Refinement is exact before any child packet is discarded.
Exact mathematical passage
P_bTP_a=\sum_{\sigma\in F_b}\sum_{\rho\in F_a}Q_\sigma TQ_\rho.
\sum_\ellUse in source · 14 passages
Search aliases: ∑_ℓ sum ell
Use in source · Signal_Reconstruction_Traditional.tex, line 664
Let U_K,L consist of real periodic signals f=s+j, where s∈ S_K and j is right-continuous, piecewise constant, mean zero, and has at most L distinct active jumps. An active jump has nonzero amplitude. For an actual jump count r, write Dj=∑_ℓ=1^r a_ℓδ_τ_ℓ, z_ℓ=e^-iτ_ℓ, ∑_ℓ a_ℓ=0. The sum-zero condition is required by periodicity. A single nonzero active jump cannot occur, so the nonconstant real class starts at r=2. The case r=0 is allowed and will be handled explicitly. The smooth-edge representa…
Exact mathematical passage
Dj=\sum_{\ell=1}^r a_\ell\delta_{\tau_\ell},\quad z_\ell=e^{-i\tau_\ell},\quad\sum_\ell a_\ell=0.
\sum_{\ell=1}^rUse in source · 5 passages
Search aliases: ∑_ℓ=1^r sum ell
Use in source · Signal_Reconstruction_Traditional.tex, line 664
Let U_K,L consist of real periodic signals f=s+j, where s∈ S_K and j is right-continuous, piecewise constant, mean zero, and has at most L distinct active jumps. An active jump has nonzero amplitude. For an actual jump count r, write Dj=∑_ℓ=1^r a_ℓδ_τ_ℓ, z_ℓ=e^-iτ_ℓ, ∑_ℓ a_ℓ=0. The sum-zero condition is required by periodicity. A single nonzero active jump cannot occur, so the nonconstant real class starts at r=2. The case r=0 is allowed and will be handled explicitly. The smooth-edge representa…
Exact mathematical passage
Dj=\sum_{\ell=1}^r a_\ell\delta_{\tau_\ell},\quad z_\ell=e^{-i\tau_\ell},\quad\sum_\ell a_\ell=0.
\setminusUse in source · 1 passage
Search aliases: ∖ setminus
Use in source · Signal_Reconstruction_Traditional.tex, line 636
aq-widetilde awidetilde q _2^2 · =|a-widetilde a|^2|suppq∩suppwidetilde q| · +|a|^2|suppq∖suppwidetilde q| +|widetilde a|^2|suppwidetilde q∖suppq|. For a=widetilde a=1 the squared norm is the cardinality of the symmetric support difference.
Exact mathematical passage
\|aq-\widetilde a\widetilde q\|_2^2 &=|a-\widetilde a|^2|\operatorname{supp}q\cap\operatorname{supp}\widetilde q|\\ &\quad+|a|^2|\operatorname{supp}q\setminus\operatorname{supp}\widetilde q| +|\widetilde a|^2|\operatorname{supp}\widetilde q\setminus\operatorname{supp}q|.
\inftyUse in source · 3 passages
Search aliases: ∞ infty
Use in source · Signal_Reconstruction_Traditional.tex, line 418
The reconstruction error is exactly f- R_Mf=s-S_Ms, and f- R_Mf _∞ ≤frac2Ce^-ρ(M+1)1-e^-ρ. The bound is uniform across the jumps of j.
Exact mathematical passage
\|f-\mathcal R_Mf\|_\infty \le\frac{2Ce^{-\rho(M+1)}}{1-e^{-\rho}}.
\capUse in source · 4 passages
Search aliases: ∩ cap
Use in source · Signal_Reconstruction_Traditional.tex, line 298
If Mf_0=Mf_1 but J(f_0)≠ J(f_1), no map R:Y→ J recovers the correct edge for both. If S∩ J≠{0}, the sum decomposition f=s+j is not unique. In particular, finitely many point samples on a periodic interval cannot recover every arbitrary interval packet.
