Lidia Fernández

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.

Orthogonal polynomials in several variables for measures with mass points

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 further specialized in the case of Jacobi measure on the simplex, with mass points added on the vertices, which are then used to study the asymptotics kernel functions for dν.

A matrix Rodrigues formula for classical orthogonal polynomials in two variables

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.

Sobolev orthogonal polynomials on product domains

Orthogonal polynomials on the product domain a 1 , b 1 × a 2 , b 2 with respect to the inner product { f , g } S = ? a 1 b 1 ? a 2 b 2 ? f ( x , y ) ? ? g ( x , y ) w 1 ( x ) w 2 ( y ) d x d y + λ 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 , λ 0 , and ( c 1 , c 2 ) is a fixed point. The main result shows how an orthogonal basis for such an inner product can be constructed for certain weight functions, in particular, for product Laguerre and product Gegenbauer weight functions, which serve as primary examples.

Multivariate Orthogonal Polynomials and Modified Moment Functionals

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 total degree 2, respectively. Orthogonal polynomials associated with modified moment functionals will be studied, as well as the impact of the modification in useful properties of the orthogonal polynomials. Finally, some illustrative examples will be given.

Bivariate orthogonal polynomials in the Lyskova class

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 equation of the same type.

New steps on Sobolev orthogonality in two variables

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 between the coefficients of the second-order partial differential operator and the moment functionals defining the Sobolev inner product. Finally, some old and new examples are given.