Kenier Castillo

Monotonicity of zeros for a class of polynomials including hypergeometric polynomials

We study the monotonicity of zeros in connection with perturbed recurrence coefficients of polynomials satisfying certain three-term recurrence relations of Frobenius-type. These recurrence relations are the key ingredient for the tridiagonal approach developed by Delsarte and Genin to solve the standard linear prediction problem. As a particular case, we consider the Askey para-orthogonal polynomials on the unit circle, 2 F 1 ( - n , a + b i ; 2 a ; 1 - z ) , a , b ? R , extending a recent result about the monotonicity of their zeros with respect to the parameter b. Finally, the consequences of our results in the theory of orthogonal polynomials on the real line are discussed.

Perturbations on the antidiagonals of Hankel matrices

Given a strongly regular Hankel matrix, and its associated sequence of moments which defines a quasi-definite moment linear functional, we study the perturbation of a fixed moment, i.e., a perturbation of one antidiagonal of the Hankel matrix. We define a linear functional whose action results in such a perturbation and establish necessary and sufficient conditions in order to preserve the quasi-definite character. A relation between the corresponding sequences of orthogonal polynomials is obtained, as well as the asymptotic behavior of their zeros. We also study the invariance of the Laguerre-Hahn class of linear functionals under such perturbation, and determine its relation with the so-called canonical linear spectral transformations.

A new linear spectral transformation associated with derivatives of Dirac linear functionals

In this contribution, we analyze the regularity conditions of a perturbation on a quasi-definite linear functional by the addition of Dirac delta functionals supported on N points of the unit circle or on its complement. We also deal with a new example of linear spectral transformation. We introduce a perturbation of a quasi-definite linear functional by the addition of the first derivative of the Dirac linear functional when its support is a point on the unit circle or two points symmetric with respect to the unit circle. Necessary and sufficient conditions for the quasi-definiteness of the new linear functional are obtained. Outer relative asymptotics for the new sequence of monic orthogonal polynomials in terms of the original ones are obtained. Finally, we prove that this linear spectral transform can be decomposed as an iteration of Christoffel and Geronimus linear transformations.

Generators of rational spectral transformations for nontrivial -functions

In this paper we consider transformations of sequences of orthogonal polynomials associated with a Hermitian linear functional L using spectral transformations of the corresponding C-function FL. We show that a rational spectral transformation of FL with polynomial coefficients is a finite composition of four canonical spectral transformations.

On a moment problem associated with Chebyshev polynomials

Abstract Given a sequence { μ n } n = 0 ∞ of real numbers, we find necessary and sufficient conditions for the existence and uniqueness of a distribution function ϕ on ( 1 , ∞ ) , such that μ n = ∫ 1 ∞ T n ( x ) d ϕ ( x ) , n = 0 , 1 , 2 , … . Here T n ( x ) are the Chebyshev polynomials of the first kind.

Asymptotic behaviour of Sobolev orthogonal polynomials on the unit circle

In this contribution, we deal with analytic properties of sequences of polynomials orthogonal with respect to a Sobolev-type inner product where μ is a non-trivial probability measure supported on the unit circle. We focus our attention on the outer relative asymptotics of these polynomials in terms of those associated with the measure μ. The behaviour of their zeros in terms of the parameter λ is studied in some illustrative examples.

On perturbed orthogonal polynomials on the real line and the unit circle via Szegź's transformation

By using the Szegźs transformation we deduce new relations between the recurrence coefficients for orthogonal polynomials on the real line and the Verblunsky parameters of orthogonal polynomials on the unit circle. Moreover, we study the relation between the corresponding S -functions and C -functions.

Zeros of para---orthogonal polynomials and linear spectral transformations on the unit circle

We study the interlacing properties of zeros of para–orthogonal polynomials associated with a nontrivial probability measure supported on the unit circle dµ and para–orthogonal polynomials associated with a modification of dµ by the addition of a pure mass point, also called Uvarov transformation. Moreover, as a direct consequence of our approach, we present some results related with the Christoffel transformation.

On a Direct Uvarov-Chihara Problem and Some Extensions

In this paper, we analyze a perturbation of a nontrivial probability measure dμ supported on an infinite subset on the real line, which consists on the addition of a time-dependent mass point. For the associated sequence of monic orthogonal polynomials, we study its dynamics with respect to the time parameter. In particular, we determine the time evolution of their zeros in the special case when the measure is semiclassical. We also study the dynamics of the Verblunsky coefficients, i.e., the recurrence relation coefficients of a polynomial sequence, orthogonal with respect to a nontrivial probability measure supported on the unit circle, induced from dμ through the Szegő transformation.

Zeros of Sobolev orthogonal polynomials on the unit circle

AbstractIn this contribution, we study the sequences of orthogonal polynomials with respect to the Sobolev inner product