Sharp approximation theorems and Fourier inequalities in the Dunkl setting

Abstract In this paper we study direct and inverse approximation inequalities in L p ( R d ) , 1 p ∞ , 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.