Linear Spectral Transformations and Laurent Polynomials

Garza, Luis E.; Marcellán, Francisco

doi:10.1007/s00009-009-0008-5

Cited by 8 publications

(18 citation statements)

References 13 publications

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“…-(21) are a direct consequence of (12)- (14). The uniqueness of X j ∈ J r for j ≥ r follows from Corollary 2.2 and the fact that (15) …”

Section: Hermitian Polynomial Modificationsmentioning

confidence: 83%

“…Bearing in mind Theorem 2.4, it is enough to prove (iii)⇒(ii)⇒ (12), (13), (14). Suppose that only (iii) holds.…”

Section: Hermitian Polynomial Modificationsmentioning

confidence: 93%

“…Other direct perturbations of degree 1 are analyzed in [39,5,14,15]. Our method yields a complementary approach which unifies the discussion of these cases and, at the same, allows us to deal with non-quasi-definite functionals.…”

Section: Corollary 34mentioning

confidence: 98%

“…Laurent polynomial perturbations of hermitian functionals have been considered previously (see for instance [1,5,6,9,12,13,11,[14][15][16][18][19][20][24][25][26][27]39]). The most detailed analysis in the literature corresponds to perturbations of degree one, among them the so-called Christoffel and Geronimus transformations on the unit circle due to the analogy with the transformations with the same name on the real line.…”

Section: Introductionmentioning

confidence: 99%

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Direct and inverse polynomial perturbations of hermitian linear functionals

Cantero

Ituarte

Velázquez

2011

Journal of Approximation Theory

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This paper is devoted to the study of direct and inverse (Laurent) polynomial modifications of moment functionals on the unit circle, i.e., associated with hermitian Toeplitz matrices. We present a new approach which allows us to study polynomial modifications of arbitrary degree.The main objective is the characterization of the quasi-definiteness of the functionals involved in the problem in terms of a difference equation relating the corresponding Schur parameters. The results are presented in the general framework of (non-necessarily quasi-definite) hermitian functionals, so that the maximum number of orthogonal polynomials is characterized by the number of consistent steps of an algorithm based on the referred recurrence for the Schur parameters.The non-uniqueness of the inverse problem makes it more interesting than the direct one. Due to this reason, special attention is paid to the inverse modification, showing that different approaches are possible depending on the data about the polynomial modification at hand. These different approaches are translated as different kinds of initial conditions for the related inverse algorithm.Some concrete applications to the study of orthogonal polynomials on the unit circle show the effectiveness of this new approach: an exhaustive and instructive analysis of the functionals coming from a general inverse polynomial perturbation of degree one for the Lebesgue measure; the classification of those pairs of orthogonal polynomials connected by a certain type of linear relation with constant polynomial coefficients; and the determination of those orthogonal polynomials whose associated ones are related to a degree one polynomial modification of the original orthogonality functional.

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“…-(21) are a direct consequence of (12)- (14). The uniqueness of X j ∈ J r for j ≥ r follows from Corollary 2.2 and the fact that (15) …”

Section: Hermitian Polynomial Modificationsmentioning

confidence: 83%

“…Bearing in mind Theorem 2.4, it is enough to prove (iii)⇒(ii)⇒ (12), (13), (14). Suppose that only (iii) holds.…”

Section: Hermitian Polynomial Modificationsmentioning

confidence: 93%

Section: Corollary 34mentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

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Direct and inverse polynomial perturbations of hermitian linear functionals

Cantero

Ituarte

Velázquez

2011

Journal of Approximation Theory

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“…Then, in [32], a Geronimus perturbation of a nontrivial positive measure supported on the unit circle is studied as well as the connection between the associated Hessenberg matrices. Finally, in [33] linear spectral transformations of Hermitian linear functionals using the multiplication by some class of Laurent polynomials are considered and the behavior of the Verblunsky parameters of the perturbed linear functional is found.…”

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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Christoffel Transformation for a Matrix of Bi-variate Measures

García-Ardila

Mañas

Marcellán

2019

Complex Anal. Oper. Theory

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We consider the sequences of matrix bi-orthogonal polynomials with respect to the bilinear forms ·, · R and ·, · L P(z 1 ), Q(z 2 ) R = T×T P(z 1 ) † L(z 1 )dµ(z 1 , z 2 )Q(z 2 ),is the set of matrix Laurent polynomials of size p × p and L(z) is a special polynomial in L p×p [z] . A connection formula between the sequences of matrix Laurent bi-orthogonal polynomials with respect to ·, · R, (resp. ·, · L) and the sequence of matrix Laurent bi-orthogonal polynomials with respect to dµ(z 1 , z 2 ) is given. (2010). 42C05 15A23 30C10. Keywords. Matrix biorthogonal polynomials, matrix-valued measures, nondegenerate continuous bilinear forms, Gauss-Borel factorization, matrix Christoffel transformations, quasideterminants, block CMV Matrices. Mathematics Subject ClassificationRecently, Cantero, Marcellán, Moral and Velázquez [14] presented an approach to the Darboux transformations for CMV matrices. In particular, for the Christoffel transformation they show that given a Hermitian polynomial L(z), a linear functional µ (supported on the unit circle) and the perturbed oneμ = L(z)µ, if L(C) has Cholesky factorization L(C) = AA † , then L(Ĉ) = A † A, where C andĈ are the CMV matrices associated with µ andμ, respectively.On the other hand, in [38] the authors deal with a measure supported on the unit circle multiplied by a non-negative trigonometric polynomial g(θ). Using the fact that for g(θ) there exists a positive integer number m ∈ N and a polynomial G 2m (z) of degree 2m such that g(θ) = z −m G 2m (z), they give a determinantal expression for the perturbed monic orthogonal polynomial of degree n multiplied by G 2m (z). In this expression, the original orthogonal polynomials, from degree n until degree n + m, their corresponding reversed polynomials (see Eq. (1)) as well as the zeros of the polynomial G 2m (z), are involved.

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Linear Spectral Transformations and Laurent Polynomials

Cited by 8 publications

References 13 publications

Direct and inverse polynomial perturbations of hermitian linear functionals

Direct and inverse polynomial perturbations of hermitian linear functionals

CMV Biorthogonal Laurent Polynomials: Perturbations and Christoffel Formulas

Christoffel Transformation for a Matrix of Bi-variate Measures

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