Polynomial approximation with doubling weights having finitely many zeros and singularities

TL;DR

This paper establishes matching direct and inverse theorems for polynomial approximation using a broad class of doubling weights with finitely many zeros and singularities, including classical weights, in weighted $L_p$ norms.

Contribution

It extends polynomial approximation theory to include weights with zeros and singularities, providing new equivalence results and covering many classical weight functions.

Findings

Proved matching direct and inverse theorems for polynomial approximation.

Included classical Jacobi and generalized weights in the analysis.

Discussed equivalence results involving realization functionals.

Abstract

We prove matching direct and inverse theorems for (algebraic) polynomial approximation with doubling weights $w$ having finitely many zeros and singularities (i.e., points where $w$ becomes infinite) on an interval and not too ``rapidly changing'' away from these zeros and singularities. This class of doubling weights is rather wide and, in particular, includes the classical Jacobi weights, generalized Jacobi weights and generalized Ditzian-Totik weights. We approximate in the weighted $L_p$ (quasi) norm $\|f\|_{p, w}$ with $0<p<\infty$, where $\|f\|_{p, w} := \left(\int_{-1}^1 |f(u)|^p w(u) du \right)^{1/p}$. Equivalence type results involving related realization functionals are also discussed.