On coincidence of classes of functions defined by a generalised modulus of smoothness and the appropriate inverse theorem

TL;DR

This paper establishes a connection between classes of functions characterized by a generalized modulus of smoothness and those defined by polynomial approximation, along with an inverse theorem in approximation theory.

Contribution

It introduces a theorem showing the equivalence of these function classes and proves an inverse theorem, advancing understanding in approximation theory.

Findings

Coincidence of function classes defined by smoothness and polynomial approximation

Proof of an inverse theorem in approximation theory

Enhanced characterization of function smoothness and approximation

Abstract

We give the theorem of coincidence of a class of functions defined by a generalised modulus of smoothness with a class of functions defined by the order of the best approximation by algebraic polynomials. We also prove the appropriate inverse theorem in approximation theory.