PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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Biorthogonal coefficient truncation error is controlled by Riesz bounds

Statement

Let $V$ be a finite Riesz synthesis matrix with bounds $A, B$ as in Finite Riesz synthesis matrix and biorthogonal coefficient coordinates, let $c = V^{- 1} x$ , and retain coefficients in $S \subseteq {0, \dots, N - 1}$ . Define $S^{c} = {0, \dots, N - 1} ∖ S$ , and define $(P_{S} c)_{i} = c_{i}$ for $i \in S$ and $0$ otherwise. Then $A \sum_{i \notin S} ∣ c_{i} ∣^{2} \leq ∥ x - V P_{S} c ∥_{2}^{2} \leq B \sum_{i \notin S} ∣ c_{i} ∣^{2} .$ Thus a biorthogonal transform has coefficient-tail error constants given by its Riesz bounds; an orthonormal identity with constant $1$ is the special case $A = B = 1$ .

Facts & Assumptions

Given: An invertible synthesis matrix $V$ , positive Riesz bounds $A, B$ , a signal $x$ , its coefficient vector $c = V^{- 1} x$ , and a retained index set $S$ .

[F1]

Finite Riesz synthesis matrix and biorthogonal coefficient coordinates: for every coefficient vector $z$ , $A ∥ z ∥_{2}^{2} \leq ∥ V z ∥_{2}^{2} \leq B ∥ z ∥_{2}^{2}$ .

[F2]

The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ : for a real coordinate vector $z$ , $∥ z ∥_{2}^{2} = \sum_{i} ∣ z_{i} ∣^{2}$ .

Proof

1.1givenF1algebra

Since $x = V c$ , the retained approximation is $V P_{S} c$ , so the residual is $V (c - P_{S} c) = V P_{S^{c}} c$ .

2.1givenF1F2step 1.1algebra∎

Apply [F1] to $z = P_{S^{c}} c$ . By [F2], $∥ P_{S^{c}} c ∥_{2}^{2} = \sum_{i \notin S} ∣ c_{i} ∣^{2}$ . This gives both inequalities. If $S$ is the full index set, both sides are zero; if $S$ is empty, the same calculation uses $z = c$ .

Depends on

Used by

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Sources

Stéphane Mallat, A Wavelet Tour of Signal Processing, 3rd ed., Chapter 9 (standard reference, not scraped)
Gilbert Strang and Kevin Amaratunga, Wavelets and Filter Banks, MIT OpenCourseWare 18.327, Handout 11 (standard reference, not scraped)