Representations of Central Matrix-Valued Carathéodory Functions in Both Nondegenerate and Degenerate Cases

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.     

On Rank Invariance of Schwarz-Pick-Potapov Block Matrices of Matricial Schur Functions

We derive statements on rank invariance of Schwarz-Pick-Potapov block matrices of matrix-valued Schur functions. The rank of these block matrices coincides with the rank of some block matrices built from the corresponding section matrices of Taylor coefficients. These results are applied to the discussion of a matrix version of the classical Schur-Nevanlinna algorithm.     

Positive Carathéodory interpolation on the polydisc

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.     

Carathéodory interpolation on the non-commutative polydisk

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.     

Solution of a multiple Nevanlinna-Pick problem via orthogonal rational functions

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.     

矩阵值Caratheodory函数广义块Pick矩阵的秩不变性

Lasarow^[1]推导出矩阵值Carath\'{e}odory函数的第一、第二型广义块Pick矩阵及其变型的秩不变性. 这些矩阵由同一个Carath\'{e}odory函数的值与它的直到某阶的导数值确定. 利用文献[2]中提出的块Toeplitz向量方法, 该文断言,这些块矩阵的秩分别相关并重合于具有秩不变性的块Toeplitz矩阵的秩, 从而改进了这两类广义块Pick矩阵的秩不变性结论的证明.     

Classes of pairs of meromorphic matrix valued functions generated by nonnegative kernels and associated Nevanlinna-Pick problems

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.     

On the sequences of the Weyl semi-radii associated with matricial Carathéodory and Schur sequences in both nondegenerate and degenerate cases

In this paper we discuss Weyl matrix balls in the context of the matricial versions of the classical interpolation problems named after Carathéodory and Schur. Our particular focus will be on studying the monotonicity of suitably normalized semi-radii of the corresponding Weyl matrix balls. We, furthermore, devote a fair bit of attention to characterizing the case in which equality holds for particular matricial inequalities. Solving these problems will provide us with a new perspective on the role of the central functions for the classes of Carathéodory and Schur.     

Caratheodory interpolation kernels

Kernels over the unit disk for which a version of Carathéodory interpolation is true are characterized in a simple computationally verifiable manner.     

Tangential Nevanlinna-Pick Interpolation and Its Connection with Hamburger Matrix Moment Problem

This paper is concerned with the solution of a certain tangential Nevanlinna-Pick interpolation for Nevanlinna functions. We use the so-called block Hankel vector method to establish two intrinsic connections between the tangential Nevanlinna-Pick interpolation in the Nevanlinna class and the truncated Hamburger matrix moment problem associated with the block Hankel vector under consideration: one is a congruent relationship between their information matrices, and the other is a divisor-remainder connection between their solutions. These investigations generalize our previous work on the Nevanlinna-Pick interpolation and power matrix moment problem.     

On the Nevanlinna-Pick interpolation problem for generalized stieltjes functions

The solutions of the Nevanlinna-Pick interpolation problem for generalized Stieltjes matrix functions are parametrized via a fractional linear transformation over a subset of the class of classical Stieltjes functions. The fractional linear transformation of some of these functions may have a pole in one or more of the interpolation points, hence not all Stieltjes functions can serve as a parameter. The set of excluded parameters is characterized in terms of the two related Pick matrices.Dedicated to the memory of M. G. Kreîn     

Interpolation and prediction problems for connected compact abelian groups

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.     

Minimum w-entropy interpolants for matricial Carathéodory functions and maximum determinant completions of associated block Pick matrix

The so-called modified block Toeplitz vector approach is used to connect a class of particular solutions G_w for w∈D of a nondegenerate interpolation problem of the Nevanlinna-Pick type with a class of particular solutions F_w 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 F_w. It turns out that G_w(w∈D) is exactly or provides us with the unique solution to these two extremal problems in a manner.     

A band method approach to a positive expansion problem in a unitary dilation setting

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.     

On three Krein extension problems and some generalizations

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.     

Multiple point interpolation in Nevanlinna classes

A Nevanlinna-Pick type interpolation problem for generalized Nevanlinna functions is considered. We prescribe the values of the function and its derivatives up to a certain order at finitely many points of the upper half plane. An operator theoretic approach is used to parametrize the solutions of this interpolation problem by means of selfadjoint extensions of a certain symmetry.     

On the Carathéodory–Fejér Interpolation Problem for Generalized Schur Functions

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     

On rank variation of block matrices generated by Nevanlinna matrix functions

Within the framework of the multiple Nevanlinna–Pick matrix interpolation and its related matrix moment problem, we study the rank of block moment matrices of various kinds, generalized block Pick matrices and generalized block Loewner matrices, as well as their Potapov modifications, generated by Nevanlinna matrix functions, and derive statements either on rank (or inertia) invariance in different senses or on rank variation of such types of block matrices (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Bitangential Inverse Input Impedance Problem for Canonical Systems,I: Weyl-Titchmarsh Classification,Existence and Uniqueness

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.     

Nevanlinna-Pick interpolation with boundary data

Versions of the Nevanlinna-Pick interpolation problem with boundary interpolation nodes and boundary interpolated values are investigated.