Direct and inverse theorems of approximation theory for a 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 it.

Contribution

The paper presents a novel asymmetric translation operator and develops a generalized modulus of smoothness with corresponding approximation theorems.

Findings

Established direct theorems relating approximation quality to smoothness.

Proved inverse theorems characterizing smoothness from approximation rates.

Introduced a new operator extending classical translation concepts.

Abstract

An asymmetric operator of generalised translation is introduced in this paper. Using this operator, we define a generalised modulus of smoothness and prove direct and inverse theorems of approximation theory for it.