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We derive left and right quotient representations for central q × q matrix-valued Carathéodory functions. Moreover, we obtain recurrent formulas for the matrix polynomials involved in the quotient representations. These formulas are the starting point for getting recurrent formulas for those matrix polynomials which occur in the Arov-Krein resolvent matrix for the nondegenerate matricial Carathéodory problem. 相似文献
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《Journal of Functional Analysis》2005,229(2):241-276
The Carathéodory problem in the N-variable non-commutative Herglotz-Agler class and the Carathéodory-Fejér problem in the N-variable non-commutative Schur-Agler class are posed. It is shown that the Carathéodory (resp., Carathéodory-Fejér) problem has a solution if and only if the non-commutative polynomial with given operator coefficients (the data of the problem indexed by an admissible set Λ) takes operator values with positive semidefinite real part (resp., contractive operator values) on N-tuples of Λ-jointly nilpotent contractive n×n matrices, for all n∈N. 相似文献
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This paper studies various completion problems for a subclass ofj
pq-inner functions. Special attention is drawn to so-calledA-normalizedj
pq-elementary factors of full-rank, which are closely related to the matricial Schur problem. Finally, as an application an inverse problem for Carathéodory sequences is answered. 相似文献
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Gong-Ning Chen 《Linear algebra and its applications》2011,434(8):1851-1878
The so-called modified block Toeplitz vector approach is used to connect a class of particular solutions Gw for w∈D of a nondegenerate interpolation problem of the Nevanlinna-Pick type with a class of particular solutions Fw of a certain matricial Carathéodory coefficient problem in a transparent way. This will suggest a simple approach to the minimum w-entropy interpolants and the maximum determinant completions of the associated block Pick matrix within the framework of that Nevanlinna-Pick type interpolation problem by using the known assertions corresponding to Fw. It turns out that Gw(w∈D) is exactly or provides us with the unique solution to these two extremal problems in a manner. 相似文献
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Andreas Lasarow 《Linear algebra and its applications》2006,413(1):36-58
In view of a multiple Nevanlinna-Pick interpolation problem, we study the rank of generalized Schwarz-Pick-Potapov block matrices of matrix-valued Carathéodory functions. Those matrices are determined by the values of a Carathéodory function and the values of its derivatives up to a certain order. We derive statements on rank invariance of such generalized Schwarz-Pick-Potapov block matrices. These results are applied to describe the case of exactly one solution for the finite multiple Nevanlinna-Pick interpolation problem and to discuss matrix-valued Carathéodory functions with the highest degree of degeneracy. 相似文献
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The inverse input impedance problem is investigated in the class
of canonical integral systems with matrizants that are strongly regular J-inner
matrix valued functions in the sense introduced in [ArD1]. The set of
solutions for a problem with a given input impedance matrix (i.e., Weyl-
Titchmarsh function) is parameterized by chains of associated pairs of entire
inner p × p matrix valued functions. In our considerations the given data for
the inverse bitangential input impedance problem is such a chain and an input
impedance matrix, i.e., a p × p matrix valued function in the Carathéodory
class. Existence and uniqueness theorems for the solution of this problem are
obtained by consideration of a corresponding family of generalized bitangential
Carathéodory interpolation problems. The connection between the inverse
bitangential input scattering problem that was studied in [ArD4] and the bitangential
input impedance problem is also exploited. The successive sections
deal with: 1. The introduction, 2. Domains of linear fractional transformations,
3. Associated pairs of the first and second kind, 4. Matrix balls, 5. The
classification of canonical systems via the limit ball, 6. The Weyl-Titchmarsh
characterization of the input impedance, 7. Applications of interpolation to
the bitangential inverse input impedance problem. Formulas for recovering
the underlying canonical integral systems, examples and related results on the
inverse bitangential spectral problem will be presented in subsequent publications.D. Z. Arov thanks the Weizmann Institute of Science for hospitality and support, partially as a
Varon Visiting Professor and partially through the Minerva Foundation. H. Dym thanks Renee
and Jay Weiss for endowing the chair which supports his research and the Minerva Foundation. 相似文献
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Hugo J. Woerdeman 《Integral Equations and Operator Theory》2002,42(2):229-242
The positive Carathéodory interpolation problem in the Agler-Herglotz class on the polydisc is solved, along with a several variable version of the Naimark dilation theorem. In addition, the positive Carathéodory interpolation problem for general holomorphic functions is discussed and numerical results are presented. 相似文献
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Scott McCullough 《Integral Equations and Operator Theory》1992,15(1):43-71
Kernels over the unit disk for which a version of Carathéodory interpolation is true are characterized in a simple computationally verifiable manner. 相似文献
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Adhemar Bultheel 《Journal of Computational and Applied Mathematics》2010,235(4):927-949
