A theorem on approximation by algebraic polynomials of functions with given generalised modulus of smoothness

TL;DR

This paper introduces a new asymmetric translation operator to define a generalized modulus of smoothness, and establishes direct and inverse approximation theorems based on this concept.

Contribution

It presents a novel asymmetric translation operator and develops approximation theorems using the generalized modulus of smoothness.

Findings

Established direct approximation theorems.

Proved inverse approximation theorems.

Defined a new generalized modulus of smoothness.

Abstract

We introduce an asymmetric operator of generalised translation, define the generalised modulus of smoothness by its means, and obtain the direct and inverse theorems in approximation theory for it.