Polynomial approximation on a compact subset of the real line

TL;DR

This paper establishes a new Bernstein-type theorem that characterizes the rate at which polynomial approximations converge for piecewise analytic functions on compact real subsets.

Contribution

It introduces an analogue of Bernstein's theorem specifically for piecewise analytic functions on compact subsets of the real line, extending classical approximation theory.

Findings

Proves a Bernstein-type approximation rate theorem.

Extends classical polynomial approximation results.

Provides theoretical bounds for approximation of piecewise analytic functions.

Abstract

We prove an analogue of the classical Bernstein theorem concerning the rate of polynomial approximation of piecewise analytic functions on a compact subset of the real line.