On a spectral theorem in paraorthogonality theory

Kenier Castillo, Ruymán Cruz-Barroso and Francisco Perdomo-Pío

Vol. 280 (2016), No. 2, 327–347

DOI: 10.2140/pjm.2016.280.327

Abstract

Motivated by the works of Delsarte and Genin (1988, 1991), who studied paraorthogonal polynomials associated with positive definite Hermitian linear functionals and their corresponding recurrence relations, we provide paraorthogonality theory, in the context of quasidefinite Hermitian linear functionals, with a recurrence relation and the analogous result to the classical Favard’s theorem or spectral theorem. As an application of our results, we prove that for any two monic polynomials whose zeros are simple and strictly interlacing on the unit circle, with the possible exception of one of them which could be common, there exists a sequence of paraorthogonal polynomials such that these polynomials belong to it. Furthermore, an application to the computation of Szegő quadrature formulas is also discussed.

Keywords

paraorthogonal polynomials, quasidefinite Hermitian linear functionals, spectral theorem, Geronimus–Wendroff theorem, Szegő quadrature formulas

Mathematical Subject Classification 2010

Primary: 42C05, 30C15, 26C10

Milestones

Received: 9 February 2015

Revised: 3 July 2015

Accepted: 12 July 2015

Published: 28 January 2016