Leseprobe
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Szegő's Theorem and Its Descendants
Barry Simon
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Naturwissenschaften, Medizin, Informatik, Technik / Analysis
Beschreibung
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.
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Existential quantification, Riemann surface, Jost function, Monic polynomial, Transfer matrix, QR decomposition, Support (mathematics), Special case, Taylor series, Function (mathematics), Triangular matrix, Integrable system, Abel's theorem, Theorem, Riemann mapping theorem, Equivalence class, Spectral theorem, Covering space, Coefficient, Green's function, Orthogonal polynomials, Riemann sphere, Degeneracy (mathematics), Plancherel theorem, Asymptote, Cauchy–Schwarz inequality, Semi-continuity, Analytic function, Blaschke product, Monotonic function, Subset, Continuous function, Lebesgue measure, Probability measure, QR algorithm, Schwarz lemma, Equation, Corollary, Toda lattice, Limit point, Harmonic measure, Moment problem, Elliptic function, Division by zero, Analytic continuation, Parameter, Summation, Variable (mathematics), Branch point, Uniform convergence, Spectral theory, Determinant, Argument principle, Poisson bracket, Calculation, Mathematical induction, Eigenvalues and eigenvectors, Absolute continuity, Compact space, Subgroup, Torus, Bijection, Polynomial, Topology of uniform convergence, Lecture, Kullback–Leibler divergence, Dimension, Jacobi matrix, Essential spectrum, Meromorphic function