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

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Finite wavelet-coefficient Besov quasi-norm on a dyadic multilevel transform

Definition

For an integer $J \geq 3$ , a coarse block $a = (a_{0}, a_{1}, a_{2}, a_{3})$ , a real $s > 0$ , and $p > 0$ , define the finite wavelet-coefficient Besov quasi-norm $∥ (a, d) ∥_{b_{p, p}^{s} (J)}^{p} = \sum_{ν = 0}^{3} ∣ a_{ν} ∣^{p} + \sum_{j = 2}^{J - 1} 2^{j p (s + 1 / 2 - 1 / p)} \sum_{k = 0}^{2^{j} - 1} ∣ d_{j, k} ∣^{p} .$ This is the wavelet coefficient convention, with one coarse block and detail blocks indexed by their number $2^{j}$ of coefficients. At the critical relation $p = (s + 1 / 2)^{- 1}$ , every scale weight equals $1$ , so the quasi-norm is exactly the $ℓ^{p}$ quasi-norm of the full coefficient vector. This finite coefficient definition does not assert equivalence with an intrinsic function-space Besov norm.

Used by

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

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, Handout 13 (standard reference, not scraped)