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

Abstract

In weighted Orlicz-type spaces Sp, µ 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 isshown that the constant obtained in the inverse approximation theorem is the best in a certain sense. Some applicationsof the results are also proposed. In particular, the constructive characteristics of functional classes defined by such moduliof smoothness are given. Equivalence between moduli of smoothness and certain Peetre K -functionals is shown in thespaces Sp, µ .