Multivariate orthogonal polynomials and integrable systems

Abstract

Abstract. Multivariate orthogonal polynomials in D real dimensions are considered from the perspective of the Cholesky factorization of a moment matrix. The approach allows for the construction of corresponding multivariate orthogonal polynomials, associated second kind functions, Jacobi type matrices and associated three term relations and also Christoffel-Darboux formulae. The multivariate orthogonal polynomials, its second kind functions and the corresponding Christoffel-Darboux kernels are shown to be quasi-determinants -as well as Schur complements-of bordered truncations of the moment matrix; quasi-tau functions are introduced. It is proven that the second kind functions are multivariate Cauchy transforms of the multivariate orthogonal polynomials. Discrete and continuous deformations of the measure lead to Toda type integrable hierarchy, being the corresponding flows described through Lax and Zakharov-Shabat equations; bilinear equations are found. Varying size matrix nonlinear partial difference and differential equations of the 2D Toda lattice type are shown to be solved by matrix coefficients of the multivariate orthogonal polynomials. The discrete flows, which are shown to be connected with a Gauss-Borel factorization of the Jacobi type matrices and its quasideterminants, lead to expressions for the multivariate orthogonal polynomials and its second kind functions in terms of shifted quasi-tau matrices, which generalize to the multidimensional realm those that relate the Baker and adjoint Baker functions with ratios of Miwa shifted τ -functions in the 1D scenario. In this context, the multivariate extension of the elementary Darboux transformation is given in terms of quasi-determinants of matrices built up by the evaluation, at a poised set of nodes lying in an appropriate hyperplane in R D , of the multivariate orthogonal polynomials. The multivariate Christoffel formula for the iteration of m elementary Darboux transformations is given as a quasi-determinant. It is shown, using congruences in the space of semi-infinite matrices, that the discrete and continuous flows are intimately connected and determine nonlinear partial difference-differential equations that involves only one site in the integrable lattice behaving as a Kadomstev-Petviashvili type system. Finally, a brief discussion of measures with a particular linear isometry invariance and some of its consequences for the corresponding multivariate polynomials are given. In particular, it is shown that the Toda times that preserve the invariance condition lay in a secant variety of the Veronese variety of the fixed point set of the linear isometry.

Multivariate Orthogonal Polynomials and Modified Moment Functionals

“…When dealing with the Christoffel modification, and in the case where both moment functionals, the original and the modified, are quasi-definite, some of the results are similar to those obtained in [3,4,5] using a different technique. In particular, the necessary condition in our Theorem 4.3 was proven there for an arbitrary degree polynomial, but the sufficient condition was not discussed there.…”

CMV Biorthogonal Laurent Polynomials: Perturbations and Christoffel Formulas

“…(iv) For matrix orthogonal Laurent polynomials on the unit circle, CMV orderings, and non-Abelian lattices on the circle [28]. (v) Multivariate orthogonal polynomials in several real variables and corresponding multispectral integrable Toda hierarchy [66,67]. Multivariate orthogonal polynomials on the multidimensional unit torus, the multivariate extension of the CMV ordering, and integrable Toda hierarchies [68].…”

Christoffel Transformations for Matrix Orthogonal Polynomials in the Real Line and the non-Abelian 2D Toda Lattice Hierarchy

“…ii) In [15] we dealt with the case of matrix orthogonal Laurent polynomials on the circle and CMV orderings. iii) For orthogonal polynomials in several real variables see [16,17] and [18] for orthogonal polynomials on the unit torus and the multivariate extension of the CMV ordering. It is well known that there is a deep connection between discrete integrable systems and Darboux transformations of continuous integrable systems, see for example [41].…”

“…It is well known that there is a deep connection between discrete integrable systems and Darboux transformations of continuous integrable systems, see for example [41]. Finally, let us comment that, in the realm of several variables, in [17,18,19] one can find extensions of the Christoffel formula to the multivariate scenario with real variables and on the unit torus, respectively.…”