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Two D4 vanishing moments give a local coefficient bound for smooth periodic samples

Statement

Let $J \geq 3$ , $N = 2^{J}$ , and let $f : R \to R$ be $1$ -periodic and $C^{2}$ . Its restriction to an open interval $U \supset [0, 1]$ belongs to $S^{2}$ ; write $M_{2} = M_{2} (f ∣_{U})$ . Form $x_{n} = N^{- 1 / 2} f (n / N)$ for $0 \leq n < N$ , and analyze $x$ with the multilevel D4 transform. For a detail coefficient $d_{j, k}$ at scale $j = J - ℓ$ ( $1 \leq ℓ \leq J - 2$ ), one has $∣ d_{j, k} ∣ \leq \frac{9 \sqrt{3}}{2} M_{2} 2^{- 5 j / 2} .$

Facts & Assumptions

Given: $J \geq 3$ , a $1$ -periodic $C^{2}$ function $f : R \to R$ , its restriction $f ∣_{U} \in S^{2}$ on an open interval $U \supset [0, 1]$ , and the normalized samples $x_{n} = 2^{- J / 2} f (n 2^{- J})$ .

[F1]

Local support, affine cancellation, and normalization of multilevel D4 detail rows: each stage- $ℓ$ detail row is supported on at most $3 \cdot 2^{ℓ}$ consecutive lifted samples, has $ℓ^{1}$ norm at most the square root of this support bound, and annihilates affine data on that lift.

[F2]

Smooth functions on the unit interval with bounded derivative data: for $q = 2$ , the second derivative of $f$ is continuous and bounded by $M_{2}$ on a full period; periodicity gives the same bound on the periodic extension.

[F3]

Taylor's Lagrange remainder follows from repeated Rolle steps: for a $C^{2}$ function and a center $c$ , $f (t)$ differs from its degree-one Taylor polynomial at $c$ by $f^{''} (ξ) (t - c)^{2} / 2$ for an intermediate point $ξ$ .

Proof

1.1givenF1F2choose

Fix a detail row at stage $ℓ$ and lift its fewer-than- $N$ periodic support to a consecutive interval of integer indices. Periodicity makes the wrapped samples equal to $f (n / N)$ at these lifted indices. Let $c$ be the midpoint of the support interval in physical coordinates. Its length is at most $3 \cdot 2^{ℓ} / N = 3 \cdot 2^{- j}$ , so every support point is within $3 \cdot 2^{- j}$ of $c$ . The row's affine cancellation from [F1] removes the degree-one Taylor polynomial of $f$ at $c$ from the coefficient sum.

2.1givenF1F3step 1.1algebra∎

By [F3], each remaining Taylor remainder is bounded by $\frac{1}{2} M_{2} (3 \cdot 2^{- j})^{2}$ . The row coefficient is the sum of these remainders weighted by $2^{- J / 2}$ times the detail-row entries. Using the $ℓ^{1}$ bound $\sqrt{3 \cdot 2^{ℓ}}$ from [F1] gives $∣ d_{j, k} ∣ \leq 2^{- J / 2} \sqrt{3 \cdot 2^{ℓ}} \frac{9 M_{2}}{2} 2^{- 2 j} = \frac{9 \sqrt{3}}{2} M_{2} 2^{- j / 2} 2^{- 2 j},$ because $ℓ = J - j$ . This is the asserted estimate.

Depends on

Used by

Smooth periodic samples have a best m-term D4 Besov-tail rate Corollary

Dependency tree · two levels

9 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, Handout 13 (standard reference, not scraped)
Stéphane Mallat, A Wavelet Tour of Signal Processing, 3rd ed., Chapter 9 (standard reference, not scraped)