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

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Orthogonality, inverse, and Parseval identity for the finite periodic Haar transform

Statement

For every $J \geq 1$ and $x \in R^{2^{J}}$ , the transform $T_{H}$ of Finite periodic Haar transform is an orthogonal linear map. Its inverse is $T_{H}^{T}$ and $∥ T_{H} x ∥_{2}^{2} = ∥ x ∥_{2}^{2}$ .

Facts & Assumptions

Given: An integer $J \geq 1$ and $x \in R^{2^{J}}$ .

[F1]

Finite periodic Haar transform: at each stage the approximation/detail pair from $(u, v)$ is $((u + v) / \sqrt{2}, (u - v) / \sqrt{2})$ , and the final output lists the coarse coefficient followed by detail blocks.

[F2]

The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ : the squared Euclidean norm of a vector is the finite sum of the squares of its coordinates.

Proof

1.1givenF1F2algebra

For any real $u, v$ , the two outputs of one Haar pair satisfy $((u + v) / \sqrt{2})^{2} + ((u - v) / \sqrt{2})^{2} = u^{2} + v^{2}$ by expansion and cancellation of the cross terms. Thus the pair map has matrix $2^{- 1 / 2} (\begin{array}{cc} 1 & 1 \\ 1 & - 1 \end{array})$ whose transpose times itself is the identity; applying it to disjoint coordinate pairs preserves the squared Euclidean norm.

2.1givenstep 1.1F1algebra∎

At stage $r$ , the approximation coordinates are replaced by their pair-map outputs, with the second outputs recorded as $d^{(r)}$ and the first outputs passed to stage $r + 1$ . Therefore the map from the original vector to all stored stage outputs is a composition of the orthogonal pair maps and a coordinate permutation. It is orthogonal; hence its transpose is its inverse and it preserves squared norm.

Depends on

Used by

Exact two-coefficient Haar compression of a four-entry step vector Example

Dependency tree · two levels

15 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)