Explicit Bernstein type inequalities for wavelet coefficients in $L_p(R^n)$

TL;DR

This paper establishes Bernstein type inequalities for wavelet coefficients in certain function spaces, providing sharp bounds for splines and analyzing asymptotic behaviors of Daubechies and spline wavelets.

Contribution

It introduces explicit Bernstein inequalities for wavelet coefficients in $L_p$ spaces and compares spline and Daubechies wavelets through a new quantitative measure.

Findings

Sharp Bernstein inequalities for splines were derived.

A lower bound for the quantity $C_{k,p}( ext{semiorthogonal spline wavelets})$ was established.

Asymptotic behaviors of wavelet coefficients for Daubechies and spline wavelets were analyzed.

Abstract

In this paper, we investigate the wavelet coefficients for function spaces $\mathcal{A}_k^p=\{f: \|(i \omega)^k\hat{f}(\omega)\|_p\leq 1\}, k\in N, p\in(1,\infty)$ using an important quantity $C_{k,p}(\psi)$. In particular, Bernstein type inequalities associated with wavelets are established. We obtained a sharp inequality of Bernstein type for splines, which induces a lower bound for the quantity $C_{k,p}(\psi)$ with $\psi$ being the semiorthogonal spline wavelets. We also study the asymptotic behavior of wavelet coefficients for both the family of Daubechies orthonormal wavelets and the family of semiorthogonal spline wavelets. Comparison of these two families is done by using the quantity $C_{k,p}(\psi)$.