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arXiv · 1912.03743

Sharp approximation theorems and Fourier inequalities in the Dunkl setting

D. V. Gorbachev·V. I. Ivanov·S. Yu. Tikhonov
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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.

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BibTeXRIS

D. V. Gorbachev, V. I. Ivanov, S. Yu. Tikhonov. 2019-12-08. Sharp approximation theorems and Fourier inequalities in the Dunkl setting. https://arxiv.org/abs/1912.03743

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