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## Publications

Publications (72)

We study families of multivariate orthogonal polynomials with respect to the symmetric weight function in d variables Bγ(x)=∏i=1dω(xi)∏i-1, where ω(t) is an univariate weight function in t∈(a,b) and x=(x1,x2,…,xd) with xi∈(a,b). Applying the change of variables xi,i=1,2,…,d, into ur,r=1,2,…,d, where ur is the r-th elementary symmetric function, we...

Orthogonal polynomials with respect to the weight function $w_{\beta,\gamma}(t) = t^\beta (1-t)^\gamma$, $\gamma > -1$, on the conic surface $\{(x,t): \|x\| = t, \, x \in \mathbb{R}^d, \, t \le 1\}$ are studied recently, and they are shown to be eigenfunctions of a second order differential operator $\mathcal{D}_\gamma$ when $\beta =-1$. We extend...

The so-called Koornwinder bivariate orthogonal polynomials are generated by means of a non-trivial procedure involving two families of univariate orthogonal polynomials and a function ρ(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usep...

Sobolev orthogonal polynomials of d variables on the product domain \(\Omega :=[a_1,b_1]\times \cdots \times [a_d,b_d]\) with respect to the inner product $$\begin{aligned} \left\langle f,g\right\rangle _S= c\int _\Omega \nabla ^\kappa f({\mathbf {x}})\cdot \nabla ^\kappa g({\mathbf {x}})W({\mathbf {x}}){\mathrm{d}}{\mathbf {x}}+ \sum _{i=0}^{\kapp...

In this work, a Sobolev inner product on the unit ball of ℝd involving the outward normal derivative is considered. A basis of mutually orthogonal polynomials associated with this inner product is constructed in terms of spherical harmonics and a radial part obtained from a family of univariate polynomials orthogonal with respect to a Sobolev inner...

Given a linear functional u in the linear space of polynomials in two variables with real coefficients and a polynomial λ(x,y), in this contribution we deal with Geronimus transformations of u, i.e., those linear functionals v such that u=λ(x,y)v. The connection formulas between the sequences of bivariate orthogonal polynomials with respect to u an...

Coherent pairs of measures were introduced in 1991 and constitute a very useful tool in the study of Sobolev orthogonal polynomials on the real line. In this work, coherence and partial coherence in two variables appear as the natural extension of the univariate case. Given two families of bivariate orthogonal polynomials expressed as polynomial sy...

We present a family of mutually orthogonal polynomials on the unit ball with respect to an inner product which includes a mass uniformly distributed on the sphere. First, connection formulas relating these multivariate orthogonal polynomials and the classical ball polynomials are obtained. Then, using the representation formula for these polynomial...

We deduce new characterizations of bivariate classical orthogonal polynomials associated with a quasi-definite moment functional, and we revise old properties for these polynomials. More precisely, new characterizations of classical bivariate orthogonal polynomials satisfying a diagonal Pearson-type equation are proved: they are solutions of two se...

We consider Koornwinder's method for constructing orthogonal polynomials in
two variables from orthogonal polynomials in one variable. If semiclassical
orthogonal polynomials in one variable are used, then Koornwinder's
construction generates semiclassical orthogonal polynomials in two variables.
We consider two methods for deducing matrix Pearson...

We study matrix three term relations for orthogonal polynomials in two variables constructed from orthogonal polynomials in one variable. Using the three term recurrence relation for the involved univariate orthogonal polynomials, the explicit expression for the matrix coefficients in these three term relations are deduced. These matrices are diago...

Let $E_n(f)_\mu$ be the error of best approximation by polynomials of degree at most $n$ in the space $L^2(\varpi_\mu, \mathbb{B}^d)$, where $\mathbb{B}^d$ is the unit ball in $\mathbb{R}^d$ and $\varpi_\mu(x) = (1-\|x\|^2)^\mu$ for $\mu > -1$. Our main result shows that, for $s \in \mathbb{N}$, $$ E_n(f)_\mu \le c n^{-2s}[E_{n-2s}(\Delta^s f)_{\mu...

