Minimal representations of unitary operators and orthogonal polynomials on the unit circle

Cantero, Maŕıa José; Ituarte, Leandro del Moral; Velázquez, L.

doi:10.1016/j.laa.2005.04.025

Cited by 54 publications

(111 citation statements)

References 25 publications

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“…, α n−2 ∈ D and α n−1 ∈ S 1 so that U is the matrix given by Definition 1.1. This result is not new and can be found in [7]. We prove this result differently and obtain it as a corollary of the following: PROPOSITION 3.3 Let B and C be two unitary matrices in CMV shape that have the same spectral measure with respect to the vector e 1 .…”

Section: Reduction To CMV Shapementioning

confidence: 85%

CMV: The unitary analogue of Jacobi matrices

Killip

Nenciu

2006

Comm Pure Appl Math

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We discuss a number of properties of CMV matrices, by which we mean the class of unitary matrices studied recently by Cantero, Moral, and Velázquez. We argue that they play an equivalent role among unitary matrices to that of Jacobi matrices among all Hermitian matrices. In particular, we describe the analogues of well-known properties of Jacobi matrices: foliation by co-adjoint orbits, a natural symplectic structure, algorithmic reduction to this shape, Lax representation for an integrable lattice system (Ablowitz-Ladik), and the relation to orthogonal polynomials. As offshoots of our analysis, we will construct action/angle variables for the finite Ablowitz-Ladik hierarchy and describe the long-time behavior of this system.

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Section: Reduction To CMV Shapementioning

confidence: 85%

CMV: The unitary analogue of Jacobi matrices

Killip

Nenciu

2006

Comm Pure Appl Math

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“…Proof.-As we have seen, the nodes are the zeros of R n (z) given by (5)- (6). Setting X n (z) = (χ 0 (z), .…”

Section: Now Taking Into Account That For Anymentioning

confidence: 99%

“…On the other hand, the rapidly growing interest on problems on the unit circle, like quadratures, Szegő polynomials and the trigonometric moment problem has suggested to develop a theory of orthogonal Laurent polynomials on the unit circle introduced by Thron in [36], continued in [26], [21], [11] and where the recent contributions of Cantero, Moral and Velázquez in [4], [3] and [6] has meant an important and definitive impulse for the spectral analysis of certain problems on the unit circle. Here, it should be remarked that the theory of orthogonal Laurent polynomials on the unit circle establishes features totally different to the theory on the real line because of the close relation between orthogonal Laurent polynomials and the orthogonal polynomials on the unit circle (see [9]).…”

Section: Introductionmentioning

confidence: 99%

A matrix approach to the computation of quadrature formulas on the unit circle

Cantero¹,

Cruz-Barroso²,

González-Vera³

2008

Applied Numerical Mathematics

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In this paper we consider a general sequence of orthogonal Laurent polynomials on the unit circle and we first study the equivalences between recurrences for such families and Szegő's recursion and the structure of the matrix representation for the multiplication operator in Λ when a general sequence of orthogonal Laurent polynomials on the unit circle is considered. Secondly, we analyze the computation of the nodes of the Szegő quadrature formulas by using Hessenberg and five-diagonal matrices. Numerical examples concerning the family of Rogers-Szegő q-polynomials are also analyzed.

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“…This analogy is illustrated in many fields of application such as random matrix theory and integrable systems [52], Dirichlet data of a circular periodic problem [53] and scattering problem [58]. The spectral analysis of CMV matrices has also attracted much attention in the last years [2,12,13,31,60].…”

Section: Matrixmentioning

confidence: 99%

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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Minimal representations of unitary operators and orthogonal polynomials on the unit circle

Cited by 54 publications

References 25 publications

CMV: The unitary analogue of Jacobi matrices

CMV: The unitary analogue of Jacobi matrices

A matrix approach to the computation of quadrature formulas on the unit circle

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

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