We study particular sequences of rational matrix functions with poles outside the unit circle. These Schur-Nevanlinna-Potapov sequences are recursively constructed based on some complex numbers with norm less than one and some strictly contractive matrices. The main theme of this paper is a thorough analysis of the matrix functions belonging to the sequences in question. Essentially, such sequences are closely related to the theory of orthogonal rational matrix functions on the unit circle. As a further crosslink, we explain that the functions belonging to Schur-Nevanlinna-Potapov sequences can be used to describe the solution set of an interpolation problem of Nevanlinna-Pick type for matricial Schur functions. 相似文献
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In this paper a positive commuting expansion problem is presented and solved. The problem is set up in the framework of a minimal unitary dilation of a contraction acting on a Hilbert space and includes the Carathéodory and other classical interpolation problems. By combining the geometry of the minimal unitary dilation with state space techniques from system theory, a special solution is constructed. Next using the band method approach and spectral factorizations of this special solution a linear fractional parameterization of all solutions is obtained. Explicit state space formulas and applications to some classical interpolation problems are given. 相似文献
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Three basic extension problems which were initiated by M. G. Krein are discussed and further developed. Connections with interpolation problems in the Carathéodory class are explained. Some tangential and bitangential versions are considered. Full characterizations of the classes of resolvent matrices for these problems are given and formulas for the resolvent matrices of left tangential problems are obtained using reproducing kernel Hilbert space methods.Dedicated to the memory of M. G. Krein, a beacon for us both.The authors wish to acknowledge the partial support of the Israel-Ukraine Exchange Program. D. Z. Arov also wishes to thank the Weizmann Institute of Science for partial support and hospitality; H. Dym wishes to thank Renee and Jay Weiss for endowing the chair which supports his research. 相似文献
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Ricardo Uribe-Vargas 《Journal of Geometry》2003,77(1-2):184-192
We discuss three classes of closed curves in the Euclidean space $\mathbb{R}^{3}$ which have non-vanishing
curvature and at least 4 flattenings (points at which the torsion vanishes). Calling these classes (de.ned below)
Barner, Segre and Carathéodory, we prove that Barner $\subset$ (Segre $\cap$ Carathéodory). We also prove that (Segre)\
(Segre $\cap$ Carathéodory) and (Carathéodory)\(Segre $\cap$ Carathéodory) are open sets in the space of closed smooth
curves with the C-topology. Finally, we define a class of closed curves containing the class of Segre curves
and -based on contact topology considerations, as the Huygens principle- we establish the conjecture that any
curve of our class has at least 4 flattenings. 相似文献
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Marisela Dominguez 《Integral Equations and Operator Theory》2001,40(2):212-230
Extensions of the Nehari theorem and of the Sarason commutation theorem are given for compact abelian groups whose dual have a complete linear order compatible with the group structure. As a special case a version of the classical interpolation theorem due to Carathéodory — Féjer is obtained.For these groups an extension of the Helson — Szegö theorem and integral representations for positive definite generalized Toeplitz kernels are given.Partially supported by the CDCH of the Universidad Central de Venezuela, and by CONICIT grant G-97000668. 相似文献
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We consider a Nevanlinna-Pick type interpolation problem for Carathéodory functions, where the values of the function and its derivatives up to certain orders are given at finitely many points of the unit disk. The set of all solutions of this problem is described by means of the orthogonal rational functions which play here a similar role as the orthogonal polynomials on the unit circle in the classical case of the trigonometric moment problem. In particular, we use a connection between Szegö and Schur parameters which in the classical situation was discovered by Ja.L. Geronimus. 相似文献
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Nikolai Nikolov 《Journal of Mathematical Analysis and Applications》2008,341(1):140-148
Estimates for the Carathéodory metric on the symmetrized polydisc are obtained. It is also shown that the Carathéodory and Kobayashi distances of the symmetrized three-disc do not coincide. 相似文献
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Carathéodory class functionsf(z) are described having the property that the self-adjoint part off(A) is positive definite for every contractionA whose spectral radius is less than 1. Analogous results are obtained for bounded analytic functions in the unit disc, and for the Nevanlinna class. Applications to Markov chains are indicated.Partially supported by the US Air Force Grant AFOSR-94-0293.Partially supported by the NSF Grant DMS-9500924. 相似文献
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Families of pairs of matrix-valued meromorphic functions
(,P) depending on two parameters andP are introduced. They are the projective analogues of classes of functions studied in [1] and include as special cases the projective Schur, Nevanlinna and Carathéodory classes. A two sided Nevanlinna-Pick interpolation problem is defined and solved in
(,P), using the fundamental matrix inequality method. 相似文献
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The solutions of the Carathéodory–Fejér interpolation problem for generalized Schur functions can be parametrized via a linear fractional transformation over the class of classical Schur functions. The linear fractional transformation of some of these functions may have a pole (simple or multiple) in one or more of the interpolation points or not satisfy one or more interpolation conditions, hence not all Schur functions can serve as a parameter. The set of excluded parameters is characterized in terms of the related Pick matrix.Research was supported by the Summer Research Grant from the College of William and MarySubmitted: June 26, 2002 Revised: January 31, 2003 相似文献