An Analysis and Extension of V.P. Potapov’s Approach to Interpolation Problems with Applications to the Generalized Bi-Tangential Schur-Nevanlinna-Pick Problem and J-Inner-Outer Factorization

Kheifets, Alexander; Yuditskii, Peter

doi:10.1007/978-3-0348-8532-4_6

Cited by 33 publications

(24 citation statements)

References 4 publications

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“…This will be done using the so called transformed matrix inequality which was introduced and applied to continuous interpolation problems by V. Katsnelson in [20] and [21]. See also [22], [23] and [25] for various applications of this idea.…”

Section: A Fundamental Matrix Inequalitymentioning

confidence: 99%

On degenerate interpolation, entropy and extremal problems for matrix Schur functions

Bolotnikov

Dym

1998

Integr equ oper theory

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We consider a general bitangential interpolation problem for matrix Schur functions and focus mainly on the case when the associated Pick matrix is singular (and positive semidefinite). Descriptions of the set of all solutions are given in terms of special linear fractional transformations which are obtained using two quite different approaches. As applications of the obtained results we consider the maximum entropy and the maximum determinant extension problems suitably adapted to the degenerate situation.

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Section: A Fundamental Matrix Inequalitymentioning

confidence: 99%

On degenerate interpolation, entropy and extremal problems for matrix Schur functions

Bolotnikov

Dym

1998

Integr equ oper theory

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“…The parametrization formula (7.6) follows from (7.4) upon applying Theorem 5.1. The fact that in the present situation the Redheffer transform Σ is one-to-one was established in [27] (see also [30,Theorem 5.8]). Thus the parameter ℰ such that = Σ [ℰ] is uniquely determined.…”

Section: Homogeneous Interpolation and Toeplitz Kernelsmentioning

confidence: 71%

“…Thus the parameter ℰ such that = Σ [ℰ] is uniquely determined. It is also shown in [30,Proposition 5.9] that…”

Section: Homogeneous Interpolation and Toeplitz Kernelsmentioning

confidence: 97%

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Abstract Interpolation in Vector-Valued de Branges–Rovnyak Spaces

Ball

Bolotnikov

Horst

2010

Integr. Equ. Oper. Theory

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Abstract. Following ideas from the Abstract Interpolation Problem of [26] for Schur class functions, we study a general metric constrained interpolation problem for functions from a vector-valued de BrangesRovnyak space ℋ( ) associated with an operator-valued Schur class function . A description of all solutions is obtained in terms of functions from an associated de Branges-Rovnyak space satisfying only a bound on the de Branges-Rovnyak-space norm. Attention is also paid to the case that the map which provides this description is injective. The interpolation problem studied here contains as particular cases (1) the vector-valued version of the interpolation problem with operator argument considered recently in [4] (for the nondegenerate and scalar-valued case) and (2) a boundary interpolation problem in ℋ( ). In addition, we discuss connections with results on kernels of Toeplitz operators and nearly invariant subspaces of the backward shift operator.Mathematics Subject Classification (2010). 46E22, 47A57, 30E05.

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“…It is obvious that the multiplication byv = We now recall the definition of the characteristic function of a unitary node and its functional model. An extensive discussion of the subject and its application to interpolation problems can be found in [8,9,10].…”

Section: Unitary Node Imentioning

confidence: 99%

Inverse Scattering Problem for a Special Class of Canonical Systems and Non-linear Fourier Integral. Part I: Asymptotics of Eigenfunctions

Kupin¹,

Peherstorfer

Volberg

et al.

Methods of Spectral Analysis in Mathematical Physics

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Abstract. An original approach to the inverse scattering for Jacobi matrices was recently suggested in [17]. The authors considered quite sophisticated spectral sets (including Cantor sets of positive Lebesgue measure), however they did not take into account the mass point spectrum. This paper follows similar lines for the continuous setting with an absolutely continuous spectrum on the half-axis and a pure point spectrum on the negative half-axis satisfying the Blaschke condition. This leads us to the solution of the inverse scattering problem for a class of canonical systems that generalizes the case of SturmLiouville (Schrödinger) operator. Faddeev-Marchenko space in Szegő/Blaschke settingOne of the important aspects of the spectral theory of differential operators is the scattering theory [13,14] and, in particular, the inverse scattering [11]. An original approach to the inverse scattering was recently suggested in [17]. The paper focused on classical Jacobi matrices and connections between the scattering and properties of a special Hilbert transform.In this paper, we carry out the plan of [17] in the continuous situation. Compared with [17], a completely new feature is that the scattering data incorporate the pure point spectrum with infinitely many mass points. Of course, this is a natural and important step in the developing the theory. The discussion leads us to the solution of the inverse scattering problem for a class of canonical systems that include the Sturm-Liouville (Schrödinger) equations. At present, though, we are unable to characterize the scattering data corresponding to the last important special case.This part of the work is mainly devoted to the asymptotic behavior of certain reproducing kernels (the generalized eigenfunctions). It is organized as follows. Section 1 contains definitions, some general facts and formulations of results on asymptotics. The asymptotic properties of reproducing kernels from certain model spaces are studied in Section 2. Special operator nodes arising from our construction are discussed in Sections 3 and 4. One of the nodes generates a canonical system we are interested in. Its properties and connections to the de Branges spaces of entire functions [3] are also in Section 4. The Sturm-Liouville (Schrödinger) equations are considered in Section 5. An example is given in the first appendix (Section 6). The second appendix (Section 7) relates the whole construction to the matrix A 2 Hunt-Muckenhoupt-Wheeden condition.

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An Analysis and Extension of V.P. Potapov’s Approach to Interpolation Problems with Applications to the Generalized Bi-Tangential Schur-Nevanlinna-Pick Problem and J-Inner-Outer Factorization

Cited by 33 publications

References 4 publications

On degenerate interpolation, entropy and extremal problems for matrix Schur functions

On degenerate interpolation, entropy and extremal problems for matrix Schur functions

Abstract Interpolation in Vector-Valued de Branges–Rovnyak Spaces

Inverse Scattering Problem for a Special Class of Canonical Systems and Non-linear Fourier Integral. Part I: Asymptotics of Eigenfunctions

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