A Geometric Approach to Polynomial and Rational Approximation

TL;DR

This paper enhances classical approximation theorems by demonstrating that polynomial and rational approximants can be constructed with specific geometric properties, controlling their critical points and values within designated domains.

Contribution

It introduces a geometric method to strengthen approximation theorems, allowing precise placement of critical points and values of approximants.

Findings

Critical points of approximants can be placed in any chosen domain containing the set.

Critical values can be confined within neighborhoods of the polynomially convex hull.

The approach applies to both polynomial and rational approximations.

Abstract

We strengthen the classical approximation theorems of Weierstrass, Runge and Mergelyan by showing the polynomial and rational approximants can be taken to have a simple geometric structure. In particular, when approximating a function $f$ on a compact set $K$, the critical points of our approximants may be taken to lie in any given domain containing $K$, and all the critical values in any given neighborhood of the polynomially convex hull of $f(K)$.