A Basic Interpolation Problem for Generalized Schur Functions and Coisometric Realizations

Alpay, Daniel; Azizov, T. Ya.; Dijksma, Aad; Langer, H.; Wanjala, Gerald

doi:10.1007/978-3-0348-8077-0_2

Cited by 13 publications

(22 citation statements)

References 5 publications

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“…The essential tool was the theory of reproducing kernel Pontryagin spaces and the Schur algorithm for generalized Schur functions as developed in [2][3][4][5][6][7]12,14,17,19].…”

Section: Introductionmentioning

confidence: 99%

Factorization of J-unitary matrix polynomials on the line and a Schur algorithm for generalized Nevanlinna functions

Alpay

Dijksma

Langer

2004

Linear Algebra and its Applications

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“…The essential tool was the theory of reproducing kernel Pontryagin spaces and the Schur algorithm for generalized Schur functions as developed in [2][3][4][5][6][7]12,14,17,19].…”

Section: Introductionmentioning

confidence: 99%

Factorization of J-unitary matrix polynomials on the line and a Schur algorithm for generalized Nevanlinna functions

Alpay

Dijksma

Langer

2004

Linear Algebra and its Applications

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“…Azizov, A. Dijksma, H. Langer and G. Wanjala (see [30][31][32][33][34]). Now we will sketch some recent developments on multivariable analogues of the Schur class in the unit disk.…”

Section: On Some Generalizations Of Schur Functions and The Classicalmentioning

confidence: 97%

“…This provides a modern system theory setting for the relationship between invariant subspaces and factorization, operator models, Kreȋn-Langer factorizations, and other topics. Various interpolation problems for generalized Schur functions (as well as for their matrix-and operator-valued analogues) were considered in Alpay/Azizov/Dijksma/Langer/Wanjala [33], Alpay/Constantinescu/Dijksma/Rovnyak [13,14], Alpay/Dijksma/Langer/Wanjala [15], Ball/ Gohberg/Rodman [50], Ball/Helton [51], Ball [42,43], Bolotnikov [59], Bolotnikov/Kheifets [61], Constantinescu/Gheondea [74,75], Golinskiȋ [105,106], Nudelman [130].…”

Section: On Some Generalizations Of Schur Functions and The Classicalmentioning

confidence: 99%

The Schur–Potapov Algorithm for Sequences of Complex p × q Matrices. II

Bogner

Fritzsche

Kirstein

2006

Complex anal.oper.theory

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This is the second and final part of a paper which appeared in a preceding issue of this journal. Herein the methods developed in the earlier sections of this paper are used first to develop a number of applications. A central theme of this paper is to study the interplay between functions from Sp×q,∞(D) and their sequence of SP-parameters. In particular, we describe how certain summability properties of the SP-parameters are expressed in terms of the associated functions from Sp×q,∞(D). As a byproduct of our investigations on the interplay between the SP-algorithm for p × q Schur functions and the SP-algorithm for sequences of complex p×q matrices we present a new approach to the nondegenerate matricial Schur problem. Our method complies with the basic strategy in Schur's classical paper [138] because it does not make use of any tools outside of the theory of Schur functions and Schur sequences. A closer look at the behaviour of distinguished subclasses of Sp×q(D) with respect to the SP-algorithm enables us to handle the corresponding versions of the matricial Schur problem restricted to these subclasses. Mathematics Subject Classification (2000). Primary 30E05, 47A57.Keywords. Schur function, matricial Schur problem, Schur-Potapov matrix polynomials, Arov-Kreȋn matrix polynomials. IntroductionIn the first part [57] of this paper we have established an appropriate adaptation of the Schur-Potapov algorithm for functions belonging to the matricial Schur class S p×q (D) to infinite sequences of complex p × q matrices. Using the fact that the Taylor coefficients sequences of functions from S p×q (D) are infinite p × q Schur sequences we mainly focused on the subclass S p×q,∞ (D) of S p×q (D) which consists of all p × q Schur functions for which the corresponding Taylor coefficient sequences are nondegenerate p × q Schur sequences. Thus, an one-to-one correspondence between infinite nondegenerate p × q Schur sequences and the set of all

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“…In Section 6 we consider the composite Schur transform, which consists possibly of two steps in order to stay in the class of functions holomorphic at z 1 . In these sections we explain the geometric meaning of the Schur transformation: it corresponds to a restriction to a subspace of the state space and to the compression of the operator or relation to this subspace.…”

Section: The Function N(z) Is a Generalized Nevanlinna Function If Anmentioning

confidence: 99%

“…Theorem 6.3. Assume n(z) ∈ N has Taylor expansion (2.1) at z 1 with Im ν 0 = 0, n(z) is defined and ≡ ∞, and n(z) has a pole of order q at z 1 …”

Section: N(z) ∼ P a ϕ(Z)mentioning

confidence: 99%

The Schur Transformation for Generalized Nevanlinna Functions: Interpolation and Self-Adjoint Operator Realizations

Alpay

Dijksma

Langer

et al. 2006

Complex anal.oper.theory

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A Basic Interpolation Problem for Generalized Schur Functions and Coisometric Realizations

Cited by 13 publications

References 5 publications

Factorization of J-unitary matrix polynomials on the line and a Schur algorithm for generalized Nevanlinna functions

Factorization of J-unitary matrix polynomials on the line and a Schur algorithm for generalized Nevanlinna functions

The Schur–Potapov Algorithm for Sequences of Complex p × q Matrices. II

The Schur Transformation for Generalized Nevanlinna Functions: Interpolation and Self-Adjoint Operator Realizations

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