arXiv subjects

Juan C. García-Ardila

Publications and source records attributed to Juan C. García-Ardila.

6 recordsLinked to original sources

arXiv2024

Generalized Gauss-Rys orthogonal polynomials

Let $(P_n(x;z;λ))_{n\geq 0}$ be the sequence of monic orthogonal polynomials with respect to the symmetric linear functional $\mathbf{s}$ defined by $$\langle\mathbf{s},p\rangle=\int_{-1}^1 p(x)(1-x^2)^{(λ-1/2)} e^{-zx^2}dx,\qquadλ>-1/2, \quad z>0.$$ In this contribution, several properties of the polynomials $P_n(x;z;λ)$ are studied taking into account the relation between the parameters of the three-term recurrence relation that they satisfy. Asymptotic expansions of these coefficients are given. Discrete Painlevé and Painlevé equations associated with such coefficients appear naturally. An electrostatic interpretation of the zeros of such polynomials as well as the dynamics of the zeros in terms of the parameters $z$ and $λ$ are given.

math.CA↗

arXiv2024

A note on Laguerre truncated polynomials and quadrature formula

In this contribution we deal with Gaussian quadrature rules based on orthogonal polynomials associated with a weight function $w(x)= x^α e^{-x}$ supported on an interval $(0,z)$, $z>0.$ The modified Chebyshev algorithm is used in order to test the accuracy in the computation of the coefficients of the three-term recurrence relation, the zeros and weights, as well as the dependence on the parameter $z.$

math.NA↗

arXiv2023

Uvarov Perturbations for Matrix Orthogonal Polynomials

Additive perturbations, specifically, matrix Uvarov transformations for matrix orthogonal polynomials, are under consideration. Christoffel-Uvarov formulas are deduced for the perturbed biorthogonal families, along with their matrix norms. These formulations are expressed in terms of the spectral jets of the Christoffel-Darboux kernels.

math.CA↗

arXiv2023

Symmetrization process and truncated orthogonal polynomials

We define the family of truncated Laguerre polynomials $P_n(x;z)$, orthogonal with respect to the linear functional $\ell$ defined by $$\langle{\ell,p\rangle}=\int_{0}^zp(x)x^αe^{-x}dx,\qquadα>-1.$$ The connection between $P_n(x;z)$ and the polynomials $S_n(x;z)$ (obtained through the symmetrization process) constitutes a key element in our analysis. As a consequence, several properties of the polynomials $P_n(x;z)$ and $S_n(x;z)$ are studied taking into account the relation between the parameters of the three-term recurrence relations that they satisfy. Asymptotic expansions of these coefficients are given. Discrete Painlevé and Painlevé equations associated with such coefficients appear in a natural way. An electrostatic interpretation of the zeros of such polynomials as well as the dynamics of the zeros in terms of the parameter $z$ are given.

math.CA↗

arXiv2021

Associated orthogonal polynomials of the first kind and Darboux transformations

Let $\un,$ be a quasi-definite linear functional defined on the space of polynomials $\mathbb{P}.$ For such a functional we can define a sequence of monic orthogonal polynomials (SMOP in short) $(P_n)_{n\geq 0},$ which satisfies a three term recurrence relation. Shifting one unity the recurrence coefficient indices given the sequence of associated polynomials of the first kind which are orthogonal with respect to a linear functional denoted by $\un^{(1)}$. In the literature two special transformations of the functional $\un$ are studied, the canonical Christoffel transformation $\widetilde \un=(x-c)\un$ and the canonical Geronimus transformation $\widehat \un=\frac{\un}{(x-c)}+M\deltan_c$ , where $c$ is a fixed complex number, $M $ is a free parameter and $\deltan_c$ is the linear functional defined on $\mathbb{P}$ as $\prodint{\deltan_c,p(x)}=p(c).$ For the Christoffel transformation with SMOP $(\widetilde P_n)_{n\geq 0}$, we are interested in analyzing the relation between the linear functionals $ \un^{(1)}$ and $\widetilde{\un}^{(1)}.$ There, the super index denotes the linear functionals associated with the orthogonal polynomial sequences of the first kind $(P_n^{(1)})_{n\geq 0}$ and $(\widetilde P_n^{(1)})_{n\geq 0},$ respectively. This problem is also studied for Geronimus transformations. Here we give close relations between their corresponding monic Jacobi matrices by using the LU and UL factorizations. To get this result, we first need to study the relation between $\un^{-1}$ (the inverse functional) and $\un^{(1)}$ which can be expressed from a quadratic Geronimus transformation.

math.CA↗

arXiv2016

Transformation theory and Christoffel formulas for matrix biorthogonal polynomials on the real line

In this paper transformations for matrix orthogonal polynomials in the real line are studied. The orthogonality is understood in a broad sense, and is given in terms of a nondegenerate continuous sesquilinear form, which in turn is determined by a quasidefinite matrix of bivariate generalized functions with a well defined support. The discussion of the orthogonality for such a sesquilinear form includes, among others, matrix Hankel cases with linear functionals, general matrix Sobolev orthogonality and discrete orthogonal polynomials with an infinite support. The first transformation considered is that of Geronimus type, two different methods are developed. A spectral one, based on the spectral properties of the perturbing polynomial, and constructed in terms of the second kind functions. Then, using spectral techniques and spectral jets, Christoffel-Geronimus formulas for the transformed polynomials and norms are presented. For this type of transformations, the paper also proposes an alternative method, which does not require of spectral techniques. A discussion on matrix spectral linear transformations is presented, as well. These transformations can be understood as rational perturbations of the matrix of bivariate generalized functions together with an addition of appropriate masses, determined these last ones by the spectral properties of the polynomial denominator. As for the Geronimus case, two techniques are applied, spectral and mixed spectral/nonspectral. Christoffel--Geronimus--Uvarov formulas are found with both approaches and some applications are given. The transformation theory is finally discussed in the context of the 2D non-Abelian Toda lattice and noncommutative KP hierarchies, understood as the theory of continuous transformations of quasidefinite sesquilinear forms. This approach allows for the finding of perturbed quasitau and Baker matrices.

math.CA↗