Polynomial Approximation and $\omega^r_\phi (f,t)$ Twenty Years Later

TL;DR

This paper surveys twenty years of research on the measure of smoothness ^r_\u03c9 (f,t) and its connection to polynomial approximation, highlighting key developments and ongoing questions.

Contribution

It provides a comprehensive overview of the evolution and current state of the measure of smoothness ^r_\u03c9 (f,t) and its role in approximation theory over the past two decades.

Findings

Significant progress in understanding the measure's properties.

New connections established with approximation rates.

Open problems and future directions identified.

Abstract

About twenty years ago the measure of smoothness $\omega ^r_\phi (f,t)$ was introduced and related to the rate of polynomial approximation. In this article we survey developments about this and related concepts since that time.