Almost optimal polynomial approximation on convex sets in $\mathbb{C}$

TL;DR

This paper introduces a sequence of nearly optimal polynomials that approximate functions on convex sets in the complex plane, achieving geometric convergence at points where the function is analytic.

Contribution

It provides a new construction of polynomial approximations that are nearly optimal and converge geometrically on convex sets in , extending classical approximation results.

Findings

Achieves geometric convergence rate for polynomial approximations.

Constructs nearly optimal polynomials for functions on convex sets.

Applicable to functions analytic in the interior of convex sets.

Abstract

For a function $f$, continuous on a compact convex set $K$ and analytic in its interior we construct a sequence of almost optimal polynomials that converge with a geometric rate at points of analyticity of $f$.