Exact mathematical passage
\mathcal S\cap\mathcal J\ne\{0\}
\intUse in source · 5 passages
Search aliases: ∫ int
Use in source · Signal_Reconstruction_Traditional.tex, line 85
The default periodic domain is T= R/(2π Z), with widehat f(n)=frac12π∫_0^2πf(x)e^-inx dx. Signal identities inferred from Fourier coefficients are identities of L^1 classes. For a piecewise-constant edge we select the right-continuous representative. This convention fixes the values at jumps, which Fourier integrals alone cannot determine. When an L^1 norm occurs it is the unnormalized integral over a period. On the unit circle we use chord distance |z-w| and circular angular distance d_ T(τ,σ).
Exact mathematical passage
\widehat f(n)=\frac1{2\pi}\int_0^{2\pi}f(x)e^{-inx}\,dx.
\int_{-a}^{a}Use in source · 1 passage
Search aliases: ∫_-a^a int
Use in source · Signal_Reconstruction_Traditional.tex, line 457
Direct integration gives widehat j(0)=a/π and widehat j(n)=sin(na)/(π n) for n≠0. The Dirichlet kernel D_M(t)=sin(λ_Mt)/sin(t/2) yields S_Mj(x)=frac12π∫_-a^aD_M(x-t) dt, (S_Mj)'(x)=fracD_M(x+a)-D_M(x-a)2π. At x=a, the coefficient formula gives S_Mj(a)=frac aπ+frac1π∑_n=1^Mfracsin(2na)n ⟶frac12. For completeness, the sine-series limit follows by summation by parts from bounded partial sums of e^2ina and by taking radial limits in ∑_n≥1r^ne^2ina/n=-log(1-re^2ia); its imaginary part tends to (π-2a)…
Exact mathematical passage
S_Mj(x)=\frac1{2\pi}\int_{-a}^{a}D_M(x-t)\,dt, \qquad (S_Mj)'(x)=\frac{D_M(x+a)-D_M(x-a)}{2\pi}.
\int_0^{2\pi}Use in source · 2 passages
Search aliases: ∫_0^2π int pi
Use in source · Signal_Reconstruction_Traditional.tex, line 85
The default periodic domain is T= R/(2π Z), with widehat f(n)=frac12π∫_0^2πf(x)e^-inx dx. Signal identities inferred from Fourier coefficients are identities of L^1 classes. For a piecewise-constant edge we select the right-continuous representative. This convention fixes the values at jumps, which Fourier integrals alone cannot determine. When an L^1 norm occurs it is the unnormalized integral over a period. On the unit circle we use chord distance |z-w| and circular angular distance d_ T(τ,σ).
Exact mathematical passage
\widehat f(n)=\frac1{2\pi}\int_0^{2\pi}f(x)e^{-inx}\,dx.
\int_0^zUse in source · 1 passage
Search aliases: ∫_0^z int
Use in source · Signal_Reconstruction_Traditional.tex, line 441
Uniformly for z in compact real intervals, S_Mj(x_M(z))⟶frac12+frac1πSi(z), Si(z)=∫_0^zfracsin tt dt. The first positive maximum of the limiting profile is 1+G, where G=fracSi(π)π-frac12 =0.089489872236….
Exact mathematical passage
S_Mj(x_M(z))\longrightarrow\frac12+\frac1\pi\operatorname{Si}(z), \qquad \operatorname{Si}(z)=\int_0^z\frac{\sin t}{t}\,dt.
\neUse in source · 16 passages
Search aliases: ≠ ne
Use in source · Signal_Reconstruction_Traditional.tex, line 139
Completeness gives S A x=x. If x_a∈ X_a, then P_ax_a=x_a and P_bx_a=0 for b≠ a. Thus the bth component of A S(x_a)_a is P_b∑_ax_a=x_b. Both inverse identities follow. For uniqueness, apply P_b to any proposed sum of components. This proof uses pairwise annihilation, not orthogonality; the packet projections can be oblique.
