Some properties of special classes of orthogonal polynomials in two variables

Abstract

Two variable analogs of the Chebyshev polynomials are considered. These are special classes of polynomials in two complex conjugate variables which are orthogonal over a region bounded by the Steiner's hypocycloid. Uniform and weighted estimates over the orthogonality region and asymptotics outside of the orthogonality region are found. Location of zeros of these polynomials is also studied. Differential relations between two classes of the polynomials are established. This work was fulfilled with the financial support of Fundamental Research Fund of Russia ( N 96-01-00875).