DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Finite Riesz synthesis matrix and biorthogonal coefficient coordinates

Definition

Let $V \in R^{N \times N}$ be invertible and suppose there are constants $0 < A \leq B < \infty$ such that $A ∥ z ∥_{2}^{2} \leq ∥ V z ∥_{2}^{2} \leq B ∥ z ∥_{2}^{2} for every z \in R^{N} .$ Call $V$ a finite Riesz synthesis matrix and call $A, B$ Riesz bounds. For a signal $x \in R^{N}$ , the analysis coefficients are $c = V^{- 1} x$ and synthesis is $x = V c$ . The inverse analysis matrix and synthesis matrix form a biorthogonal coordinate pair; unless $A = B = 1$ , coefficient truncation errors need not equal Euclidean tail energy.

Depends on

The Euclidean inner product $\langle x,y\rangle = \sum_{k<n} x_k y_k$ on $\mathbb{R}^n$

Used by

Biorthogonal coefficient truncation error is controlled by Riesz bounds Proposition

Dependency tree · two levels

14 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

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)