Discrete Fourier transform and quadrature rule associated with Bernstein-Szegö polynomials
The Bernstein-Szegö polynomials are a multi-parameter family of orthogonal polynomials generalizing the Chebyshev polynomials. It is well-known that they diagonalize a family of semi-infinite Jacobi matrices. In this talk we will explain how to glue two such families of Bernstein-Szegö polynomials so as to diagonalize a corresponding family of finite Jacobi matrices.
The symmetry of the finite Jacobi matrix gives rise to a finite-dimensional system of discrete orthogonal relations for the pertinent composite eigenbasis built of Bernstein-Szegö polynomials. We will indicate how these relations imply Gauss-type quadrature rules for the exact integration of rational functions with prescribed poles against the Chebyshev weight function. We will also discuss, if time permits, multi-variable analogs.
Based on joint work with J. F. van Diejen (Universidad de Talca).