Orthogonal Laurent polynomials on the unit circle, extended CMV ordering and 2D Toda type integrable hierarchies

Álvarez-Fernández, Carlos; Mañas, Manuel

doi:10.1016/j.aim.2013.02.020

Cited by 30 publications

(37 citation statements)

References 59 publications

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“…In [9] we gave a complete study in terms of factorization for multiple orthogonal polynomials of mixed type and characterized the integrable systems associated to them. Then, we studied Laurent orthogonal polynomials in the unit circle trough the CMV approach in [10] and find in [11] the Christoffel-Darboux formula for generalized orthogonal matrix polynomials. These methods where further extended, for example we gave an alternative Christoffel-Darboux formula for mixed multiple orthogonal polynomials [12] or developed the corresponding theory of matrix Laurent orthogonal polynomials in the unit circle and its associated Toda type hierarchy [13].…”

Section: Introductionmentioning

confidence: 99%

Christoffel transformations for multivariate orthogonal polynomials

Ariznabarreta

Mañas

2018

Journal of Approximation Theory

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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.1991 Mathematics Subject Classification. 14J70,15A23,33C45,37K10,37L60,42C05,46L55.

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Section: Introductionmentioning

confidence: 99%

Christoffel transformations for multivariate orthogonal polynomials

Ariznabarreta

Mañas

2018

Journal of Approximation Theory

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“…In [10] we gave a complete study in terms of factorization for multiple orthogonal polynomials of mixed type and characterized the integrable systems associated to them. Then, we studied Laurent orthogonal polynomials in the unit circle trough the CMV approach in [11] and find in [12] the Christoffel-Darboux formula for generalized orthogonal matrix polynomials. These methods where further extended, for example we gave an alternative Christoffel-Darboux formula for mixed multiple orthogonal polynomials [13] or developed the corresponding theory of matrix Laurent orthogonal polynomials in the unit circle and its associated Toda type hierarchy [14].…”

Section: 2mentioning

confidence: 99%

Multivariate orthogonal polynomials and integrable systems

Ariznabarreta

Mañas

2016

Advances in Mathematics

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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.

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“…Another pathology of admissible products of infinite matrices is that, in contrast to the case of finite matrices, the existence of inverse can be consistent with a non-trivial left or right kernel. This is illustrated by the matrix T given in (2) which, despite having the matrix (1) as an inverse, satisfies…”

Section: Hessenberg Type Matrices Zig-zag Bases and Darboux Factorizmentioning

confidence: 99%

“…These topics are also closely related to the unitary counterpart of Jacobi, the CMV matrices, which date back to works on the unitary eigenproblem [4,8,81], a decade before their rediscovery in the context of orthogonal polynomials on the unit circle [11,70,71]. It was later realized that CMV matrices provide the Lax pair of integrable systems known under the name of Schur flows (discrete mKdV and unitary analogue of Toda) and Ablowitz-Ladik (discrete nonlinear Schrödinger) [2,3,22,30,32,52,54,63,64,65,73]. Yet, Darboux has not been applied to CMV matrices so far, the closest precedents being on related issues for isometric Hessenberg matrices [16,25,27,34,42,80].…”

Section: Introductionmentioning

confidence: 99%

Darboux transformations for CMV matrices

Cantero

Marcellán

Ituarte

et al. 2016

Advances in Mathematics

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We develop a theory of Darboux transformations for CMV matrices, canonical representations of the unitary operators. In perfect analogy with their self-adjoint version -the Darboux transformations of Jacobi matrices -they are equivalent to Laurent polynomial modifications of the underlying measures. We address other questions which emphasize the similarities between Darboux transformations for Jacobi and CMV matrices, like their (almost) isospectrality or the relation that they establish between the corresponding orthogonal polynomials, showing also that both transformations are connected by the Szegő mapping.Nevertheless, we uncover some features of the Darboux transformations for CMV matrices which are in striking contrast with those of the Jacobi case. In particular, when applied to CMV matrices, the matrix realization of the inverse Darboux transformations -what we call 'Darboux transformations with parameters' -leads to spurious solutions whose interpretation deserves future research. Such spurious solutions are neither unitary nor band matrices, so Darboux transformations for CMV matrices are much more subject to the subtleties of the algebra of infinite matrices than their Jacobi counterparts.A key role in our theory is played by the Cholesky factorizations of infinite matrices. Actually, the Darboux transformations introduced in this paper are based on the Cholesky factorizations of degree one Hermitian Laurent polynomials evaluated on CMV matrices. These transformations are also generalized to higher degree Laurent polynomials, as well as to the extension of CMV matrices to quasi-definite functionals -what we call 'quasi-CMV' matrices.Furthermore, we show that this CMV version of Darboux transformations plays a role in integrable systems like the Schur flows or the Ablowitz-Ladik model which parallels that of Darboux for Jacobi matrices in the Toda lattice.

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Orthogonal Laurent polynomials on the unit circle, extended CMV ordering and 2D Toda type integrable hierarchies

Cited by 30 publications

References 59 publications

Christoffel transformations for multivariate orthogonal polynomials

Christoffel transformations for multivariate orthogonal polynomials

Multivariate orthogonal polynomials and integrable systems

Darboux transformations for CMV matrices

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