Generalized characteristic polynomials and Gaussian cubature rules

TL;DR

This paper introduces a class of orthogonal polynomials derived from Toeplitz matrices that have maximal real zeros, enabling the creation of Gaussian cubature rules for two-variable integration.

Contribution

It generalizes characteristic polynomials to two variables and links them to Gaussian cubature rules, including special cases like Chebyshev polynomials on the deltoid.

Findings

Orthogonal polynomials with maximal real zeros are constructed.

These polynomials generate Gaussian cubature rules in two variables.

Includes special cases such as Chebyshev polynomials of the second kind.

Abstract

For a family of near banded Toeplitz matrices, generalized characteristic polynomials are shown to be orthogonal polynomials of two variables, which include the Chebyshev polynomials of the second kind on the deltoid as a special case. These orthogonal polynomials possess maximal number of real common zeros, which generate a family of Gaussian cubature rules in two variables.