On a spectral theorem in paraorthogonality theory

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.The authors thank the referee for her/his valuable suggestions and comments which have contributed to improve the final form of this paper. The research of the first author is supported by the Portuguese Government through the Fundação para a Ciência e a Tecnologia (FCT) under the grant SFRH/BPD/101139/2014 and partially supported by the Brazilian Government through the CNPq under the project 470019/2013-1 and the Dirección General de Investigación Científica y Técnica, Ministerio de Economía y Competitividad of Spain under the project MTM2012–36732–C03–01. The work of the second and third authors is partially supported by Dirección General de Programas y Transferencia de Conocimiento, Ministerio de Ciencia e Innovación of Spain under the project MTM2011–28781

Poncelet's Theorem, Paraorthogonal Polynomials and the Numerical Range of Compressed Multiplication Operators

There has been considerable recent literature connecting Poncelet's theorem to ellipses, Blaschke products and numerical ranges, summarized, for example, in the recent book [11]. We show how those results can be understood using ideas from the theory of orthogonal polynomials on the unit circle (OPUC) and, in turn, can provide new insights to the theory of OPUC.Comment: 46 pages, 4 figures; minor revisions from v1; accepted for publication in Adv. Mat

On differential equations associated with perturbations of orthogonal polynomials on the unit circle

In this contribution, we propose an algorithm to compute holonomic second-order differential equations satisfied by some families of orthogonal polynomials. Such algorithm is based in three properties that orthogonal polynomials satisfy: a recurrence relation, a structure formula, and a connection formula. This approach is used to obtain second-order differential equations whose solutions are orthogonal polynomials associated with some spectral transformations of a measure on the unit circle, as well as orthogonal polynomials associated with coherent pairs of measures on the unit circle.Comunidad de MadridUniversidad de Alcal