Exact mathematical passage
b\ne a
\leUse in source · 71 passages
Search aliases: ≤ le
Use in source · Signal_Reconstruction_Traditional.tex, line 227
smallDomain and hypotheses. Absolutely summable sequences on Z. Fix an integer N≥1. Let h,x∈ℓ^1( Z) and y=h*x. Define x_a[q]=x[Nq+a] and y_b[q]=y[Nq+b], for 0≤ a,b<N. [SG-05 Polyphase analysis and convolution kernels]
Exact mathematical passage
0\le a,b<N
\geUse in source · 48 passages
Search aliases: ≥ ge
Use in source · Signal_Reconstruction_Traditional.tex, line 158
T=∑_a,bP_bTP_a, (ST)_ca=∑_bS_cbT_ba. The block representation is an algebra isomorphism to End(⊕_aX_a), with multiplication by contraction of intermediate packets. For every integer k≥1, iterating the product gives P_a_kT^kP_a_0 =∑_a_1,…,a_k-1 (P_a_kTP_a_k-1)⋯(P_a_1TP_a_0).
Exact mathematical passage
k\ge1
\subsetUse in source · 2 passages
Search aliases: ⊂ subset
Use in source · Signal_Reconstruction_Traditional.tex, line 317
smallDomain and hypotheses. Two-jump periodic packets of fixed known amplitudes. Fix a finite set F⊂ Z of derivative-moment indices. For 0<δ<π/4, compare locations (0,δ) and (π,π+δ), both with amplitudes (1,-1). [SG-08 No uniform location stability without separation]
Exact mathematical passage
F\subset\mathbb Z
\subseteqUse in source · 2 passages
Search aliases: ⊆ subseteq
Use in source · Signal_Reconstruction_Traditional.tex, line 520
smallDomain and hypotheses. Finite real or complex Euclidean spaces. Let S⊆ F^m be a subspace, P its orthogonal projector, R=I-P, D∈ F^m× d, and A=RD. Let s≥0 be an integer. [SG-14 Residualized sparse injectivity]
Exact mathematical passage
\mathcal S\subseteq\mathbb F^m
\bigoplusUse in source · 2 passages
Search aliases: ⊕ bigoplus
Use in source · Signal_Reconstruction_Traditional.tex, line 132
The maps A x=(P_ax)_a, S(x_a)_a=∑_ax_a are inverse isomorphisms between X and ⊕_aX_a. In particular, x=∑_aP_ax uniquely.
Exact mathematical passage
\bigoplus_aX_a
\bigoplus_aUse in source · 2 passages
Search aliases: ⊕_a bigoplus
Use in source · Signal_Reconstruction_Traditional.tex, line 132
The maps A x=(P_ax)_a, S(x_a)_a=∑_ax_a are inverse isomorphisms between X and ⊕_aX_a. In particular, x=∑_aP_ax uniquely.
Exact mathematical passage
\bigoplus_aX_a
\perpUse in source · 1 passage
Search aliases: ⊥ perp
Use in source · Signal_Reconstruction_Traditional.tex, line 1522
f-P_k+1f ≤ f-P_kf , f-P_kf⊥ V_k. If ⋃_kV_k is dense, then P_kf→ f in norm.
Exact mathematical passage
\|f-P_{k+1}f\|\le\|f-P_kf\|, \qquad f-P_kf\perp V_k.
\bigcupUse in source · 1 passage
Search aliases: ⋃ bigcup
Use in source · Signal_Reconstruction_Traditional.tex, line 1526
f-P_k+1f ≤ f-P_kf , f-P_kf⊥ V_k. If ⋃_kV_k is dense, then P_kf→ f in norm.
