Luis E. Garza

Orthogonal polynomials and measures on the unit circle. The Geronimus transformations

In this paper we analyze a perturbation of a nontrivial positive measure supported on the unit circle. This perturbation is the inverse of the Christoffel transformation and is called the Geronimus transformation. We study the corresponding sequences of monic orthogonal polynomials as well as the connection between the associated Hessenberg matrices. Finally, we show an example of this kind of transformation.

Spectral transformations of measures supported on the unit circle and the Szegő transformation

In this paper we analyze spectral transformations of measures supported on the unit circle with real moments. The connection with spectral transformations of measures supported on the interval [−1,1] using the Szegő transformation is presented. Some numerical examples are studied.

Szegő transformations and rational spectral transformations for associated polynomials

In this contribution we analyze rational spectral transformations related to associated polynomials with respect to probability measures supported on the interval [-1, 1]. The connection with rational spectral transformations of measures supported on the unit circle using the Szego transformation is presented.

Moment perturbation of matrix polynomials

We consider a sequence of perturbed matrix orthogonal polynomials caused by some perturbation on the moments. We find an explicit relation between the perturbed polynomials and the original ones. We also analyse the perturbed second kind polynomials via its associated orthogonal polynomials and the non-perturbed second kind polynomials.

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.

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.

Szegő transformations and Nth order associated polynomials on the unit circle

In this paper we analyze the Stieltjes functions defined by the Szego inverse transformation of a nontrivial probability measure supported on the unit circle such that the corresponding sequence of orthogonal polynomials is defined by either backward or forward shifts in their Verblunsky parameters. Such polynomials are called anti-associated (respectively associated) orthogonal polynomials. Thus, rational spectral transformations appear in a natural way.

Jacobi–Sobolev-type orthogonal polynomials: holonomic equation and electrostatic interpretation – a non-diagonal case

Consider the Sobolev-type inner product where p and q are polynomials with real coefficients, α, β>−1, ℙ(x)=(p(x), p′(x)) t , and is a positive semidefinite matrix, with M 0, M 1≥0, and λ∈ℝ. We obtain an expression for the family of polynomials , orthogonal with respect to the above inner product, a connection formula that relates with some family of Jacobi polynomials and the holonomic equation that they satisfy, as well as an electrostatic interpretation of their zeros.

A higher order Sobolev-type inner product for orthogonal polynomials in several variables

We consider polynomials in several variables orthogonal with respect to a Sobolev-type inner product, obtained from adding a higher order gradient evaluated in a fixed point to a standard inner product. An expression for these polynomials in terms of the orthogonal family associated with the standard inner product is obtained. A particular case using polynomials in the unit ball is analyzed, and some asymptotic results are derived.