Direct and inverse approximation theorems in the Besicovitch-Museilak-Orlicz spaces of almost periodic functions

TL;DR

This paper establishes direct and inverse approximation theorems for Besicovitch almost periodic functions within Orlicz spaces, focusing on the behavior of Fourier exponents and the optimality of constants involved.

Contribution

It introduces new approximation theorems for Besicovitch almost periodic functions in Orlicz spaces, analyzing the sharpness of constants in these theorems.

Findings

Theorems are proved relating best approximations and moduli of smoothness.

Constants in the theorems are shown to be unimprovable in certain cases.

Results extend approximation theory to a broader class of almost periodic functions.

Abstract

In terms of the best approximations of functions and generalized moduli of smoothness, direct and inverse approximation theorems are proved for Besicovitch almost periodic functions whose Fourier exponent sequences have a single limit point in infinity and their Orlicz norms are finite. Special attention is paid to the study of cases when the constants in these theorems are unimprovable.