Like the previous volume with the same title, this is a collection of papers in the field of Schur analysis, a mathematical research fiels involving moment problems and more generally interpolation in the complex plane by bounded analytic (Schur) functions or positive real (Carathéodry) functions. The papers are produced by the team of the U. Leipzig and one coworker who are well known ambassadors of Schur analysis. This volume contains a coherent set of papers related to matrix valued moment problems on the complex unit circle and on real line, a half line or a finite interval.
Interpolation, Schur Functions and Moment Problems (Part I) was edited by D. Alpay and I. Gohberg and published by Birkhäuser in 2006 as volume 165 in the same series on Operator Theory: Advances and Applications. This book is volume 226 in the same series. Many (in fact all) other volumes in this series are also related to topics in operator theory and analysis that play a role in the interaction between pure mathematics and applications in systems theory, signal processing, linear algebra, etc. By the number of volumes in the series, founded in 1979 by I. Gohberg, it is seen that this is a very productive and active research area. That is why the series has now two subseries: Linear Operators and Linear Systems (to which the present book belongs) and Advances in partial Differential Equations.
The topics of this volume, as well as the ones in volume 1 dwell in the realm of Schur analysis. The name stems from I. Schur who published two papers in 1917/18 where he proposes an algorithm to solve a coefficient problem (i.e. the trigonometric moment problem), which is a kind of continued fraction-like decomposition of a function analytic in the complex unit disk that is bounded by 1 (now called a function in the Schur class). This involves a recurrence that coincides with the recurrence relation of polynomials orthogonal on the unit circle (studied by Ya. Geronimus and G. Szegő) and can be interpreted as a discretized transmission line in circuit theory (V. Belevich) or a digital prediction filter in signal processing (N. Wiener and P. Masani). It were the applications that revived the interest in Schur analysis in the 1960s and it hasn't stopped since. This has been generalized much further in many different directions and linked to other work by some great mathematicians like R. Nevanlinna, G. Pick, G. Herglotz, T. Stieltjes, H. Weyl, etc. A translation of Schur's original papers into English can be found e.g. in volume 18 of the OT series ( I. Schur methods in operator theory and signal processing (I. Gohberg (ed.), 1986) and many other of the basic papers in Schur analysis are collected in Ausgewählte Arbeiten zu den Urspüngen de Schur-Analysis Band 16 of Teubner Archiv zur Mathematik, (B. Fritzsche, B. Kirstein (eds.), 1991).
The six papers of this volume are mainly focussing on block generalizations of moment problems and Nevanlinna's theory. Moment theory asks for the existence and characterization of a measure when a sequence of moments are given. Nevanlinna's analogue is a generalization when the moments are not given at one point, but at more than one point, i.e. instead of Taylor series coefficients at just one point, a number of Taylor coefficients are given at different points, leading to a form of Hermite interpolation at these points. The block generalization refers to the fact that the functions and the moments are matrix valued, and hence also the measure one is looking for. As a consequence, the theory does not involve (matrix valued) orthogonal polynomials but more general matrix valued orthogonal rational functions.
Bernd Fritzsche and Bernd Kirstein from Leibniz University have been passionate ambassadors of Schur analysis. The papers in this volume are all written by them and their coworkers at their institute (C. Mädler, T. Schwartz, A. Lasarow, A. Rahn) with one exception, the paper by A.E. Choque Rivero (Mexico) is a continuation of his earlier collaboration with the Leipzig team.
Here is a short summary.
Papers 1-3 involve reciprocal sequences and their applications, papers 4-5 discuss power moment problems on the real line and the last papers is about orthogonal rational functions.
The contributors of this volume are very productive and they have published a large number of papers in many journals and books. Bringing a number of papers together and publish them as a book is a good idea. They have a particular style of writing, involving very precise formulations that also require complex notation. It may take a while for the reader to get used to it, but once familiar with these habits and the constructs involved, it is good reading. It will be of high interest to anyone who is involved from far or near with Schur analysis and the whole universe of related topics that I sketched in the beginning. Given the special character of this book (sub)series, it is clear that whoever was interested in volumes of this OT subseries before will be interested in practically all of them, a fortiori in this one. Be warned though that this book is at an advanced level, treating particular and specialized results in the domain, so this is not the right place to start learning the subject.