On weighted approximation with Jacobi weights

TL;DR

This paper establishes precise direct and inverse approximation theorems for polynomial approximation in weighted Lp spaces with Jacobi weights, providing a constructive way to characterize function smoothness classes.

Contribution

It offers matching direct and inverse theorems for weighted polynomial approximation with Jacobi weights, enhancing understanding of function smoothness characterization.

Findings

Matching direct and inverse theorems for Jacobi-weighted polynomial approximation.

Constructive characterization of smoothness classes via approximation degree.

Enhanced understanding of weighted approximation in Lp spaces.

Abstract

We obtain matching direct and inverse theorems for the degree of weighted $L_p$-approximation by polynomials with the Jacobi weights $(1-x)^\alpha (1+x)^\beta$. Combined, the estimates yield a constructive characterization of various smoothness classes of functions via the degree of their approximation by algebraic polynomials.