Title: Orthogonal Laurent polynomials on the unit circle and snake-shaped matrix factorizations
Authors: Cruz Barroso, Ruyman ×
Delvaux, Steven #
Issue Date: Nov-2009
Publisher: Academic Press
Series Title: Journal of approximation theory vol:161 issue:1 pages:65-87
Abstract: Let there be given a probability measure μ on the unit circle T of the complex plane and consider the inner product induced by μ. In this paper we consider the problem of orthogonalizing a sequence of monomials {z^(r_k)}_k, for a certain order of the r_k in Z, by means of the Gram–Schmidt orthogonalization process. This leads to a sequence of orthonormal Laurent polynomials {ψ_k}_k. We show that the matrix representation with respect to {ψ_k}_k of the operator of multiplication by z is an infinite unitary or isometric matrix allowing a ‘snake-shaped’ matrix factorization. Here the ‘snake shape’ of the factorization is to be understood in terms of its graphical representation via sequences of little line segments, following an earlier work of S. Delvaux and M. Van Barel. We show that the shape of the snake is determined by the order in which the monomials {z^(r_k)}_k are orthogonalized, while the ‘segments’ of the snake are canonically determined in terms of the Schur parameters for μ. Isometric Hessenberg matrices and unitary five-diagonal matrices (CMV matrices) follow as a special case of the presented formalism.
Description: published online 5 November 2008
ISSN: 0021-9045
Publication status: published
KU Leuven publication type: IT
Appears in Collections:Numerical Analysis and Applied Mathematics Section
Analysis Section
× corresponding author
# (joint) last author

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