Approximation properties of Bernstein singular integrals in variable exponent Lebesgue spaces on the real axis

TL;DR

This paper investigates how Bernstein singular integrals can be used to approximate functions in variable exponent Lebesgue spaces, providing new inequalities and estimates for such approximations.

Contribution

It introduces novel approximation inequalities and estimates for Bernstein singular integrals in variable exponent Lebesgue spaces, expanding understanding of their approximation capabilities.

Findings

Established inequalities for approximation by integral functions in L^{p(.)}

Derived estimates for simultaneous approximation in variable exponent spaces

Enhanced understanding of Bernstein singular integrals in non-uniform Lebesgue spaces

Abstract

In generalized Lebesgue spaces L^{p(.)} with variable exponent p(.) defined on the real axis, we obtain several inequalities of approximation by integral functions of finite degree. Approximation properties of Bernstein singular integrals in these spaces are obtained. Estimates of simultaneous approximation by integral functions of finite degree in L^{p(.)} are proved.