In this paper we study direct and inverse approximation inequalities in $L^p\left({\mathbb{R}}^d\right)$, $1<p<\infty$, 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.

中文翻译：

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Dunkl设置中的尖锐逼近定理和傅立叶不等式

在本文中，我们研究了方程中的正和逆近似不等式 ${\mathrm{大号}}^p（{\mathbb{[R}}^d）$， $\mathrm{1个}<p<\infty$，加上Dunkl的体重。我们以敏锐的形式获得了这些估计，从而大大改善了先前的结果。我们还建立了函数平滑度的新估计$F$ 通过Dunkl Laplacian近似值的分数次方 $F$。此外，我们根据Dunkl变换获得了有关光滑度模量的新Lebesgue类型估计。导出了所需的Pitt型和Kellogg型Fourier-Dunkl不等式。