Miguel A. Piñar

Laguerre-Sobolev orthogonal polynomials

Abstract In this paper, polynomials that are orthogonal with respect to the inner product (f,g)s=∫ +∞ 0 f(x)g(x)x α e −x dx+λ∫ +∞ 0 f′(x)g′(x)x α e −x dx 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 the zeros of these polynomials are deduced.

Sobolev orthogonality for the Gegenbauer polynomials { C n (-N+1/2) } n ≥0

Abstract In this work, we obtain the property of Sobolev orthogonality for the Gegenbauer polynomials {C n (−N+ 1 2 ) } 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 instance, we can obtain linear relations with the classical Gegenbauer polynomials.

Gegenbauer-Sobolev Orthogonal Polynomials

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 Gegenbauer polynomials Finally, some properties concerning the localization and separation of the zeros of these polynomials are deduced.

An asymptotic result for Laguerre—Sobolev orthogonal polynomials

Abstract Let { S n } denote the sequence of polynomials orthogonal with respect to the Sobolev inner product (f,g)s = ∫ 0 +∞ f(x)g(x)x α e −x d x+λ ∫ 0 +∞ f′(x)g′(x)x α e −x d x where α > − 1, λ > 0 and the leading coefficient of the S n is equal to the leading coefficient of the Laguerre polynomial L n ( α ) . Then, if x ∈C s [0,+∞), lim n→∞ S n (x) L n (α−1) (x) is a constant depending on λ.

What is beyond coherent pairs of orthogonal polynomials

Abstract 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 corresponding orthogonal polynomial sequences constitute a coherent pair.

Global properties of zeros for Sobolev-type orthogonal polynomials

Perez, T.E. and M.A. Pifiar, Global properties of zeros for Sobolev-type orthogonal polynomials, Journal of Computational and Applied Mathematics 49 (1993) 22.5-232. In this paper we analyze some properties concerning the zeros of orthogonal polynomials Q,(x), associated to the inner product (f, g) = /,f(xlg(x) dp(x1-t Mf(c)g(c)+ Nf’(c)g’(c), where I is a (not necessarily bounded) real interval, p is a positive measure on I, c E [w and M, N 2 0. In particular, some properties concerning the localization and separation for the roots of Q,(x) are obtained.

Classical orthogonal polynomials in two variables: a matrix approach ∗

Abstract 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 differential equation.

On Koornwinder classical orthogonal polynomials in two variables

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 polynomials on the unit ball, on the simplex or the tensor product of Jacobi polynomials in one variable, but the remaining cases are not considered classical by other authors. The definition of classical orthogonal polynomials considered in this work provides a different perspective on the subject. We analyze in detail Koornwinder polynomials and using the Koornwinder tools, new examples of orthogonal polynomials in two variables are given.

Krall-type orthogonal polynomials in several variables

For a bilinear form obtained by adding a Dirac mass to a positive definite moment functional defined in the linear space of polynomials 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 classical orthogonal polynomials.

Orthogonal polynomials in two variables as solutions of higher order partial differential equations

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 as well as some examples are studied.