Polynomial approximation of piecewise analytic functions on quasi-smooth arcs

TL;DR

This paper develops polynomial approximation methods for piecewise analytic functions on quasi-smooth arcs, achieving near-optimal convergence rates and demonstrating geometric convergence on specific arc classes.

Contribution

It introduces a sequence of near-best polynomials with exponential convergence rates for functions on quasi-smooth arcs, extending approximation theory.

Findings

Convergence rate of $e^{-n^{\sigma}}$ for approximants at points of analyticity.

Existence of quasi-smooth arcs with geometric convergence rate $e^{-cn}$.

Near-best polynomials closely approximate the best polynomial approximants.

Abstract

For a function $f$ that is piecewise analytic on a quasi-smooth arc $\mathcal{L}$ and any $0<\sigma<1$ we construct a sequence of "near-best" polynomials that converge at a rate $e^{-n^{\sigma}}$ at each point of analyticity of $f$ and are close to the best polynomial approximants on the whole $\mathcal{L}$. Also we give examples of quasi-smooth arcs for which convergence of "near-best" approximants at points of analiticity of $f$ is geometric, i.e. has a rate $e^{-cn}$ with some $c>0$.