Darboux transformations for multivariate orthogonal polynomials

Abstract

Darboux transformations for polynomial perturbations of a real multivariate measure are found. The 1D Christoffel formula is extended to the multidimensional realm: multivariate orthogonal polynomials are expressed in terms of last quasi-determinants and sample matrices. The coefficients of these matrices are the original orthogonal polynomials evaluated at a set of nodes, which is supposed to be poised. A discussion for the existence of poised sets is given in terms of algebraic hypersufaces in the complex affine space.

In this version we have not only added two more bibliographic references but also performed major changes in Section 3 on poised sets. This was motivated by our recent finding that full column rank of the Vandermonde matrix is not only necessary but sufficient. arXiv admin note: text overlap with arXiv:1409.0570

References in corpus (2)

• Matrix Orthogonal Laurent Polynomials on the Unit Circle and Toda Type Integrable Systems
• Multivariate orthogonal polynomial and integrable systems

Cited by in corpus (8)

• Multivariate orthogonal polynomial and integrable systems
• Christoffel transformations for matrix orthogonal polynomials in the real line and the non-Abelian 2D Toda lattice hierarchy
• Transformation theory and Christoffel formulas for matrix biorthogonal polynomials on the real line
• Multivariate Orthogonal Polynomials and Modified Moment Functionals
• Linear spectral transformations for multivariate orthogonal polynomials and multispectral Toda hierarchies
• CMV biorthogonal Laurent polynomials: Christoffel formulas for Christoffel and Geronimus perturbations
• CMV biorthogonal Laurent polynomials. II: Christoffel formulas for Geronimus-Uvarov perturbations
• Christoffel transformations for (partial-)skew-orthogonal polynomials and applications