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 periodic Haar transform

Definition

Let $J \geq 1$ , $N = 2^{J}$ , and $x \in R^{N}$ , with the Euclidean inner product and norm of The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ . Put $a^{(0)} = x$ . For $r = 1, \dots, J$ and $0 \leq k < 2^{J - r}$ define $a_{k}^{(r)} = \frac{a_{2 k}^{(r - 1)} + a_{2 k + 1}^{(r - 1)}}{\sqrt{2}}, d_{k}^{(r)} = \frac{a_{2 k}^{(r - 1)} - a_{2 k + 1}^{(r - 1)}}{\sqrt{2}} .$ The finite periodic Haar transform $T_{H} x$ is the length- $N$ vector obtained by listing $a_{0}^{(J)}$ , then $d_{0}^{(J)}$ , then the entries of $d^{(J - 1)}$ , and continuing through $d^{(1)}$ . The input is already of even dyadic length; this convention uses no padding and no endpoint extension.

Depends on

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

Used by

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

Gilbert Strang and Kevin Amaratunga, Wavelets and Filter Banks, MIT OpenCourseWare 18.327, Handouts 3 and 7 (standard reference, not scraped)
Stéphane Mallat, A Wavelet Tour of Signal Processing, 3rd ed., Chapter 9 (standard reference, not scraped)