Approximation by random complex polynomials and random rational functions

TL;DR

This paper explores how randomness in complex polynomials and rational functions can enhance convergence properties and provides new insights into the behavior of random functions on compact sets.

Contribution

It introduces the use of randomness to improve convergence in sequences of complex functions, extending classical approximation theory.

Findings

Randomness can improve convergence of polynomial and rational function sequences.

Analysis of random functions on compact sets reveals new approximation behaviors.

The study connects probabilistic methods with complex approximation theory.

Abstract

We investigate random compact sets with random functions defined thereon, such as polynomials, rational functions, the pluricomplex Green function and the Siciak extremal function. One surprising consequence of our study is that randomness can be used to `improve' convergence for sequences of functions.