Estimation of wavelet coefficients on some classes of functions

TL;DR

This paper investigates the asymptotic behavior of wavelet coefficients for certain function classes, providing precise limits as the wavelet order increases, which advances understanding of wavelet approximation properties.

Contribution

It establishes the limiting behavior of wavelet coefficients for Daubechies wavelets on specific function classes as the wavelet order tends to infinity.

Findings

Derived explicit limit formulas for wavelet coefficient ratios

Analyzed behavior of Daubechies wavelets with increasing zero moments

Provided theoretical bounds for wavelet coefficient estimates

Abstract

Let $\Psi_m^D$ be orthogonal Daubechies wavelets that have m zero moments and let $$ W_{2,p}^k=\{f \in L_2(R):\|(I \omega)^k\hat f(\omega)\|_p\leq 1\}, \, k \in N. $$ We prove that $$ \lim_{m \to \infty}\, \sup\left\{\frac{|(\Psi_m^D)|}{\|(\hat \Psi_m^D)\|_q}: f \in W_{2, p'}^k\right\}=\frac{\frac{(2\pi)^{1/p-1/2}}{\pi^k}\left(\frac{1-2^{1-pk}}{pk-1}\right)^{1/p}}{(2\pi)^{1/q-1/2}}. $$