On approximation of functions by algebraic polynomials in H\"older spaces

TL;DR

This paper investigates how well functions can be approximated by algebraic polynomials within H"older spaces, introducing improved theorems and criteria for approximation accuracy, with applications to Bernstein polynomial operators.

Contribution

It provides new direct and inverse approximation theorems in H"older spaces using generalized moduli of smoothness, enhancing understanding of approximation rates.

Findings

Improved direct and inverse approximation theorems.

Criteria for the precise order of approximation.

Strong converse inequalities for approximation methods.

Abstract

We study approximation of functions by algebraic polynomials in the H\"older spaces corresponding to the generalized Jacobi translation and the Ditzian-Totik moduli of smoothness. By using modifications of the classical moduli of smoothness, we give improvements of the direct and inverse theorems of approximation and prove the criteria of the precise order of decrease of the best approximation in these spaces. Moreover, we obtain strong converse inequalities for some methods of approximation of functions. As an example, we consider approximation by the Durrmeyer-Bernstein polynomial operators.