The European Mathematical Society

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.

1. B. Fritzsche, B. Kirstein, C. Mädler and T. Schwarz. On the Concept of Invertibility for Sequences of Complex p × q-matrices Application to Holomorphic p×q-matrix-valued Functions
   
   This involves the construction the a q× p-sequence which is the reciprocal of a p×q-matrix sequence f(z). This is a matrix version of constructing the scalar coefficients in a series 1/f(z), given the coefficients of the series f(z). It applies to investigating the holomorphicity of the Moore-Penrose inverse of a matrix valued holomorphic function.
2. B. Fritzsche, B. Kirstein, A. Lasarow and A. Rahn. On Reciprocal Sequences of Matricial Carathéodory Sequences and Associated Matrix Functions
   
   This studies a particular case of the previous paper. It concentrates on the case p = q for Carathéodory functions (functions holomorphic is the disc and whose Hermitian part is nonnegative definite). The reciprocal of such Carathéodory function is again a Carathéodory function.
3. B. Fritzsche, B. Kirstein, C. Mädler and T. Schwarz. On a Schur-type Algorithm for Sequences of Complex p×q-matrices and its Interrelations with the Canonical Hankel Parametrization
   
   This is another application of reciprocal sequences. The Hamburger moment problem involves power moments for a (matrix-valued) measure on the real line, and the measure has to be recovered. The moments are coefficients in a Taylor series and a second sequence of moments can be defined as the Taylor coefficients in the reciprocal series. On the other hand partially defined (block) Hankel matrices that have a positive definite extension play an important role in the existence of a solution. By recursively constructing Schur complements, a sequence of matrices can be defined characterizing them. This allows to formulate a Schur-type algorithm to find this so called canonical characterization of Hankel matrices and hence finding solutions of truncated Hamburger moment problems.
4. A.E. Choque Rivero. Multiplicative Structure of the Resolvent Matrix for the Truncated Hausdorff Matrix Moment Problem
   
   The Hausdorff moment problem has to find a measure, supported on a finite interval, given the power moments. This involves two sequences of nested (block) Hankel matrices corresponding to an even or an odd number of prescribed moments. The solution of the moment problem is given in terms of a resolvent matrix.
   In this paper, the resolvent matrix is constructed and its factorization in elementary factors (Blaschke-Potapov factors). Again this corresponds to some recurrence relation for the (matrix-valued) orthogonal polynomials on a finite interval. This factorization of the resolvent has been treated before in the case of an even number of moments, here the odd case is analysed.
5. B. Fritzsche, B. Kirstein and C. Mädler. On a Special Parametrization of Matricial α-Stieltjes One-sided Non-negative Definite Sequences
   
   The Stieltjes moment problem looks for a measure supported on the positive real line, given its power moment sequence. Like in the Hamburger case, the existence of a solution involves (block) Hankel matrices containing the moments that have to be in a certain class. Being a member or not can be easily checked in the Hausdorff case by their canonical parametrization that can be constructed by a recursive Schur-like algorithm. Here a similar characterization is derived for the Stieltjes case and the relation between both is established.
6. B. Fritzsche, B. Kirstein and A. Lasarow. On Maximal Weight Solutions of a Moment Problem for Rational Matrix-valued Functions
   
   Truncated moment problems on the unit circle with moments given at several points involves rational interpolation (rather than matching the initial terms in series expansions). For a finite number of moments, a measure solving the moment problem can be described as a quadrature formula, i.e. it is a discrete measure with mass at a finite number of mass points. In this paper such particular solutions are investigated that have extremal mass at a prescribed point on the unit circle.

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.