# Bernstein-Walsh theory associated to convex bodies and applications to multivariate approximation theory

## TL;DR

This paper extends Bernstein-Walsh theorem to multivariate polynomial approximation on convex bodies, providing new insights and validating previous observations in multivariate approximation theory.

## Contribution

It introduces a version of Bernstein-Walsh theorem for convex bodies in several complex variables, enhancing understanding of multivariate polynomial approximation.

## Key findings

- Validated and clarified Trefethen's observations in multivariate approximation
- Extended Bernstein-Walsh theorem to subclasses of polynomial spaces associated with convex bodies
- Provided theoretical foundations for multivariate uniform polynomial approximation

## Abstract

We prove a version of the Bernstein-Walsh theorem on uniform polynomial approximation of holomorphic functions on compact sets in several complex variables. Here we consider subclasses of the full polynomial space associated to a convex body P. As a consequence, we validate and clarify some observations of Trefethen in multivariate approximation theory.