Multivariate orthogonal polynomials can be introduced by using a moment functional defined on the linear space of polynomials in several variables with real coefficients. We study the so--called Uvarov and Christoffel modifications obtained by adding to the moment functional a finite set of mass points, or by multiplying it times a polynomial of to...

We analyse a family of mutually orthogonal polynomials on the unit ball with
respect to an inner product which involves the outward normal derivatives on
the sphere. Using their representation in terms of spherical harmonics,
algebraic and analytic properties will be deduced. First, we deduce explicit
connection formulas relating classical multivar...

The purpose of this work is to analyse a family of mutually orthogonal
polynomials on the unit ball with respect to an inner product which includes an
additional term on the sphere. First, we will get connection formulas relating
classical multivariate orthogonal polynomials on the ball with our family of
orthogonal polynomials. Then, using the rep...

Orthogonal polynomials on the product domain $[a_1,b_1] \times [a_2,b_2]$
with respect to the inner product $$
\langle f,g \rangle_S = \int_{a_1}^{b_1} \int_{a_2}^{b_2} \nabla f(x,y)\cdot
\nabla g(x,y)\, w_1(x)w_2(y)
\,dx\, dy + \lambda f(c_1,c_2)g(c_1,c_2) $$ are constructed, where $w_i$ is a
weight function on $[a_i,b_i]$ for $i = 1, 2$, $\lambda...

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 usi...

Classical orthogonal polynomials can be characterized in terms of the corresponding Stieltjes function.We consider the construction of a Stieltjes function in terms of the falling factorials for discrete classical orthogonal polynomials (Charlier, Krawtchouk, Meixner, and Hahn). This Stieltjes function associated with classical orthogonal polynomia...

Multivariate orthogonal polynomials associated with a Sobolev-type inner product, that is, an inner product defined by adding the evaluation of derivatives at several points to a measure, are studied. Orthogonal polynomials and kernel functions associated with this new inner product can be explicitly expressed in terms of those corresponding to the...

For the weight function $W_\mu(x) = (1-|x|^2)^\mu$, $\mu > -1$, $\lambda > 0$
and $b_\mu$ a normalizing constant, a family of mutually orthogonal polynomials
on the unit ball with respect to the inner product $$
\la f,g \ra = {b_\mu [\int_{\BB^d} f(x) g(x) W_\mu(x) dx +
\lambda \int_{\BB^d} \nabla f(x) \cdot \nabla g(x) W_\mu(x) dx]} $$ are
constru...

In 1975, Tom Koornwinder studied examples of two variable analogues of the Jacobi polynomials in two variables. Those orthogonal polynomials are eigenfunctions of two commuting and algebraically independent partial differential operators. Some of these examples are well known classical orthogonal polynomials in two variables, such as orthogonal pol...

We present a Uvarov modification of the two variable classical measure on the unit disk by adding a finite set of equally spaced mass points on the border. In such a case, both orthogonal polynomials and reproducing kernels associated with this new measure can be explicitly expressed in terms of those corresponding with the classical one. Then, asy...

Multivariate orthogonal polynomials associated with a Sobolev-type inner product, that is, an inner product defined by adding to a measure the evaluation of the gradients in a fixed point, are studied. Orthogonal polynomials and kernel functions associated with this new inner product can be explicitly expressed in terms of those corresponding with...

Classical orthogonal polynomials in two variables can be characterized as the polynomial solutions of a second order partial differential equation involving polynomial coefficients. We study orthogonal polynomials in two variables which satisfy higher order partial differential equations. In particular, fourth order partial differential equations a...

Tesis Univ. de Granada. Departamento de Matemática Aplicada. Leída el 12 de junio de 1992

Sobolev orthogonal polynomials in two variables are defined via inner products involving gradients. Such a kind of inner product appears in connection with several physical and technical problems. Matrix second-order partial differential equations satisfied by Sobolev orthogonal polynomials are studied. In particular, we explore the connection betw...

