A Survey of Weighted Approximation for Exponential Weights

TL;DR

This survey reviews historical and recent developments in weighted polynomial approximation on the real line, focusing on Bernstein's problem, inequalities, and orthogonal expansions, highlighting key ideas and proof techniques.

Contribution

It provides a comprehensive overview of weighted approximation theory, including classical and modern results, with insights into proof strategies and techniques for even weights on the real line.

Findings

Summarizes key results in weighted Jackson and Bernstein theorems.

Discusses inequalities like Markov-Bernstein and Nikolskii for weighted approximation.

Highlights main ideas and proof techniques used in the field.

Abstract

Let W: R to (0,1] be continuous. Bernstein's approximation problem, posed in 1924, deals with approximation by polynomials in the weighted uniform norm ||fW|| Linfinity(R) . The qualitative form of this problem was solved by Achieser, Mergelyan, and Pollard, in the 1950's. Quantitative forms of the problem were actively investigated starting from the 1960's. We survey old and recent aspects of this topic, including the Bernstein problem, weighted Jackson and Bernstein Theorems, Markov-Bernstein and Nikolskii inequalities, orthogonal expansions and Lagrange interpolation. We present the main ideas used in many of the proofs, and different techniques of proof, though not the full proofs. The class of weights we consider is typically even, and supported on the whole real line, so we exclude Laguerre type weights on [0,infinity). Nor do we discuss Saff's weighted approximation problem, nor the asymptotics of orthogonal polynomials.