The Jacobi matrices approach to Nevanlinna–Pick problems

Derevyagin, Maxim

doi:10.1016/j.jat.2010.08.001

Cited by 6 publications

(19 citation statements)

References 41 publications

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“…Actually, it would be rather efficient to use specifics of the problem than applying the standard machinery of pencils. Say, repeating the reasoning from [10], one can verify that the recurrence relation (5.1) can be renormalized to the following one…”

Section: The Underlying Linear Pencilsmentioning

confidence: 79%

“…Here we will consider the recurrence relations (4.5) as a particular case of the theory of linear pencils of tridiagonal matrices that was elaborated in [5], [10], and [12], which, in turn, had their origin in [31].…”

Section: The Underlying Linear Pencilsmentioning

confidence: 99%

“…However, it still has to be done a bit more carefully than in the ordinary case. The rough idea is to replace the problem (H − λJ) u = 0 with an ordinary one such as (J −1/2 HJ −1/2 − λI) u = 0 (see [10]) or (J −1 H − λI) u = 0. The task is not evident as there might be pitfalls and later on we will discuss an example with a drastic change in comparison with Jacobi matrices and that only happens for the linear pencil case.…”

Section: The Underlying Linear Pencilsmentioning

confidence: 99%

“…At first, the results from [10] are applicable to Wall pencils but under an additional restriction that the measure in the integral representation of the function ϕ is a probability measure. This condition simply means that the underlying pencil can be reduced to a self-adjoint operator.…”

Section: The Underlying Linear Pencilsmentioning

confidence: 99%

“…This is what is actually done in Section 3 and Section 4 through the findings of H. S. Wall. The section after that is where we simply adopt a more general theory developed in [5], [10], and [12] to this very particular case of Wall's continued fraction representation of Nevanlinna functions. Besides, Section 5 reveals the relation between the spectral theories of OPUC and linear pencils of Jacobi matrices.…”

Section: Introductionmentioning

confidence: 99%

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A note on Wall’s modification of the Schur algorithm and linear pencils of Jacobi matrices

Derevyagin

2017

Journal of Approximation Theory

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Abstract. In this note we revive a transformation that was introduced by H. S. Wall and that establishes a one-to-one correspondence between continued fraction representations of Schur, Carathéodory, and Nevanlinna functions. This transformation can be considered as an analog of the Szegő mapping but it is based on the Cayley transform, which relates the upper half-plane to the unit disc. For example, it will be shown that, when applying the Wall transformation, instead of OPRL, we get a sequence of orthogonal rational functions that satisfy three-term recurrence relation of the form (H −λJ)u = 0, where u is a semi-infinite vector, whose entries are the rational functions. Besides, J and H are Hermitian Jacobi matrices for which a version of the Denisov-Rakhmanov theorem holds true. Finally we will demonstrate how pseudo-Jacobi polynomials (aka Routh-Romanovski polynomials) fit into the picture.

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Section: The Underlying Linear Pencilsmentioning

confidence: 79%

Section: The Underlying Linear Pencilsmentioning

confidence: 99%

Section: The Underlying Linear Pencilsmentioning

confidence: 99%

Section: The Underlying Linear Pencilsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A note on Wall’s modification of the Schur algorithm and linear pencils of Jacobi matrices

Derevyagin

2017

Journal of Approximation Theory

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CMV Matrices and Little and Big −1 Jacobi Polynomials

2012

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Abstract. We introduce a new map from polynomials orthogonal on the unit circle to polynomials orthogonal on the real axis. This map is closely related with the theory of CMV matrices. It contains an arbitrary parameter λ which leads to a linear operator pencil. We show that the little and big -1 Jacobi polynomials are naturally obtained under this map from the Jacobi polynomials on the unit circle.

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A generalization of Schur functions: Applications to Nevanlinna functions, orthogonal polynomials, random walks and unitary and open quantum walks

Grünbaum

Velázquez

2018

Advances in Mathematics

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Recent work on return properties of quantum walks (the quantum analogue of random walks) has identified their generating functions of first returns as Schur functions. This is connected with a representation of Schur functions in terms of the operators governing the evolution of quantum walks, i.e. the unitary operators on Hilbert spaces.In this paper we propose a generalization of Schur functions by extending the above operator representation to arbitrary closed operators on Banach spaces. Such generalized 'Schur functions' meet the formal structure of first return generating functions, thus we call them FR-functions. We derive some general properties of FR-functions, among them a simple relation with an operator version of Stieltjes functions which generalizes the renewal equation already known for random and quantum walks. We also prove that FRfunctions satisfy splitting properties which extend useful factorizations of Schur functions.When specialized to self-adjoint operators on Hilbert spaces, we show that FR-functions become Nevanlinna functions. This allows us to obtain properties of Nevanlinna functions which, as far as we know, seem to be new. The FR-function structure leads to a new operator representation of Nevanlinna functions in terms of self-adjoint operators, whose spectral measures provide also new integral representations of such functions. This allows us to characterize each Nevanlinna function by a measure on the real line, which we refers to as 'the measure of the Nevanlina function'. In contrast to standard operator and integral representations of Nevanlinna functions, these new ones are exact analogues of those already known for Schur functions. The above results are also the source of a very simple 'Schur algorithm' for Nevanlinna functions based on interpolations at points on the real line, which we refer to as the 'Schur algorithm on the real line'.The paper is completed with several applications of FR-functions to orthogonal polynomials and random and quantum walks which illustrate their wide interest: an analogue for orthogonal polynomials on the real line of the Khrushchev formula for orthogonal polynomials on the unit circle, and the use of FR-functions to study recurrence in random walks, quantum walks and open quantum walks. These applications provide numerous explicit examples of FR-functions, clarifying the meaning of these functions -as first return generating functions-and their splittings -which become recurrence splitting rules. They also show that these new tools, despite being extensions of very classical ones, play an important role in the study of physical problems of a highly topical nature.

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The Jacobi matrices approach to Nevanlinna–Pick problems

Cited by 6 publications

References 41 publications

A note on Wall’s modification of the Schur algorithm and linear pencils of Jacobi matrices

A note on Wall’s modification of the Schur algorithm and linear pencils of Jacobi matrices

CMV Matrices and Little and Big −1 Jacobi Polynomials

A generalization of Schur functions: Applications to Nevanlinna functions, orthogonal polynomials, random walks and unitary and open quantum walks

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