# Direct and inverse theorems of approximation of functions in weighted Orlicz type spaces with variable exponent

## TL;DR

This paper establishes direct and inverse approximation theorems in weighted Orlicz spaces with variable exponents, linking smoothness measures to approximation quality, and demonstrates the optimality of constants involved.

## Contribution

It introduces new approximation theorems in weighted Orlicz spaces with variable exponents, including the equivalence of moduli of smoothness and Peetre K-functionals, and discusses applications and optimal constants.

## Key findings

- Proved direct and inverse approximation theorems in weighted Orlicz spaces.
- Established the equivalence between moduli of smoothness and Peetre K-functionals.
- Demonstrated the optimality of the constants in inverse theorems.

## 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 in a certain sense the best. 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 ${\mathcal S}_{_{\scriptstyle \mathbf p,\,\mu}}$.