On Complex (Non-Analytic) Chebyshev Polynomials in ℂ2

Abstract

We consider the problem of finding a best uniform approximation to the standard monomial on the unit ball in C 2 by polynomials of lower degree with complex coefficients. We reduce the problem to a onedimensional 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. P = sup(z 1 ,z 2 )∈B |P (z 1 , z 2 )|, B = {(z 1 , z 2 ) ∈ C 2 : |z 1 | 2 + |z 2 | 2 ≤ 1}.

Comb functions

“…Such polynomials turn out to be useful in the description of multidimensional polynomials of least deviation from zero [27]. As an example we formulate the following theorem.…”