Author:

Keywords:

Science & Technology, Physical Sciences, Mathematics, Applied, Mathematics, Szego polynomials, Orthogonal Laurent polynomials, Orthogonal rational functions, Szego quadrature formulas, Hessenberg matrices, CMV matrices, ORTHOGONAL-LAURENT-POLYNOMIALS, HAMBURGER MOMENT PROBLEM, GAUSSIAN QUADRATURE, RATIONAL QUADRATURE, SZEGO QUADRATURES, VALUED FUNCTIONS, WEIGHT-FUNCTIONS, OPERATORS, COMPUTATION, THEOREM, MOMENTS, 0102 Applied Mathematics, 0103 Numerical and Computational Mathematics, 0906 Electrical and Electronic Engineering, Numerical & Computational Mathematics, 4613 Theory of computation, 4901 Applied mathematics, 4903 Numerical and computational mathematics

Abstract:

© 2014 Elsevier B.V. All rights reserved. In this paper we give a survey of some results concerning the computation of quadrature formulas on the unit circle. Like nodes and weights of Gauss quadrature formulas (for the estimation of integrals with respect to measures on the real line) can be computed from the eigenvalue decomposition of the Jacobi matrix, Szego{combining double acute accent} quadrature formulas (for the approximation of integrals with respect to measures on the unit circle) can be obtained from certain unitary five-diagonal or unitary Hessenberg matrices that characterize the recurrence for an orthogonal (Laurent) polynomial basis. These quadratures are exact in a maximal space of Laurent polynomials. Orthogonal polynomials are a particular case of orthogonal rational functions with prescribed poles. More general Szego{combining double acute accent} quadrature formulas can be obtained that are exact in certain spaces of rational functions. In this context, the nodes and the weights of these rules are computed from the eigenvalue decomposition of an operator Möbius transform of the same five-diagonal or Hessenberg matrices.