This book is an important one. Barry Simon has revolutionized the study of orthogonal polynomials since he entered the field. Many of the fundamental advances he and his students pioneered appear here in book form for the first time. There is no question of his profound scholarship and expertise on this topic. -- Doron Lubinsky, Georgia Institute of Technology Simon is a leading specialist in orthogonal polynomials and spectral theory, with a very wide mathematical and physical background. This book contains a huge amount of new material found only in research papers. Those interested in orthogonal polynomials will find here many new results and techniques, while specialists in spectral theory will discover deep connections with topics from classical analysis and other areas. -- Andrei Martinez-Finkelshtein, University of Almeria, Spain

This book presents a comprehensive overview of the sum rule approach to spectral analysis of orthogonal polynomials, which derives from Gabor 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.