Multivariate Orthogonal Polynomials and Modified Moment Functionals

Abstract

Abstract. 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.

Perturbations of Christoffel–Darboux Kernels: Detection of Outliers

“…The latter addresses measures on the unit circle, but the same applies to orthogonality on the real line. In case of several variables, quite a few authors have considered classical measures µ with explicit formulas for the corresponding orthogonal polynomials like Jacobitype measures on the unit ball, and σ being one [11] or several mass points [12], or the Lebesgue measure on the sphere [30, Theorem 3.3].…”

Bivariate orthogonal polynomials, 2D Toda lattices and Lax-type pairs

“…We consider the functional v obtained from a polynomial modification of a reflexive functional u, given by v, p(x, y) = λ(x, y)u, p(x, y) = u, λ(x, y) p(x, y) , for p(x, y) ∈ Π. The Christoffel modification on a multivariate functional, using multiplication by a polynomial of degree 2, was studied in [10].…”

“…From [10,Th. 4.1], it is known that, for n 1, there exist real matrices R n and S n of respective sizes (n + 1) × n and n × (n − 1), with S 2 ≡ 0, such that…”

“…We now verify that the matrices R n and S n are centrosymmetric. First we denote R n = (r Comparing expressions (6.4) and (6.6), and using that Q n is reflexive polynomial vector, we have r Conversely, from [10,Th. 4.3], consider {Q n } n 0 a MOPS associated with a moment functional u and two sequences of matrices {R n } n 1 and {S n } n 2 of size (n + 1) × n and (n + 1) × (n − 1) respectively, with S 2 ≡ 0.…”

Fourth order partial differential equations for Krall-type orthogonal polynomials on the triangle

“…In [4], the authors presented 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. A general framework of Uvarov modifications in the multivariate case was studied in [5].…”