# Direct and inverse approximation theorems of functions in the Orlicz type spaces S_M

## TL;DR

This paper establishes direct and inverse approximation theorems in Orlicz type spaces, linking best approximation, moduli of smoothness, and Peetre K-functionals, advancing the theoretical understanding of function approximation in these spaces.

## Contribution

It introduces new approximation theorems in Orlicz spaces and demonstrates the equivalence between moduli of smoothness and Peetre K-functionals.

## Key findings

- Proved direct and inverse approximation theorems in Orlicz spaces.
- Established the equivalence between moduli of smoothness and Peetre K-functionals.
- Enhanced theoretical framework for function approximation in Orlicz spaces.

## Abstract

In the Orlicz type spaces ${\mathcal S}_{M}$, we prove direct and inverse approximation theorems in terms of the best approximations of functions and moduli of smoothness of fractional order. We also show the equivalence between moduli of smoothness and Peetre $K$-functionals in the spaces ${\mathcal S}_{M}$.