Victor Katsnelson

An abstract interpolation problem and the extension theory of isometric operators

The algebraic structure of V.P. Potapov’s Fundamental Matrix Inequality (FMI) is discussed and its interpolation meaning is analyzed. Functional model spaces are involved. A general Abstract Interpolation Problem is formulated which seems to cover all the classical and recent problems in the field and the solution set of this problem is described using the Arov-Grossman formula. The extension theory of isometric operators is the proper language for treating interpolation problems of this type.

On the theory of matrix-valued functions belonging to the Smirnov class

A theory of matrix-valued functions from the matricial Smirnov class is systematically developed. In particular, the maximum principle of V.I. Smirnov, inner-outer factorization, the Smirnov-Beurling characterization of outer functions and an analogue of Frostman’s theorem are presented for matrix-valued functions from the Smirnov class. We also consider a family of functions belonging to the matricial Smirnov class which is indexed by a complex parameter λ. We show that with the exception of a “very small” set of such λ the corresponding inner factor in the inner-outer factorization of the function Fλ is a Blaschke-Potapov product.

Topics in interpolation theory

This book is devoted primarily to topics in interpolation for scalar, matrix and operator valued functions. About half the papers are based on lectures which were delivered at a conference held at Leipzig University in August 1994 to commemorate the 80th anniversary of the birth of Vladimir Petrovich Potapov. The volume also contains the English translation of several papers relatively unknown in the West, two expository papers written especially for this volume, and historical material based on reminiscences of former colleagues, students and associates of Mr. Potapov. Several examples of interpolation problems of the Nevanlinna-Pick and Caratheodory-Fejer type are included as well as moment problems and problems of integral representation in several settings. The major themes covered include applications of the Potapov method of fundamental matrix inequalities, multiplicative decompositions of J-inner matrix valued functions, the abstract interpolation problem, canonical systems of differential equations and interpolation in spaces with an indefinite metric, and this text should appeal to mathematicians specializing in pure and applied mathematics and engineers who work in systems theory and control.

LEFT AND RIGHT BLASCHKE-POTAPOV PRODUCTS AND AROV-SINGULAR MATRIX-VALUED FUNCTIONS

IfJ is an indefinite signature matrix, then there exists aJ contractive holomorphic matrix valued functionW(z) in the open unit disc which can be expressed as a left Blaschke-Potapov product:W(z)=B(l)(z), but not as a right Blaschke-Potapov product:W(z)=E(z)B(r)(z), whereB(r)(z) is a right Blaschke-Potapov product andE(z) is a so called Arov singular matrix function. In factB(l)(z) may be chosen to obtain any Arov singular matrix functionE(z) in the second representation. This phenomenon and multiplicative representations of Arov singular functions are discussed.

On transformations of Potapov’s fundamental matrix inequality

According to V.P. Potapov, a classical interpolation problem can be reformulated in terms of a so-called Fundamental Matrix Inequality (FMI). To show that every solution of the FMI satisfies the interpolation problem, we usually have to transform the FMI in some special way. In this paper a number of the transformations of the FMI which come into play are motivated and demonstrated by simple, but typical examples.

Deformations of Fuchsian Systems of Linear Differential Equations and the Schlesinger System

We consider holomorphic deformations of Fuchsian systems parameterized by the pole loci. It is well known that, in the case when the residue matrices are non-resonant, such a deformation is isomonodromic if and only if the residue matrices satisfy the Schlesinger system with respect to the parameter. Without the non-resonance condition this result fails: there exist non-Schlesinger isomonodromic deformations. In the present article we introduce the class of the so-called isoprincipal deformations of Fuchsian systems. Every isoprincipal deformation is also an isomonodromic one. In general, the class of the isomonodromic deformations is much richer than the class of the isoprincipal deformations, but in the non-resonant case these classes coincide. We prove that a deformation is isoprincipal if and only if the residue matrices satisfy the Schlesinger system. This theorem holds in the general case, without any assumptions on the spectra of the residue matrices of the deformation. An explicit example illustrating isomonodromic deformations, which are neither isoprincipal nor meromorphic with respect to the parameter, is also given.

Sampling and Interpolation for Functions with Multi-Band Spectrum: The Mean Periodic Continuation Method

Abstract. Functions f belonging to L2(ℝ) are considered in which spectrum is contained in a ‘multi-band’ set E, i.e. in a subset of the real axis, which is the union of finitely many intervals. For such functions, a generalization of the Whittaker-Shannon-Kotelnikov sampling formula is done. The considered problem is also related to Riesz bases of exponentials in L2(E). No additional restrictions concerning the form of the set E are imposed. We reduce the problem in question to the problem of invertibility of a certain operator. This operator is the composition of two operators. The first one is the operator which realizes a mean periodic continuation of a function from L2([0, mes E]) to a function defined on L2(ℝ). (This operator acts from L2([0, mes E]) into Lloc2(IR) . It could be constructed as the solution of the Cauchy problem for an appropriate difference equation with the initial data on ([0, mes E]).) The second one is the truncation operator from Lloc2(IR) onto L2(E). The sampling sequence appears as the zero set of some almost periodic exponential polynomial, which could be constructed effectively from the set E. The sampling series could also be constructed effectively.

The Schur Algorithm in Terms of System Realizations

The main goal of this paper is to demonstrate the usefulness of certain ideas from System Theory in the study of problems from complex analysis. With this paper, we also aim to encourage analysts, who might not be familiar with System Theory, colligations or operator models to take a closer look at these topics. For this reason, we present a short introduction to the necessary background. The method of system realizations of analytic functions often provides new insights into and interpretations of results relating to the objects under consideration. In this paper we will use a well-studied topic from classical analysis as an example. More precisely, we will look at the classical Schur algorithm from the perspective of System Theory. We will confine our considerations to rational inner functions. This will allow us to avoid questions involving limits and will enable us to concentrate on the algebraic aspects of the problem at hand. Given a non-negative integer n, we describe all system realizations of a given rational inner function of degree n in terms of an appropriately constructed equivalence relation in the set of all unitary (n+1)×(n+1)-matrices. The concept of Redheffer coupling of colligations gives us the possibility to choose a particular representative from each equivalence class. The Schur algorithm for a rational inner function is, consequently, described in terms of the state space representation.

Rational Solutions of the Schlesinger System and Isoprincipal Deformations of Rational Matrix Functions I

In this second article of the series we study holomorphic families of generic rational matrix functions parameterized by the pole and zero loci. In particular, the isoprincipal deformations of generic rational matrix functions are proved to be isosemiresidual. The corresponding rational solutions of the Schlesinger system are constructed and the explicit expression for the related tau function is given. The main tool is the theory of joint system representations for rational matrix functions with prescribed pole and zero structures.

Self-adjoint Boundary Conditions for the Prolate Spheroid Differential Operator

We consider the formal prolate spheroid differential operator on a finite symmetric interval and describe all its self-adjoint boundary conditions. Only one of these boundary conditions corresponds to a self-adjoint differential operator which commute with the Fourier operator truncated on the considered finite symmetric interval.