Direct and inverse approximation theorems in the weighted Orlicz-type spaceswith a variable exponent

Abstract

In weighted Orlicz-type spaces ${\mathcal S}_{_{\scriptstyle \mathbf p,\,\mu}}$ with a variable summation exponent, the direct and inverse approximation theorems are proved in terms of best approximations of functions and moduli of smoothness of fractional order. It is shown that the constant obtained in the inverse approximation theorem is the best in a certain sense. Some applications of the results are also proposed. In particular, the constructive characteristics of functional classes defined by such moduli of smoothness are given. Equivalence between moduli of smoothness and certain Peetre $K$-functionals is shown in the spaces ${\mathcal S}_{_{\scriptstyle \mathbf p,\,\mu}}$.