Let dν be a measure in ℝd
obtained from adding a set of mass points to another measure dμ. Orthogonal polynomials in several variables associated with dν can be explicitly expressed in terms of orthogonal polynomials associated with dμ, so are the reproducing kernels associated with these polynomials. The explicit formulas that are obtained are fur...

Szegő type polynomials with respect to a linear functional M for which the moments M[tn]=μ−n are all complex, μ−n=μn and Dn≠0 for n⩾0, are considered. Here, Dn are the associated Toeplitz determinants. Para-orthogonal polynomials are also studied without relying on any integral representation. Relation between the Toeplitz determinants of two diffe...

In this contribution we present a survey concerning orthogonal polynomials in several variables. We emphasize two questions in order to do a comparison with the one variable case. First, according to different orderings in the monomial basis {x n y m } n,m∈ℕ , we analyze the existence of recurrence relations for the corresponding sequences of ortho...

Classical orthogonal polynomials in two variables can be characterized as the polynomial solutions of a matrix second-order partial differential equation involving matrix polynomial coefficients. In this work, we study classical orthogonal polynomials in two variables whose partial derivatives satisfy again a second-order partial differential equat...

Classical orthogonal polynomials in one variable can be characterized as the only orthogonal polynomials satisfying a Rodrigues formula. In this paper, using the second kind Kronecker power of a matrix, a Rodrigues formula is introduced for classical orthogonal polynomials in two variables.

For a bilinear form obtained by adding a Dirac mass to a positive definite moment functional in several variables, explicit formulas of orthogonal polynomials are derived from the orthogonal polynomials associated with the moment functional. Explicit formula for the reproducing kernel is also derived and used to establish certain inequalities for c...

Orthogonal polynomials of degree $n$ with respect to the weight function $W_\mu(x) = (1-\|x\|^2)^\mu$ on the unit ball in $\RR^d$ are known to satisfy the partial differential equation $$ [ \Delta - \la x, \nabla \ra^2 - (2 \mu +d) \la x, \nabla \ra \right ] P = -n(n+2 \mu+d) P $$ for $\mu > -1$. The singular case of $\mu = -1,-2, ...$ is studied i...

In this work, semiclassical orthogonal polynomials in two variables are defined as the orthogonal polynomials associated with a quasi definite linear functional satisfying a matrix Pearson-type differential equation. Semiclassical functionals are characterized by means of the analogue of the structure relation in one variable. Moreover, non trivial...

In this paper, we consider bivariate orthogonal polynomials associated with a quasi-definite moment functional which satisfies
a Pearson-type partial differential equation. For these polynomials differential properties are obtained. In particular, we
deduce some structure and orthogonality relations for the successive partial derivatives of the pol...

In this paper we deal with a family of nonstandard polynomials orthogonal with respect to an inner product involving differences.
This type of inner product is the so-called Δ-Sobolev inner product. Concretely, we consider the case in which both measures appearing in the inner product correspond
to the Pascal distribution (the orthogonal polynomial...

In this work, we introduce the classical orthogonal polynomials in two variables as the solutions of a matrix second order partial differential equation involving matrix polynomial coefficients, the usual gradient operator, and the divergence operator. Here we show that the successive gradients of these polynomials also satisfy a matrix second orde...

Differential properties for orthogonal polynomials in several variables are studied. We consider multivariate orthogonal polynomials whose gradients satisfy some quasi--orthogonality conditions. We obtain several characterizations for these polynomials including the analogous of the semiclassical Pearson differential equation, the structure relatio...

Classical orthogonal polynomials in two variables are defined as the orthogonal polynomials associated to a two-variable moment functional satisfying a matrix analogue of the Pearson differential equation. Furthermore, we characterize classical orthogonal polynomials in two variables as the polynomial solutions of a matrix second order partial diff...

Orthogonal polynomials in two variables constitute a very old subject in approximation theory. Usually they are studied as solutions of second-order partial differential equations. In this work, we study two-variable orthogonal polynomials associated with a moment functional satisfying the two-variable analogue of the Pearson differential equation....

