Linear spectral transformations for multivariate orthogonal polynomials and multispectral Toda hierarchies

Abstract

Linear spectral transformations of orthogonal polynomials in the real line, and in particular Geronimus transformations, are extended to orthogonal polynomials depending on several real variables. Multivariate Christoffel-Geronimus-Uvarov formulae for the perturbed orthogonal polynomials and their quasi-tau matrices are found for each perturbation of the original linear functional. These expressions are given in terms of quasi-determinants of bordered truncated block matrices and the 1D Christoffel-Geronimus-Uvarov formulae in terms of quotient of determinants of combinations of the original orthogonal polynomials and their Cauchy transforms, are recovered. A new multispectral Toda hierarchy of nonlinear partial differential equations, for which the multivariate orthogonal polynomials are reductions, is proposed. This new integrable hierachy is associated with non-standard multivariate biorthogonality. Wave and Baker functions, linear equations, Lax and Zakharov-Shabat equations, KP type equations, appropriate reductions, Darboux/linear spectral transformations, and bilinear equations involving linear spectral transformations are presented.

A better treatment of generalized functions framework is implemented. An Appendix discussing 0D and 1D Uvarov transformations for the multivariate scenario is also included

References in corpus (5)

• Matrix Orthogonal Laurent Polynomials on the Unit Circle and Toda Type Integrable Systems
• Multivariate orthogonal polynomial and integrable systems
• Darboux transformations for multivariate orthogonal polynomials
• Multivariate Orthogonal Polynomials and Modified Moment Functionals
• On linearly related orthogonal polynomials in several variables

Cited by in corpus (4)

• 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