On the error of best polynomial approximation of composite functions

TL;DR

This paper characterizes the approximation error of best polynomial approximations for composite functions in weighted spaces, aiding convergence analysis of numerical methods for nonlinear problems with low smoothness.

Contribution

It provides a new estimate of derivatives of composite functions in weighted norms, crucial for analyzing polynomial approximation errors.

Findings

Derived a characterization of approximation error in weighted spaces

Established estimates for derivatives of composite functions

Facilitated convergence analysis of numerical methods for nonlinear problems

Abstract

The purpose of the paper is to provide a characterization of the error of the best polynomial approximation of composite functions in weighted spaces. Such a characterization is essential for the convergence analysis of numerical methods applied to non-linear problems or for numerical approaches that make use of regularization techniques to cure low smoothness of the solution. This result is obtained through an estimate of the derivatives of composite functions in weighted uniform norm.