Exact mathematical passage
\bigcup_kV_k
\bigcup_kUse in source · 1 passage
Search aliases: ⋃_k bigcup
Use in source · Signal_Reconstruction_Traditional.tex, line 1526
f-P_k+1f ≤ f-P_kf , f-P_kf⊥ V_k. If ⋃_kV_k is dense, then P_kf→ f in norm.
Exact mathematical passage
\bigcup_kV_k
\cdotsUse in source · 3 passages
Search aliases: ⋯ cdots
Use in source · Signal_Reconstruction_Traditional.tex, line 159
T=∑_a,bP_bTP_a, (ST)_ca=∑_bS_cbT_ba. The block representation is an algebra isomorphism to End(⊕_aX_a), with multiplication by contraction of intermediate packets. For every integer k≥1, iterating the product gives P_a_kT^kP_a_0 =∑_a_1,…,a_k-1 (P_a_kTP_a_k-1)⋯(P_a_1TP_a_0).
Exact mathematical passage
P_{a_k}T^kP_{a_0} =\sum_{a_1,\ldots,a_{k-1}} (P_{a_k}TP_{a_{k-1}})\cdots(P_{a_1}TP_{a_0}).
\langleUse in source · 2 passages
Search aliases: ⟨ langle
Use in source · Signal_Reconstruction_Traditional.tex, line 598
For an unrestricted scalar amplitude the least-squares profile is widehat a(q)=frac⟨ Rq,Ry⟩ Rq _2^2, J(q)= Ry-widehat a(q)Rq _2^2. The true mask has widehat a(q_*)=a_* and J(q_*)=0. It is the unique zero-residual positive-amplitude mask exactly when no other residualized mask lies on its positive ray. Define the closed rays C_q={aRq:a≥0}. If Γ=inf_q≠ q_*dist(Ry,C_q)>0, with the convention inf∅=+∞, then every perturbed residualized datum Ry+e with e _2<Γ/2 selects q_* uniquely by nearest-ray dist…
Exact mathematical passage
\widehat a(q)=\frac{\langle Rq,Ry\rangle}{\|Rq\|_2^2}, \qquad J(q)=\|Ry-\widehat a(q)Rq\|_2^2.
\rangleUse in source · 2 passages
Search aliases: ⟩ rangle
Use in source · Signal_Reconstruction_Traditional.tex, line 598
For an unrestricted scalar amplitude the least-squares profile is widehat a(q)=frac⟨ Rq,Ry⟩ Rq _2^2, J(q)= Ry-widehat a(q)Rq _2^2. The true mask has widehat a(q_*)=a_* and J(q_*)=0. It is the unique zero-residual positive-amplitude mask exactly when no other residualized mask lies on its positive ray. Define the closed rays C_q={aRq:a≥0}. If Γ=inf_q≠ q_*dist(Ry,C_q)>0, with the convention inf∅=+∞, then every perturbed residualized datum Ry+e with e _2<Γ/2 selects q_* uniquely by nearest-ray dist…
Exact mathematical passage
\widehat a(q)=\frac{\langle Rq,Ry\rangle}{\|Rq\|_2^2}, \qquad J(q)=\|Ry-\widehat a(q)Rq\|_2^2.
\longrightarrowUse in source · 2 passages
Search aliases: ⟶ longrightarrow
Use in source · Signal_Reconstruction_Traditional.tex, line 441
Uniformly for z in compact real intervals, S_Mj(x_M(z))⟶frac12+frac1πSi(z), Si(z)=∫_0^zfracsin tt dt. The first positive maximum of the limiting profile is 1+G, where G=fracSi(π)π-frac12 =0.089489872236….
Exact mathematical passage
S_Mj(x_M(z))\longrightarrow\frac12+\frac1\pi\operatorname{Si}(z), \qquad \operatorname{Si}(z)=\int_0^z\frac{\sin t}{t}\,dt.
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