Recent Advances in Inverse Scattering, Schur Analysis and Stochastic Processes
Lev Aronovich Saknovich was born on February 24, 1932 in Lugansk, Ukraine. Almost agains all odds as a Jew in the communist Russia of those days, be became a mathematician and had teachers such as D.P. Milman, V.P. Potapov, M.S. Brodskij, and M.S. Livšic and his PhD was supported a.o. by I.M. Gelfand, M.G. Krein, and M.A. Naimark.
This book is compiled on the occasion of his 80th birthday. It starts with a short biography and a list of his publications. Lev Saknovich himself gives an account of his studies and teachers. The main part of the book consists of 10 scientific papers that are of course related to the work of Sakhnovich. They fit very well in the Birkhäuser series on Operator Theory Advances and Applications founded by I. Gohberg in 1979 and more particulary in the subseries Linear Operators and Linear Systems. Contributors to this book are well known is this domain. Several of them published books in this series before like for example D. Alpay, V. Dubovoy, A. Kheifets, A.E. Frazho, M.A. Kaashoek, B. Fritzsche, B. Kirstein, J. Rovnyak, and by Lev Sakhnovich himself. Volume 84 in the OTAA series Integral equations with difference kernels on finite intervals by L.A. Sakhnovich got a second revised edition in 2015 along with the present book.
The papers are listed alphabetically, but they can be organized in 4 groups. Three papers deal with interpolation and moment problems. Infinite product representations of reproducing kernels are used to study iterated function systems, harmonic analysis, and stochastic processes (Alpay et al). Commutant lifting and solution of Riccati equations are used to generate all rational solutions to a Leech problem in state space form (Frazho et al). It is the continuation of a previously published paper generating minimum entropy solutions. The solutions for the truncated matrix Hamburger moment problem are traditionally treated separately for the even and the odd case. Here Schur analysis is used to treat both simultaneously via a Schur-type algorithm (Fritzsche et al).
Two papers fall under the flag of indefinite inner product spaces. The paper on quaternionic Krein spaces is a continuation of a published paper that treated quaternionic Pontryagin spaces (Alpay et al). In a joint paper Rovnyak and Sakhnovich continue exploring the relation between interpolation problems, operator identities and Krein-Langer representation of Carathéodory functions. In their previous paper they had treated the case of Nevanlinna functions.
Different aspects of operator-valued functions are treated in four papers. One deals with operator-valued Q-functions with positive definite boundary conditions (Arlinskii et al). Another paper gives proofs for the Radon-Nikodym theorem for measures that are vector- or operator-valued (Boiko et al). Semi-separable iintegral kernels in infinite dimensional spaces, and in particular their Fredholm determinants are analyzed using a Jost-Pais type reduction (Gesztesy et al). The relation between certain analytic function classes, their closedness under addition and multiplication, and other properties is the subject of another paper (Makarov et al).
Finally a paper by Lev Sakhnovich and his son discusses stability of nonlinear Fokker-Planck equations in inhomogeneous space.
From this short sketch of the contents, it is clear that, besides the brief biographic part, the essence is a collection of research papers that will be of interest to the mathematcians and engineers that feel at home in this Birkhäuser series. Most of the papers could as well have been published in journals, and are not of introductory of survey type