Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 090, 25 pages      arXiv:1601.07194      https://doi.org/10.3842/SIGMA.2016.090
Contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications

Multivariate Orthogonal Polynomials and Modified Moment Functionals

Antonia M. Delgado, Lidia Fernández, Teresa E. Pérez and Miguel A. Piñar
IEMath - Math Institute and Department of Applied Mathematics, University of Granada, 18071, Granada, Spain

Received January 28, 2016, in final form September 05, 2016; Published online September 10, 2016

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.

Key words: multivariate orthogonal polynomials; moment functionals; Christoffel modification; Uvarov modification; ball polynomials.

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