Generalized Schur functions are scalar- or operator-valued holomorphic functions such that certain associated kernels have a finite number of negative squares. This book develops the realization theory of such functions as characteristic functions of coisometric, isometric, and unitary colligations whose state spaces are reproducing kernel Pontryagin spaces. This provides a modern system theory setting for the relationship between invariant subspaces and factorization, operator models, Krein-Langer factorizations, and other topics. The book is intended for students and researchers in mathematics and engineering. An introductory chapter supplies background material, including reproducing kernel Pontryagin spaces, complementary spaces in the sense of de Branges, and a key result on defining operators as closures of linear relations. The presentation is self-contained and streamlined so that the indefinite case is handled completely parallel to the definite case.

Schur Functions, Operator Colligations, and Reproducing Kernel Pontryagin Spaces (Operator Theory Advances & Applications)

Science & Math

Daniel Alpay

Mathematics

Birkhauser

This is a collection of exercises in the theory of analytic functions, with completed and detailed solutions. We wish to introduce the student to applications  and aspects of the theory of analytic functions not always touched upon in a first course. Using appropriate exercises we wish to show to the students some aspects of what lies beyond a first course in complex variables. We also discuss topics of interest for electrical engineering students (for instance, the realization of rational functions and its connections to the theory of linear systems and state space representations of such systems). Examples of  important Hilbert spaces of analytic functions (in particular the Hardy space and the  Fock space) are given. The book also includes a part where relevant facts from topology, functional analysis and Lebesgue integration are reviewed.

A Complex Analysis Problem Book

COMPLEX ANALYSIS PROBLEM BOOK

Springer

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Books by Daniel Alpay