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.
"Simon can write books faster than most people can read them. The quality is very high and the level of scholarship is enormous. . . . The book is recommended to everyone who wants to broaden his or her knowledge about recent developments in orthogonal polynomials. It is a pleasure to see that many areas of mathematics are tied together."--Christian Berg, Journal of Approximation Theory
"This book is a magnificent compilation of results of which the author can be justifiably proud. It can be used on many levels, running from a reference book for experts to a manual for beginners on what one really needs to know to pursue research in spectral theory. The latter objective is enhanced by the inclusion of tutorial sections on a variety of prerequisites, which also makes the individual chapters suitable for seminars and reading projects. It is also an excellent source of examples."--Harry Dym, Mathematical Reviews
"This book can be very useful for mathematicians who are familiar with the fields of analysis such as measure theory, orthogonal polynomials, and application of the theory of matrices and bounded linear operators."--Vladimir L. Makarov, Zentralblatt MATH
"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
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