Szegö's theorem and its descendants : spectral theory for L2 perturbations of orthogonal polynomials / Barry Simon.

Format:

Book

Author/Creator:

Simon, Barry, 1946-

Series:

Porter Lectures

Porter Lectures ; 6

Language:

English

Subjects (All):

Spectral theory (Mathematics).

Orthogonal polynomials.

Physical Description:

1 online resource (663 p.)

Edition:

Course Book

Place of Publication:

Princeton, N.J. : Princeton University Press, 2010.

Language Note:

English

Summary:

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.

Contents:

Frontmatter

Contents

Preface

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

Bibliography

Author Index

Subject Index

Notes:

Description based upon print version of record.

Includes bibliographical references and index.

ISBN:

9786612821158

9781282821156

1282821156

9781400837052

1400837057

OCLC:

677162060