Szego’s Theorem and Its Descendants Spectral Theory for L2 Perturbations of Orthogonal Polynomials 1st Edition by Barry Simon ISBN 1400837057 9781400837052

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Product details:

ISBN 10: 1400837057 

ISBN 13: 9781400837052

Author: Barry Simon 

This book presents a comprehensive overview of the sum rule approach to spectral analysis of orthogonal polynomials, which derives from Gábor Szego’s classic 1915 theorem and its 1920 extension. Barry Simon emphasizes necessary and sufficient conditions, and provides mathematical background that until now has been available only in journals. Topics include background from the theory of meromorphic functions on hyperelliptic surfaces and the study of covering maps of the Riemann sphere with a finite number of slits removed. This allows for the first book-length treatment of orthogonal polynomials for measures supported on a finite number of intervals on the real line.

In addition to the Szego and Killip-Simon theorems for orthogonal polynomials on the unit circle (OPUC) and orthogonal polynomials on the real line (OPRL), Simon covers Toda lattices, the moment problem, and Jacobi operators on the Bethe lattice. Recent work on applications of universality of the CD kernel to obtain detailed asymptotics on the fine structure of the zeros is also included. The book places special emphasis on OPRL, which makes it the essential companion volume to the author’s earlier books on OPUC.

Table of contents:

Chapter One. Gems of Spectral Theory
Chapter Two. Szegő’s Theorem
Chapter Three The Killip–Simon Theorem: Szegő for OPRL
Chapter Four. Sum Rules and Consequences for Matrix Orthogonal Polynomials
Chapter Five. Periodic OPRL
Chapter Six. Toda Flows and Symplectic Structures
Chapter Seven. Right Limits
Chapter Eight. Szegő and Killip–Simon Theorems for Periodic OPRL
Chapter Nine. Szegő’s Theorem for Finite Gap OPRL
Chapter Ten. A.C. Spectrum for Bethe–Cayley Trees

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Tags: Barry Simon, Szego’s Theorem, Spectral Theory, L2 Perturbations, Orthogonal Polynomials