Cover -- Contents -- Preface -- Chapter 1. Gems of Spectral Theory -- 1.1 What Is Spectral Theory? -- 1.2 OPRL as a Solution of an Inverse Problem -- 1.3 Favard's Theorem, the Spectral Theorem, and the Direct Problem for OPRL -- 1.4 Gems of Spectral Theory -- 1.5 Sum Rules and the Plancherel Theorem -- 1.6 Pólya's Conjecture and Szegő's Theorem -- 1.7 OPUC and Szegő's Restatement -- 1.8 Verblunsky's Form of Szegő's Theorem -- 1.9 Back to OPRL: Szegő Mapping and the Shohat-Nevai Theorem -- 1.10 The Killip-Simon Theorem -- 1.11 Perturbations of the Periodic Case -- 1.12 Other Gems in the Spectral Theory of OPUC -- Chapter 2. Szegő's Theorem -- 2.1 Statement and Strategy -- 2.2 The Szegő Integral as an Entropy -- 2.3 Carathéodory, Herglotz, and Schur Functions -- 2.4 Weyl Solutions -- 2.5 Coefficient Stripping, Geronimus' and Verblunsky's Theorems, and Continued Fractions -- 2.6 The Relative Szegő Function and the Step-by-Step Sum Rule -- 2.7 The Proof of Szegő's Theorem -- 2.8 A Higher-Order Szegő Theorem -- 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 -- 4.1 Introduction -- 4.2 Basics of MOPRL -- 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 -- Chapter 5. Periodic OPRL -- 5.1 Overview -- 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 -- 5.9 Regularity: An Aside -- 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 -- 5.14 Periodic OPUC -- Chapter 6. Toda Flows and Symplectic Structures -- 6.1 Overview -- 6.2 Symplectic Dynamics and Completely Integrable Systems -- 6.3 QR Factorization -- 6.4 Poisson Brackets of OPs, Eigenvalues, and Weights -- 6.5 Spectral Solution and Asymptotics of the Toda Flow -- 6.6 Lax Pairs -- 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 -- 6.10 Flows for OPUC -- Chapter 7. Right Limits -- 7.1 Overview -- 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 -- 8.1 Overview -- 8.2 The Magic Formula -- 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 -- 9.1 Overview -- 9.2 Fractional Linear Transformations -- 9.3 Möbius Transformations -- 9.4 Fuchsian Groups -- 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 -- 9.13 Szegő Asymptotics -- Chapter 10. A.C. Spectrum for Bethe-Cayley Trees -- 10.1 Overview -- 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 -- Bibliography -- Author Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- J -- K -- L -- M -- N -- O -- P -- R -- S -- T -- U -- V -- W -- Y -- Z -- Subject Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- J -- K -- L -- M -- N -- O -- P -- Q -- R -- S -- T -- U -- V -- W.