An Operator Approach to the Potapov Scheme for the Solution of Interpolation Problems

Ivanchenko, T. S.; Sakhnovich, Lev

doi:10.1007/978-3-0348-8532-4_4

Cited by 16 publications

(6 citation statements)

References 11 publications

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“…The FMI approach of V. P. Potapov was enriched by L. A. Sakhnovich who introduced a method of operator identities which serves to unify the particular instances of V. P. Potapov's procedure under one framework (see [18,56,88]). …”

Section: Introductionmentioning

confidence: 99%

On Distinguished Solutions of Truncated Matricial Hamburger Moment Problems

Dyukarev

Fritzsche

Kirstein

et al. 2008

Complex Anal. Oper. Theory

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We study two slightly different versions of the truncated matricial Hamburger moment problem. A central topic is the construction and investigation of distinguished solutions of both moment problems under consideration. These solutions turn out to be nonnegative Hermitian q × q Borel measures on the real axis which are concentrated on a finite number of points. These points and the corresponding masses will be explicitly described in terms of the given data. Furthermore, we investigate a particular class of sequences (sj) ∞ j=0 of complex q × q matrices for which the corresponding infinite matricial Hamburger moment problem has a unique solution. Our approach is mainly algebraic. It is based on the use of particular matrix polynomials constructed from a nonnegative Hermitian block Hankel matrix. These matrix polynomials are immediate generalizations of the monic orthogonal matrix polynomials associated with a positive Hermitian block Hankel matrix. We generalize a classical theorem due to Kronecker on infinite Hankel matrices of finite rank to block Hankel matrices and discuss its consequences for the nonnegative Hermitian case. Mathematics Subject Classification (2000). Primary 44A60, 47A57; Secondary 30E05.

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Section: Introductionmentioning

confidence: 99%

On Distinguished Solutions of Truncated Matricial Hamburger Moment Problems

Dyukarev

Fritzsche

Kirstein

et al. 2008

Complex Anal. Oper. Theory

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“…To do it we apply a special transformation to the fundamental inequalities. Such a method was applied in [13], [17], [19] for quite different interpolation problems (we also refer to [18] for a general view on this method). Here we adapt the ideas from the papers just mentioned to the Stieltjes case (i.e., to the case of two fundamental matrix inequalities).…”

Section: Transformation Of Fundamental Matrix Inequalitiesmentioning

confidence: 99%

“…In [13] a general interpolation problem generated by an operator identity has been considered in the Nevanlinna class of matrix valued functions analytic and with the nonnegative imaginary part in the upper half-plane C + . It was shown that the classical NevanlinnaPick and Carathéodory-Féjer interpolation problems are particular cases of this problem as well as the Hamburger moment problem, the Krein extension problem for positive functions on the interval and so on.…”

Section: Introductionmentioning

confidence: 99%

On an operator approach to interpolation problems for Stieltjes functions

Bolotnikov

Sakhnovich²

1999

Integr equ oper theory

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A general interpolation problem for operator-valued Stieltjes functions is studied using V. P. Potapov's method of fundamental matrix inequalities and the method of operator identities. The solvability criterion is established and under certain restrictions the set of all solutions is parametrized in terms of a linear fractional transformation. As applications of a general theory, a number of classical and new interpolation problems are considered. IntroductionClassical interpolation problems (Schur, Nevanlinna-Pick, Carathéodory-Féjer problems, the moment problem et.c.) and their various generalizations were studied using several different approaches and methods (see e.g., [8] for short historical survey). It turns out that the interpolation data of each such problem satisfies a certain operator identity. The structure of this identity turns to be similar (and often, the same) for quite different problems. This allows one to consider a whole circle of problems in a unified way [19], [23]. On the other hand, the operator identities themselves are useful for applications [23], [24].In [13] a general interpolation problem generated by an operator identity has been considered in the Nevanlinna class of matrix valued functions analytic and with the nonnegative imaginary part in the upper half-plane C + . It was shown that the classical NevanlinnaPick and Carathéodory-Féjer interpolation problems are particular cases of this problem as well as the Hamburger moment problem, the Krein extension problem for positive functions on the interval and so on. In this paper we consider a similar problem for the Stieltjes functions (see Definition 2.1) which form an important subclass of the Nevanlinna class. The Stieltjes functions and their role in applications have been studied in [14]. The interpolation problems for Stieltjes functions has been considered in [21]

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“…Using this strategy, several matricial interpolation and moment problems could successfully be handled (see, e. g. [8, 9, 16, 17, 19, 20, 22, 23, 25-27, 37, 38, 42-49, 53, 61]). L. A. Sakhnovich enriched Potapov's method by unifying the particular instances of Potapov's procedure under the framework of one type of operator identities [11,40,56].…”

Section: Introductionmentioning

confidence: 99%