On Complex (non analytic) Chebyshev Polynomials in $\bbC^2$

TL;DR

This paper investigates the problem of best uniform approximation of complex monomials on the unit ball in a2^2, reducing it to a one-dimensional minimization problem and deriving asymptotics for the minimal deviation.

Contribution

It introduces a reduction of the approximation problem to a weighted minimization on an interval and provides explicit representations and asymptotics for extremal polynomials.

Findings

Reduction to a one-dimensional weighted minimization problem

Representation of extremal polynomials via conformal mappings

Asymptotic formulas for the minimal deviation

Abstract

We consider the problem of finding a best uniform approximation to the standard monomial on the unit ball in $\bbC^2$ by polynomials of lower degree with complex coefficients. We reduce the problem to a one-dimensional weighted minimization problem on an interval. In a sense, the corresponding extremal polynomials are uniform counterparts of the classical orthogonal Jacobi polynomials. They can be represented by means of special conformal mappings on the so-called comb-like domains. In these terms, the value of the minimal deviation and the representation for a polynomial of best approximation for the original problem are given. Furthermore, we derive asymptotics for the minimal deviation.