The Extremal Function for the Complex Ball for Generalized Notions of Degree and Multivariate Polynomial Approximation

TL;DR

This paper investigates the extremal function in pluripotential theory for the complex unit ball, analyzing how polynomial approximation properties vary with different notions of degree defined by convex bodies.

Contribution

It introduces a generalized framework for the extremal function based on convex bodies, extending classical polynomial approximation theory in several complex variables.

Findings

Characterization of the Siciak-Zaharjuta extremal function for generalized polynomial degrees

Analysis of approximation properties depending on the convex body P

Insights into multivariate polynomial approximation in complex analysis

Abstract

We discuss the Siciak-Zaharjuta extremal function of pluripotential theory for the unit ball in C^d for spaces of polynomials with the notion of degree determined by a convex body P. We then use it to analyze the approximation properties of such polynomial spaces, and how these may differ depending on the function f to be approximated.