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.
Cite
@article{arxiv.1503.04786,
title = {Darboux transformations for multivariate orthogonal polynomials},
author = {Gerardo Ariznabarreta and Manuel Mañas},
journal= {arXiv preprint arXiv:1503.04786},
year = {2019}
}
Comments
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