The Bernstein-Walsh-Siciak Theorem for analytic hypersurfaces

TL;DR

This paper extends the Bernstein-Walsh-Siciak Theorem to analytic hypersurfaces, providing insights into polynomial approximation of complex structures in several variables.

Contribution

It introduces a set-theoretic version of the theorem for analytic hypersurfaces, broadening the scope of polynomial approximation theory.

Findings

Established a Bernstein-Walsh-Siciak type theorem for analytic hypersurfaces

Demonstrated rapid polynomial approximation on polynomially convex sets

Extended classical results to complex hypersurface structures

Abstract

As a first step towards a general set-theoretic counterpart of the remarkable Bernstein-Walsh-Siciak Theorem concerning the rapidity of polynomial approximation of a holomorphic function on polynomially convex compact sets in $\mathbb{C}^n$, we prove a version of this theorem for analytic hypersurfaces.