Description
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.
Chapter
2.9 The Szegő Function and Szegő Asymptotics
2.10 Asymptotics for Weyl Solutions
2.11 Additional Aspects of Szegő's Theorem
2.12 The Variational Approach to Szegő's Theorem
2.13 Another Approach to Szegő Asymptotics
2.14 Paraorthogonal Polynomials and Their Zeros
2.15 Asymptotics of the CD Kernel: Weak Limits
2.16 Asymptotics of the CD Kernel: Continuous Weights
2.17 Asymptotics of the CD Kernel: Locally Szegő Weights
Chapter 3. The Killip–Simon Theorem: Szegő for OPRL
3.1 Statement and Strategy
3.2 Weyl Solutions and Coefficient Stripping
3.3 Meromorphic Herglotz Functions
3.4 Step-by-Step Sum Rules for OPRL
3.5 The P[sub(2)] Sum Rule and the Killip–Simon Theorem
3.6 An Extended Shohat–Nevai Theorem
3.7 Szegő Asymptotics for OPRL
3.8 The Moment Problem: An Aside
3.9 The Krein Density Theorem and Indeterminate Moment Problems
3.10 The Nevai Class and Nevai Delta Convergence Theorem
3.11 Asymptotics of the CD Kernel: OPRL on [–2, 2]
3.12 Asymptotics of the CD Kernel: Lubinsky's Second Approach
Chapter 4. Sum Rules and Consequences for Matrix Orthogonal Polynomials
4.3 Coefficient Stripping
4.4 Step-by-Step Sum Rules of MOPRL
4.5 A Shohat–Nevai Theorem for MOPRL
4.6 A Killip–Simon Theorem for MOPRL
5.2 m-Functions and Quadratic Irrationalities
5.3 Real Floquet Theory and Direct Integrals
5.4 The Discriminant and Complex Floquet Theory
5.5 Potential Theory, Equilibrium Measures, the DOS, and the Lyapunov Exponent
5.6 Approximation by Periodic Spectra, I. Finite Gap Sets
5.7 Chebyshev Polynomials
5.8 Approximation by Periodic Spectra, II. General Sets
5.10 The CD Kernel for Periodic Jacobi Matrices
5.11 Asymptotics of the CD Kernel: OPRL on General Sets
5.12 Meromorphic Functions on Hyperelliptic Surfaces
5.13 Minimal Herglotz Functions and Isospectral Tori
Appendix to Section 5.13: A Child's Garden of Almost Periodic Functions
Chapter 6. Toda Flows and Symplectic Structures
6.2 Symplectic Dynamics and Completely Integrable Systems
6.4 Poisson Brackets of OPs, Eigenvalues, and Weights
6.5 Spectral Solution and Asymptotics of the Toda Flow
6.7 The Symes–Deift–Li–Tomei Integration: Calculation of the Lax Unitaries
6.8 Complete Integrability of Periodic Toda Flow and Isospectral Tori
6.9 Independence of Toda Flows and Trace Gradients
7.2 The Essential Spectrum
7.3 The Last–Simon Theorem on A.C. Spectrum
7.4 Remling's Theorem on A.C. Spectrum
7.5 Purely Reflectionless Jacobi Matrices on Finite Gap Sets
7.6 The Denisov–Rakhmanov–Remling Theorem
Chapter 8. Szegő and Killip–Simon Theorems for Periodic OPRL
8.3 The Determinant of the Matrix Weight
8.4 A Shohat–Nevai Theorem for Periodic Jacobi Matrices
8.5 Controlling the ℓ[sup(2)] Approach to the Isospectral Torus
8.6 A Killip–Simon Theorem for Periodic Jacobi Matrices
8.7 Sum Rules for Periodic OPUC
Chapter 9. Szegő's Theorem for Finite Gap OPRL
9.2 Fractional Linear Transformations
9.3 Möbius Transformations
9.5 Covering Maps for Multiconnected Regions
9.6 The Fuchsian Group of a Finite Gap Set
9.7 Blaschke Products and Green's Functions
9.8 Continuity of the Covering Map
9.9 Step-by-Step Sum Rules for Finite Gap Jacobi Matrices
9.10 The Szegő–Shohat–Nevai Theorem for Finite Gap Jacobi Matrices
9.11 Theta Functions and Abel's Theorem
9.12 Jost Functions and the Jost Isomorphism
Chapter 10. A.C. Spectrum for Bethe–Cayley Trees
10.2 The Free Hamiltonian and Radially Symmetric Potentials
10.3 Coefficient Stripping for Trees
10.4 A Step-by-Step Sum Rule for Trees
10.5 The Global ℓ[sup(2)] Theorem
10.6 The Local ℓ[sup(2)] Theorem
Szegős Theorem and Its Descendants
：Spectral Theory for L2 Perturbations of Orthogonal Polynomials
( Porter Lectures )
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