Verblunsky Parameters and Linear Spectral Transformations

Garza, Luis E.; Marcellán, Francisco

doi:10.4310/maa.2009.v16.n1.a5

Cited by 12 publications

(16 citation statements)

References 17 publications

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“…More precisely, in [21], explicit expressions for polynomials orthogonal with respect to the perturbed measure have been obtained in terms of the orthogonal polynomials with respect to the initial probability measure.…”

Section: Ggt Matrixmentioning

confidence: 99%

“…For a class of perturbations of the measure dμ, algebraic and analytic properties of the sequences of polynomials orthogonal with respect to the perturbed measure dμ have been extensively studied, see [9][10][11]15,[19][20][21]26,28,41,46,48,49] among others. We will focus our attention on three examples of perturbations:…”

Section: Perturbed Measuresmentioning

confidence: 99%

“…The above perturbations are examples of linear spectral transformations (see [21]). On the other hand, rational spectral transformations for Hermitian Toeplitz matrices are introduced in [57] and they have been analyzed in [15,22], among others.…”

Section: Perturbed Measuresmentioning

confidence: 99%

“…Since the Verblunsky coefficients characterize the family of orthogonal polynomials, the following lemma illustrates them [21].…”

Section: Proposition 61 the Family Of Monic Orthogonal Polynomials mentioning

confidence: 99%

“…For fixed α, the necessary and sufficient conditions about the choices of m ∈ R + such that the linear functional L U is quasi-definite have been obtained in [21,48], which is 1 +mK n−1 (α, α) = 0, n 1, in contrast to the fact that it is unconditionally true for positive definite measures.…”

Section: Lemma 65 the Sequence Of Verblunsky Coefficientsmentioning

confidence: 99%

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OPUC, CMV matrices and perturbations of measures supported on the unit circle

Marcellán

Shayanfar

2015

Linear Algebra and its Applications

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a b s t r a c t Keywords:Orthogonal polynomials on the unit circle GGT matrix CMV matrix Fundamental matrix Canonical linear spectral transformations Let us consider a Hermitian linear functional defined on the linear space of Laurent polynomials with complex coefficients. In the literature, canonical spectral transformations of this functional are studied. The aim of this research is focused on perturbations of Hermitian linear functionals associated with a positive Borel measure supported on the unit circle. Some algebraic properties of the perturbed measure are pointed out in a constructive way. We discuss the corresponding sequences of orthogonal polynomials as well as the connection between the associated Verblunsky coefficients. Then, the structure of the Θ matrices of the perturbed linear functionals, which is the main tool for the comparison of their corresponding CMV matrices, is deeply analyzed. From the comparison between different CMV matrices, other families of perturbed Verblunsky coefficients will be considered. We introduce a new matrix, named Fundamental matrix, that is a tridiagonal symmetric unitary matrix, containing basic information about the family of orthogonal polynomials. However, we show that it is connected to another family of orthogonal polynomials through the Takagi decomposition.

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Section: Ggt Matrixmentioning

confidence: 99%

Section: Perturbed Measuresmentioning

confidence: 99%

Section: Perturbed Measuresmentioning

confidence: 99%

“…Since the Verblunsky coefficients characterize the family of orthogonal polynomials, the following lemma illustrates them [21].…”

Section: Proposition 61 the Family Of Monic Orthogonal Polynomials mentioning

confidence: 99%

Section: Lemma 65 the Sequence Of Verblunsky Coefficientsmentioning

confidence: 99%

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OPUC, CMV matrices and perturbations of measures supported on the unit circle

Marcellán

Shayanfar

2015

Linear Algebra and its Applications

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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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Spectral transformations and second kind polynomials associated with a hermitian linear functional

García-Ardila,

Marcellán

2024

Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat.

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The aim of this paper is to analyze the relation between the Christoffel transformations in the framework of hermitian linear functionals and the associated polynomials of the second kind through of their associated Carathéodory formal power series. In particular, some results are given in the way of characterization of linear spectral transformations using Christoffel and Geronimus transformations. This gives a new example of rational spectral transformation that has not yet been studied in the literature.

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Verblunsky Parameters and Linear Spectral Transformations

Abstract: In this paper we analyze the behavior of Verblunsky parameters for hermitian linear functionals deduced from canonical linear spectral transformations of a quasi-definite hermitian linear functional. Some illustrative examples are studied.

Cited by 12 publications

References 17 publications

OPUC, CMV matrices and perturbations of measures supported on the unit circle

OPUC, CMV matrices and perturbations of measures supported on the unit circle

Christoffel Transformation for a Matrix of Bi-variate Measures

Spectral transformations and second kind polynomials associated with a hermitian linear functional

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