Convergence rate for weighted polynomial approximation on the real line

TL;DR

This paper provides a quantitative analysis of weighted polynomial approximation on the real line, extending classical results by estimating approximation errors in the logarithmic scale, particularly for the Cauchy kernel.

Contribution

It offers a new quantitative version of Bernstein's approximation problem in weighted spaces, completing and extending Mergelyan's classical result from 1960.

Findings

Estimates the approximation error of the Cauchy kernel in weighted polynomial spaces.

Provides a logarithmic scale error bound for polynomial approximation.

Completes the classical Bernstein approximation problem with quantitative results.

Abstract

In this note we study a quantitative version of Bernstein's approximation problem when the polynomials are dense in weighted spaces on the real line completing a result of S.~N.~Mergelyan (1960). We estimate in the logarithmic scale the error of the weighted polynomial approximation of the Cauchy kernel.