Let {Sn} denote the sequence of polynomials orthogonal with respect to the Sobolev inner productwhere α>−1,λ>0 and the leading coefficient of the Sn is equal to the leading coefficient of the Laguerre polynomial Ln(α). In this work, a generating function for the Sobolev–Laguerre polynomials is obtained.

We study the family of polynomials which are orthogonal with respect to the discrete inner product involving difference operators: (f,g) Δ (K) =∑ m,k=0 K 〈λ m,k u,Δ m fΔ k g〉,K≥0, where u is a positive definite Δ-semiclassical discrete functional and λ m,k are polynomials, for m, k=0,1,⋯,K such that, Λ (K) =(λ m,k (x)) m,k=0 K is a symmetric and po...

In this paper we study a Jacobi block matrix and the behavior of the limit of its entries when a perturbation of its spectral matrix measure by the addition of a Dirac delta matrix measure is introduced.

Let Qn be the polynomials orthogonal with respect to the Sobolev inner product (f; g)S = fg d0 + f g d1; being (0; ;1) a coherent pair where one of the measures is the Hermite measure. The outer relative asymptotics for Qn with respect to Hermite polynomials are found. On the other hand, we consider the Sobolev scaled polynomials and we obtain the...

The asymptotic behavior of γn(dβ) γn(dα)−1 and Pn(x, dβ) P−1n(x, dα) is studied. Here (γn(.))n are the leading coefficients of the orthonormal matrix polynomials Pn(x, .) with respect to the matrix measures dβ and dα which are related by dβ(u)=dα(u)+∑Nk=1 Mk δ(u−ck), where Mk are positive definite matrices, δ is the Dirac measure and ck lies outsid...

In this work, we study algebraic and analytic properties for the polynomials { Q
n
}
n
( p,q )sò - ¥ + ¥ ( p,p¢ ) ( \frac1mml )( \fracqq¢ )e - x2 dx \left( {p,q} \right)s\int_{ - \infty }^{ + \infty } {\left( {p,p'} \right)} \left( {\frac{{1{\mu }}}{{{\mu \lambda }}}} \right)\left( {\frac{{q}}{{{q'}}}} \right){e}^{{ - x}^{2} {dx}}
where , R suc...

In this paper, we study orthogonal polynomials with respect to the bilinear form (f, g)
S
= V(f) A
V(g)
T
+ u, f
(N)
g
(N)where V(f) =(f(c
0), f "(c
0), ..., f
(n – 1)
0(c
0), ..., f(c
p
), f "(c
p
), ..., f
(n – 1)
p(c
p
))
u is a regular linear functional on the linear space P of real polynomials, c
0, c
1, ..., c
p
are distinct real numbers,...

Given the Sobolev bilinear form (f, g)
S
=>u
0, f
g< +="">u
1, f " g "u
0 and u
1 linear functionals, a characterization of the linear second–order differential operators with polynomial coefficients, symmetric with respect to (, )
S
in terms of u
0 and u
1 is obtained. In particular, several interesting functionals u
0 and u
1 are considered, reco...

Let Snn denote a sequence of polynomials orthogonal with respect to the Sobolev inner productf,gS=∫fxgxdψ0x+λ∫f′xg′xdψ1x,where λ>0 and {dψ0,dψ1} is a so-called coherent pair with at least one of the measures dψ0 or dψ1 a Laguerre measure. We investigate the asymptotic behaviour of Sn(x) outside the supports of dψ0 and dψ1, and n→+∞.

Let {S-n}(n) denote a sequence of polynomials orthogonal with respect to the Sobolev inner product (f,g)s = integral f(x)g(x) d psi(0)(x) + lambda integral f'(x)g'(x) d psi(1)(x) where lambda > 0 and (d psi(0), d psi(1)) is a so-called coherent pair with at least one of the measures d psi(0) or d psi(1) a Jacobi measure. We investigate the asymptot...

In this paper, we study orthogonal polynomials with respect to the bilinear form (Equation Presented) where u is a quasi-definite (or regular) linear functional on the linear space ℙ of real polynomials, c0, c 1,..., cN-1 are distinct real numbers, N is a positive integer number, and A is a real N x N matrix such that each of its principal submatri...

