A note on the Geronimus transformation and Sobolev orthogonal polynomials

Derevyagin, Maxim; Marcellán, Francisco

doi:10.1007/s11075-013-9788-6

Cited by 29 publications

(38 citation statements)

References 23 publications

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“…Theorem 2 extends previous results corresponding to the case d = 1. More specifically, the following corollary is an immediate consequence, which was proved in [5] with other arguments.…”

Section: Introductionmentioning

confidence: 61%

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Geronimus transformations for sequences of d-orthogonal polynomials

Rolanía

García-Ardila

2019

RACSAM

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“…Theorem 2 extends previous results corresponding to the case d = 1. More specifically, the following corollary is an immediate consequence, which was proved in [5] with other arguments.…”

Section: Introductionmentioning

confidence: 61%

“…. , d. One of our goals is to characterize the regularity of the new vectors of functionals verifying (3)-(5), which is done in the following result.…”

mentioning

confidence: 99%

Geronimus transformations for sequences of d-orthogonal polynomials

Rolanía

García-Ardila

2019

RACSAM

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“…Proof. The implications piq ñ piiiq and piq ñ pivq are already proved because we know that, up to a positive rescaling, piq gives the relation (19) between χ and ω, and A has the structure (15).…”

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confidence: 92%

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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“…Uvarov [9] found Christoffel type formulas, and the addition of a finite number of Dirac masses to a linear functional appears in the framework of the spectral analysis of fourth-order linear differential operators with polynomial coefficients and with orthogonal polynomials as eigenfunctions. Geronimus perturbations of degree two of scalar bilinear forms have been very recently treated in [12] and in the general case in [13].…”

Section: Introductionmentioning

confidence: 99%

CMV Biorthogonal Laurent Polynomials: Perturbations and Christoffel Formulas

Ariznabarreta¹,

Mañas²,

Toledano³

2018

Stud Appl Math

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Quasidefinite sesquilinear forms for Laurent polynomials in the complex plane and corresponding CMV biorthogonal Laurent polynomial families are studied. Bivariate linear functionals encompass large families of orthogonalities such as Sobolev and discrete Sobolev types. Two possible Christoffel transformations of these linear functionals are discussed. Either the linear functionals are multiplied by a Laurent polynomial, or are multiplied by the complex conjugate of a Laurent polynomial. For the Geronimus transformation, the linear functional is perturbed in two possible manners as well, by a division by a Laurent polynomial or by a complex conjugate of a Laurent polynomial, in both cases the addition of appropriate masses (linear functionals supported on the zeros of the perturbing Laurent polynomial) is considered. The connection formulas for the CMV biorthogonal Laurent polynomials, its norms, and Christoffel-Darboux kernels, in all the four cases, are given. For the Geronimus transformation, the connection formulas for the second kind functions and mixed Christoffel-Darboux kernels are also given in the two possible cases. For prepared Laurent polynomials, i.e., of the form L(z) = L n z n + · · · + L −n z −n , L n L −n = 0, these connection formulas lead to quasideterminantal (quotient of determinants) Christoffel formulas for all the four transformations, expressing an arbitrary degree perturbed biorthogonal Laurent polynomial in terms of 2n unperturbed biorthogonal Laurent polynomials, their second kind functions or Christoffel-Darboux kernels and its mixed versions. Different curves are presented as examples, such as the real line, the circle, the Cassini oval,

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A note on the Geronimus transformation and Sobolev orthogonal polynomials

Abstract: In this note we recast the Geronimus transformation in the framework of polynomials orthogonal with respect to symmetric bilinear forms. We also show that the double Geronimus transformations lead to non-diagonal Sobolev type inner products.

Cited by 29 publications

References 23 publications

Geronimus transformations for sequences of d-orthogonal polynomials

Geronimus transformations for sequences of d-orthogonal polynomials

Darboux transformations for CMV matrices

CMV Biorthogonal Laurent Polynomials: Perturbations and Christoffel Formulas

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