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Bernd Kirstein

Publications and source records attributed to Bernd Kirstein.

22 records · Page 2Linked to original sources
arXiv2009

Shift Operators Contained in Contractions, Pseudocontinuable Schur Functions and Orthogonal Systems on the Unit Circle

The main aim of this paper is to establish the connection between well-known criteria for the pseudocontinuability of a non-inner Schur function Theta in the unit disk. In a canonical way we associate a probability measure mu on the unit circle with Theta. One of the two criteria will be reformulated in the face of mu, whereas the other one is drafted in view of a completely non--unitary contraction T having Theta as its corresponding characteristic function. Our main result clarifies an immediate connection between the above-mentioned two criteria. For this reason, we construct a special orthogonal basis in the space of square-integrable functions with respect to mu and rewrite these criteria in terms of this orthogonal basis.

math.FA↗
arXiv2008

The Schur Algorithm in Terms of System Realization

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) \times (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.

math.CV↗
arXiv2007

On the Theory of Matrix Valued Functions Belonging to the Smirnov Class

A theory of matrix-valued functions from the matricial Smirnov class ${\goth N}_n^+({\Bbb D})$ 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 ${\goth N}_n^+({\Bbb D})$. We also consider a family $F_λ =F-λI$ 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 $F_λ$ is a Blaschke-Potapov product. The main goal of this paper is to provide users of analytic matrix-function theory with a standard source for references related to the matricial Smirnov class.

math.CA↗