Wednesday 18 April 2018, 16:00-17:00 in HG03.085
Erdal Emsiz (Pontificia Universidad Catolica de Chile)
Discrete Fourier transforms and cubature rules
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, based on recent joint work with Jan Felipe van Diejen, 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. We will also discuss multivariate versions, in particular cubature rules for the exact integration of symmetric rational functions with prescribed poles.