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

Statement

Let $J \geq 3$ , $N = 2^{J}$ , and let $f : R \to R$ be $1$ -periodic and $C^{2}$ , with $f ∣_{U} \in S^{2}$ for an open interval $U \supset [0, 1]$ . Form the normalized sample vector $x_{n} = N^{- 1 / 2} f (n / N)$ and its multilevel D4 coefficients $(a, d)$ . For any $0 < s < 2$ , set $p = (s + 1 / 2)^{- 1}$ . Then $∥ (a, d) ∥_{b_{p, p}^{s} (J)}^{p} \leq 4 M_{0}^{p} + \frac{(9 \sqrt{3} M_{2} / 2)^{p} 2^{- 2 p (2 - s)}}{1 - 2^{- p (2 - s)}},$ where $M_{r} = \sup_{[0, 1]} ∣ f^{(r)} ∣$ , and for $1 \leq m \leq N$ , $∥ x - U_{N}^{T} H_{m} (U_{N} x) ∥_{2} \leq \sqrt{\frac{p}{2 - p}} {(4 M_{0}^{p} + \frac{(9 \sqrt{3} M_{2} / 2)^{p} 2^{- 2 p (2 - s)}}{1 - 2^{- p (2 - s)}})}^{1 / p} m^{- s} .$ The norm is the finite wavelet-coefficient convention of Finite wavelet-coefficient Besov quasi-norm on a dyadic multilevel transform; no intrinsic Besov norm equivalence is asserted.

Facts & Assumptions

Given: A $1$ -periodic $C^{2}$ function $f : R \to R$ , its restriction $f ∣_{U} \in S^{2}$ on an open interval $U \supset [0, 1]$ , its normalized samples, and $s \in (0, 2)$ .

[F1]

Two D4 vanishing moments give a local coefficient bound for smooth periodic samples: each detail coefficient at scale $j$ is bounded by $(9 \sqrt{3} / 2) M_{2} 2^{- 5 j / 2}$ .

[F2]

Periodic multilevel transform using the four-tap Daubechies filter: the coarse block has four coefficients, and the level- $j$ detail block has $2^{j}$ coefficients.

[F3]

Inverse and Parseval identity for a periodic orthogonal compact-filter transform: the multilevel analysis matrix $U_{N}$ is orthogonal and preserves the Euclidean norm.

[F4]

Finite wavelet-coefficient Besov quasi-norm on a dyadic multilevel transform: the critical coefficient norm has no scale weights when $p = (s + 1 / 2)^{- 1}$ .

[F5]

Critical Besov coefficient control implies a best m-term squared-error rate: a finite orthogonal coefficient vector with this critical norm has best m-term error at most $\sqrt{p / (2 - p)} ∥ c ∥_{b_{p, p}^{s}} m^{- s}$ .

Proof

1.1givenF2F3F4algebra

Here $1 / p = s + 1 / 2$ , so the coefficient norm's scale weight is $1$ by [F4]. Orthogonality [F3] gives $∥ x ∥_{2}^{2} = N^{- 1} \sum_{n = 0}^{N - 1} ∣ f (n / N) ∣^{2} \leq M_{0}^{2}$ , and therefore each of the four coarse coefficients has magnitude at most $M_{0}$ .

1.2givenF1F2F4algebra

At scale $j$ , [F1] and [F2] imply $\sum_{k = 0}^{2^{j} - 1} ∣ d_{j, k} ∣^{p} \leq 2^{j} (9 \sqrt{3} M_{2} / 2)^{p} 2^{- 5 j p / 2} = (9 \sqrt{3} M_{2} / 2)^{p} 2^{- j p (2 - s)},$ because $1 - p (5 / 2) = - p (2 - s)$ . Summing over the finite range $2 \leq j \leq J - 1$ and bounding by the infinite geometric series gives the first displayed estimate. Its ratio is $2^{- p (2 - s)} < 1$ since $0 < s < 2$ .

2.1givenF3F5step 1.2algebra∎

Apply [F5] to the finite orthogonal coefficient vector of $U_{N} x$ and substitute the first bound from step 1.2. This gives the stated best m-term error rate with the declared constant.

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Sources

Stéphane Mallat, A Wavelet Tour of Signal Processing, 3rd ed., Chapter 9, §9.2.3 (standard reference, not scraped)
Gilbert Strang and Kevin Amaratunga, Wavelets and Filter Banks, MIT OpenCourseWare 18.327, Handouts 8 and 13 (standard reference, not scraped)