Orthogonal polynomials and measures on the unit circle. The Geronimus transformations

Garza, Luis E.; Hernández, Javier Domínguez; Marcellán, Francisco

doi:10.1016/j.cam.2007.11.023

Cited by 22 publications

(27 citation statements)

References 15 publications

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“…Later on, following the ideas of the spectral transformations associated with the spectral measures of Jacobi matrices (see [1], [18], [21]), in [3], [4], [13] some analog problems have been considered, mainly from the point of view of the Hessenberg matrices and their LU and QR factorization, respectively.…”

Section: Introduction and Preliminary Resultsmentioning

confidence: 99%

Verblunsky Parameters and Linear Spectral Transformations

Garza¹,

Marcellán²

2009

Methods and Applications of Analysis

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Section: Introduction and Preliminary Resultsmentioning

confidence: 99%

Verblunsky Parameters and Linear Spectral Transformations

Garza¹,

Marcellán²

2009

Methods and Applications of Analysis

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“…One of the reasons for studying this transformation is that it appears naturally when considering the Geronimus transformation (see [3,8]), defined by…”

Section: Introductionmentioning

confidence: 99%

When is the Uvarov transformation positive definite?

Humet

Barel

2011

Numer Algor

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Let L be a positive definite bilinear functional, then the Uvarov transformation of L is given byIn this paper we analyze conditions on m for U to be positive definite in the linear space of polynomials of degree less than or equal to n. In particular, we show that m has to lie inside a circle in the complex plane defined by α, n and the moments associated with L. We also give an upper bound for the radius of this circle that depends only on α and n. This and other conditions on m are visualized for some examples.

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“…The connection between the GGT matrix associated with a probability measure and the perturbed linear functional, respectively, in terms of their QR factorization instead of the LU factorization has been analyzed in [15,20,49]. 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%

“…A more general framework is presented in [28]. Besides, a special case of the Geronimus transformation has been analyzed in [20].…”

Section: Geronimus Transformationmentioning

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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Orthogonal polynomials and measures on the unit circle. The Geronimus transformations

Cited by 22 publications

References 15 publications

Verblunsky Parameters and Linear Spectral Transformations

Verblunsky Parameters and Linear Spectral Transformations

When is the Uvarov transformation positive definite?

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

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