In this work, we obtain a non-standard orthogonality property for Meixner polynomials {Mn(γ,μ)}n≥0, with 0 < μ < 1 and γ ∈ ℝ, that is, we show that they are orthogonal with respect to some discrete inner product involving difference operators. The non-standard orthogonality can be used to recover properties of these Meixner polynomials, e. g., line...

In this work, we obtain the property of Sobolev orthogonality for the Gegenbauer polynomials {Cn(−N+12)}n⩾0, with N⩾1 a given nonnegative integer, that is, we show that they are orthogonal with respect to some inner product involving derivatives. The Sobolev orthogonality can be used to recover properties of these Gegenbauer polynomials. For instan...

Strong asymptotics for the sequence of monic polynomials Q(n)(z), orthogonal with respect to the inner product (f,g)(s) = integral (f(x)g(x) d mu(1)(x) + lambda integral f'(x) g'(x) d mu(2)(x), lambda>0, with z outside of the support of the measure mu(2), is established under the additional assumption that mu(1) and mu(2) form a so-called coherent...

Let {Sn} denote the sequence of polynomials orthogonal with respect to the Sobolev inner product where α > − 1, λ > 0 and the leading coefficient of the Sn is equal to the leading coefficient of the Laguerre polynomial Ln(α). Then, if x∈Cß[0,+∞), is a constant depending on λ.

The orthogonality of the generalized Laguerre polynomials, {L(n)((alpha))(x)} (n greater than or equal to 0), is a well known fact when the parameter alpha is a real number but not a negative integer. In fact, for -1 <alpha, they are orthogonal on the interval [0 + infinity) with respect to the weight function rho(x) = x(alpha)e(-x), and for alpha...

In this paper, polynomials that are orthogonal with respect to the inner product where α > −1 and λ⩾0, are studied. For these nonstandard orthogonal polynomials algebraic and differential properties as well as the relation with the classical Laguerre polynomials are obtained. Finally, some properties concerning the localization and separation of th...

In this paper, we study orthogonal polynomials with respect to the inner product (f, g)(S)((N)) = [u, fg] + Sigma(m=1)(N) lambda(m)[u, f((m))g((m))], where lambda(m) greater than or equal to 0 for m = 1,..., N, and u is a semiclassical, positive definite linear functional. For these non-standard orthogonal polynomials, algebraic and differential pr...

Usually, coherent pairs of orthogonal polynomials have been considered in the wider context of Sobolev orthogonality. In this paper, we focus our attention on the problem of coherence between two orthogonal polynomial sequences in terms of the corresponding linear functionals. We deduce some conditions about the linear functionals in order that the...

In this paper, orthogonal polynomials in the Sobolev space W
1,2([-1,1], p
(α),λ p
(α)), where \({\rho ^{(\alpha )}} = {(1 - {x^2})^{\alpha - \frac{1}{2}}},\alpha >- \frac{1}{2}\) and λ ≥ 0, are studied. For these non-standard orthogonal polynomials algebraic and differential properties are obtained, as well as the relation with the classical Gegen...

In this paper we analyze some properties concerning the zeros of orthogonal polynomials Qn(x), associated to the inner product , where I is a (not necessarily bounded) real interval, μ is a positive measure on I,cϵ and M, N ⩾ 0. In particular, some properties concerning the localization and separation for the roots of Qn(x) are obtained.

Matrix relations for orthogonal polynomials associated to a non-definite linear functional c are found. In addition, the reproducing kernels for the functional c are introduced and the coefficients of the Gaussian quadrature formulas for the functional c are also calculated by means of a procedure similar to that developed by Brezinski for definite...

Classical orthogonal polynomials in one variable can be defined as the orthogonal polynomials associated to a moment functional satisfying a Pearson differential equation. We extend this concept to several variables defining classical multivariate orthogonal polynomials as those associated to a moment functional satisfying a matrix analogue of the...

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