Polynomial approximation avoiding values in sets II

TL;DR

This paper investigates conditions under which functions on compact sets in complex and real spaces can be approximated by polynomials that avoid specific values, extending classical approximation results to higher dimensions and infinite-dimensional spaces.

Contribution

It introduces new polynomial approximation theorems avoiding prescribed sets in complex, real, and infinite-dimensional spaces, generalizing Lavrentiev's theorem.

Findings

Polynomial approximation avoiding countable sets in $\, \\mathbb R^n$

Extension of approximation results to infinite-dimensional Banach spaces

Higher-dimensional analogues of classical theorems

Abstract

We prove some results on when functions on compact sets $K \subset \mathbb C$ can be approximated by polynomials avoiding values in given sets. We also prove some higher dimensional analogues. In particular we prove that a continuous function from a compact set $K \subset \mathbb R^n$ without interior points to $\mathbb R^n$ can be uniformly approximated by a polynomial mapping avoiding values in any given countable set $A \subset \mathbb R^n$, giving a real $n$-dimensional analogue of a recent version of Lavrentiev's theorem of Andersson and Rousu. We also prove the same result for infinite dimensional Banach spaces.