Paper VStatements and proofs
Statements and proofs
Find a statement anywhere in this work by title, printed number, type or source label. Entries follow the chapter order. Each proof link opens the written proof passage.
Statements by chapter
The signal realization of the seed calculus
Residue projectors and exact polyphase coordinates
Linear time-invariant operators as transfer matrices
Analysis banks, synthesis banks, and perfect reconstruction
Wavelet packets as addressed refinements
Known edge packets and spectral reconstruction
Mask, amplitude, and smooth-background error
One-dimensional numerical study
Two-dimensional edge packets
Sparse wavelet packets and global Fourier modes
Finite adaptive packet trees
- Definition 11.1: Complete packet pruning
- Theorem 11.2: Exact synthesis from every complete pruning Read proof
- Theorem 11.3: Finite globally optimal stable selector Read proof
- Theorem 11.4: Dynamic-programming best packet basis Read proof
- Corollary 11.5: Local perturbation certificate Read proof
- Numerical certificate 11.6: Adaptive-tree optimum
- Boundary 11.7: Finite versus asymptotic selection
Nonlinear signal packets
Smith coordinates as cyclic signal coordinates
Dyadic–odd phase factorization and boundary weighting
Boundary-localized error and claim control
Implementation and reproducibility
Signal seeds and lattice addresses
Polyphase coordinates on finite index sublattices
Convolution as a packet transfer matrix
- Definition E4.1: Packet transfer kernel
- Theorem E4.2: Exact convolution transfer law Read proof
- Proposition E4.3: Transfer norm control Read proof
- Proposition E4.4: Composition of transfer matrices Read proof
- Corollary E4.5: Finite impulse response closure Read proof
- Example E4.6: A four phase finite impulse response operator
Filter banks, frame bounds, and lifting
- Theorem E5.1: Perfect reconstruction criterion Read proof
- Theorem E5.2: Quantitative imperfect reconstruction Read proof
- Example E5.3: A reconstruction certificate
- Theorem E5.4: Stability under filter perturbation Read proof
- Theorem E5.5: Frame bounds from the polyphase symbol Read proof
- Corollary E5.6: Coefficient noise under canonical synthesis Read proof
- Theorem E5.7: Perturbation of frame bounds Read proof
- Remark E5.8
- Corollary E5.9: Paraunitary bank Read proof
- Corollary E5.10: Haar reconstruction and energy Read proof
- Theorem E5.11: Exact lifting update Read proof
- Example E5.12: A two neighbor predict and update transform
Wavelet packet trees and admissible bases
- Definition E6.1: Unitary packet tree
- Theorem E6.2: Reconstruction on a pruned packet tree Read proof
- Corollary E6.3: Coefficient perturbations in a unitary tree Read proof
- Proposition E6.4: Bounds for nonunitary local splits Read proof
- Theorem E6.5: Exact additive best basis recursion Read proof
- Theorem E6.6: Stability of the optimal cost Read proof
- Corollary E6.7: Stability of the selected basis Read proof
Singular packets and exact reconstruction
- Definition E7.1: Sobolev step packet class
- Theorem E7.2: Higher order singular packet decomposition Read proof
- Corollary E7.3: Uniqueness of the trace packet Read proof
- Remark E7.4
- Theorem E7.5: Exact retained packet identity Read proof
- Remark E7.6
- Corollary E7.7: Sobolev rate after packet separation Read proof
- Theorem E7.8: Reconstruction with an estimated packet Read proof
Exact error formulas for estimated packets
- Theorem E8.1: Exact scalar region formula Read proof
- Corollary E8.2: Location error for one step Read proof
- Theorem E8.3: Location stability for higher order packets Read proof
- Theorem E8.4: Overlap matrix formula Read proof
- Corollary E8.5: Background and packet error Read proof
- Definition E8.6: Linear tube bound
- Theorem E8.7: Geometric packet bound Read proof
- Example E8.8: A perturbed disk
Stable recovery of separated jumps
- Lemma E9.1: Contrast gap Read proof
- Theorem E9.2: Deterministic jump cell recovery Read proof
- Corollary E9.3: Jump count and sign recovery Read proof
- Corollary E9.4: Packet reconstruction after detection Read proof
- Remark E9.5: Why finite (L p ) matters
- Corollary E9.6: Deterministic consistency Read proof
- Example E9.7: A numerical threshold certificate
- Theorem E9.8: Probability recovery bound Read proof
- Corollary E9.9: Repeated measurements Read proof
- Remark E9.10
- Corollary E9.11: Probability consistency Read proof
- Proposition E9.12: Vector jump recovery Read proof
Haar localization and annihilating differences
- Theorem E10.1: Smooth Haar coefficient bound Read proof
- Theorem E10.2: Exact Haar coefficient of a step Read proof
- Corollary E10.3: Scale separation Read proof
- Theorem E10.4: Half cell shift guarantee Read proof
- Corollary E10.5: Alignment free two grid Haar detection Read proof
- Theorem E10.6: Polynomial annihilation and boundary support Read proof
- Proposition E10.7: Noise amplification by differences Read proof
- Corollary E10.8: Robust breakpoint bands Read proof