Sharp approximation theorems and Fourier inequalities in the Dunkl setting

TL;DR

This paper advances approximation theory in the Dunkl setting by establishing sharp inequalities, new smoothness estimates, and Fourier inequalities, significantly improving prior results in harmonic analysis involving Dunkl operators.

Contribution

The paper introduces sharp approximation inequalities, new modulus of smoothness estimates via Dunkl Laplacian, and Fourier inequalities, enhancing the theoretical framework of Dunkl harmonic analysis.

Findings

Sharp approximation inequalities in Dunkl setting

New estimates of modulus of smoothness via Dunkl Laplacian

Lebesgue type estimates for Dunkl transform

Abstract

In this paper we study direct and inverse approximation inequalities in $L^{p}(\mathbb{R}^{d})$, $1<p<\infty$, with the Dunkl weight. We obtain these estimates in their sharp form substantially improving previous results. We also establish new estimates of the modulus of smoothness of a function $f$ via the fractional powers of the Dunkl Laplacian of approximants of $f$. Moreover, we obtain new Lebesgue type estimates for moduli of smoothness in terms of Dunkl transforms. Needed Pitt-type and Kellogg-type Fourier--Dunkl inequalities are derived.