1 The signal realization of the seed calculus
On this page · Visual guide: Three objects, three different questions · Permalink · Source line 262
Let \(\ell^2({\mathbb Z};{\mathbb C})\), abbreviated \(\ell^2({\mathbb Z})\), be the Hilbert space of square-summable complex discrete signals. The inner product is conjugate-linear in its first argument. A superscript \(*\) on a bounded operator denotes its Hilbert adjoint. Write \(\delta_n\) for the unit impulse at address \(n\) and let the bilateral shift be
Then
The primitive signal seed is therefore \(s=\delta_0\), the address operators are the shift words \(S^n\), and the completion is the \(\ell^2\) norm.
Theorem 1.1: Impulse seed theorem
The closed shift-generated hull of \(\delta_0\) is all of \(\ell^2({\mathbb Z})\):
Every finitely supported signal has the exact finite expression
Read proof · Read statement and proof side by side · Permalink · Source line 274 · Claim record · Lean coverage · Code · Worked examples
Proof of Theorem 1.1
The shift orbit is the standard orthonormal basis \(\{\delta_n\}_{n\in{\mathbb Z}}\). For \(x\in\ell^2\), the finite truncations
satisfy \(\|x-x_F\|_2^2=\sum_{n\notin F}|x[n]|^2\to0\) as the finite sets exhaust \({\mathbb Z}\). Thus the unordered Hilbert-space sum \(\sum_nx[n]\delta_n\) converges to \(x\), and the finite-support statement is exact. The shift is a surjective isometry because \(n\mapsto n-1\) is a bijection of \({\mathbb Z}\), with inverse shift \(n\mapsto n+1\). This proves the closed-span assertion.
Back to statement · Read statement and proof side by side · Permalink · Source line 288 · Claim record · Lean coverage · Code · Worked examples
This realization fixes the language used below:
Remark 1.2: Several completions
The same algebraic seed description applies to finite sequences, \(\ell^1\), \(\ell^2\), periodic sequences, tempered distributions, and multidimensional lattices. The topology decides which infinite reconstructions are legitimate. Finite-dimensional experiments in this paper use ordinary Euclidean norms; theorem statements specify the ambient norm when it matters.