Szego polynomials: Quadrature rules on the unit circle and on [-1,1]

摘要

We consider some of the relations that exist between real Szego polynomials and certain para-orthogonal polynomials defined on the unit circle, which are again related to certain orthogonal polynomials on [-1, 1] through the transformation x = (z(1/2) +z(-1/2))/2. Using these relations we study the interpolatory quadrature rule based on the zeros of polynomials which are linear combinations of the orthogonal polynomials on [-1, 1]. In the case of any symmetric quadrature rule on [-1, 1], its associated quadrature rule on the unit circle is also given. [References: 13]