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.
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