Szegő's Theorem and Its Descendants

Barry Simon is the IBM Professor of Mathematics and Theoretical Physics at the California Institute of Technology. His books include "Methods of Modern Mathematical Physics and Orthogonal Polynomials on the Unit Circle".

• Preface ix Chapter 1. Gems of Spectral Theory 1 1.1 What Is Spectral Theory? 1 1.2 OPRL as a Solution of an Inverse Problem 4 1.3 Favard's Theorem, the Spectral Theorem, and the Direct Problem for OPRL 11 1.4 Gems of Spectral Theory 18 1.5 Sum Rules and the Plancherel Theorem 20 1.6 Polya's Conjecture and Szego's Theorem 22 1.7 OPUC and Szego's Restatement 24 1.8 Verblunsky's Form of Szego's Theorem 26 1.9 Back to OPRL: Szego Mapping and the Shohat-Nevai Theorem 30 1.10 The Killip-Simon Theorem 37 1.11 Perturbations of the Periodic Case 39 1.12 Other Gems in the Spectral Theory of OPUC 41 Chapter 2. Szego's Theorem 43 2.1 Statement and Strategy 44 2.2 The Szego Integral as an Entropy 48 2.3 Caratheodory, Herglotz, and Schur Functions 52 2.4 Weyl Solutions 66 2.5 Coefficient Stripping, Geronimus' and Verblunsky's Theorems, and Continued Fractions 74 2.6 The Relative Szego Function and the Step-by-Step Sum Rule 80 2.7 The Proof of Szego's Theorem 84 2.8 A Higher-Order Szego Theorem 86 2.9 The Szego Function and Szego Asymptotics 91 2.10 Asymptotics for Weyl Solutions 97 2.11 Additional Aspects of Szego's Theorem 98 2.12 The Variational Approach to Szego's Theorem 103 2.13 Another Approach to Szego Asymptotics 108 2.14 Paraorthogonal Polynomials and Their Zeros 113 2.15 Asymptotics of the CD Kernel: Weak Limits 118 2.16 Asymptotics of the CD Kernel: Continuous Weights 123 2.17 Asymptotics of the CD Kernel: Locally Szego Weights 132 Chapter 3. The Killip-Simon Theorem: Szego for OPRL 143 3.1 Statement and Strategy 143 3.2 Weyl Solutions and Coefficient Stripping 144 3.3 Meromorphic Herglotz Functions 151 3.4 Step-by-Step Sum Rules for OPRL 158 3.5 The P2 Sum Rule and the Killip-Simon Theorem 163 3.6 An Extended Shohat-Nevai Theorem 167 3.7 Szego Asymptotics for OPRL 173 3.8 The Moment Problem: An Aside 183 3.9 The Krein Density Theorem and Indeterminate Moment Problems 203 3.10 The Nevai Class and Nevai Delta Convergence Theorem 207 3.11 Asymptotics of the CD Kernel: OPRL on [?2, 2] 213 3.12 Asymptotics of the CD Kernel: Lubinsky's Second Approach 222 Chapter 4. Sum Rules and Consequences for Matrix Orthogonal Polynomials 228 4.1 Introduction 228 4.2 Basics of MOPRL 229 4.3 Coefficient Stripping 234 4.4 Step-by-Step Sum Rules of MOPRL 239 4.5 A Shohat-Nevai Theorem for MOPRL 244 4.6 A Killip-Simon Theorem for MOPRL 246 Chapter 5. Periodic OPRL 250 5.1 Overview 250 5.2 m-Functions and Quadratic Irrationalities 253 5.3 Real Floquet Theory and Direct Integrals 257 5.4 The Discriminant and Complex Floquet Theory 263 5.5 Potential Theory, Equilibrium Measures, the DOS, and the Lyapunov Exponent 283 5.6 Approximation by Periodic Spectra, I. Finite Gap Sets 306 5.7 Chebyshev Polynomials 312 5.8 Approximation by Periodic Spectra, II. General Sets 319 5.9 Regularity: An Aside 323 5.10 The CD Kernel for Periodic Jacobi Matrices 327 5.11 Asymptotics of the CD Kernel: OPRL on General Sets 334 5.12 Meromorphic Functions on Hyperelliptic Surfaces 344 5.13 Minimal Herglotz Functions and Isospectral Tori 360 Appendix to Section 5.13: A Child's Garden of Almost Periodic Functions 371 5.14 Periodic OPUC 377 Chapter 6. Toda Flows and Symplectic Structures 379 6.1 Overview 379 6.2 Symplectic Dynamics and Completely Integrable Systems 382 6.3 QR Factorization 387 6.4 Poisson Brackets of OPs, Eigenvalues, and Weights 390 6.5 Spectral Solution and Asymptotics of the Toda Flow 398 6.6 Lax Pairs 403 6.7 The Symes-Deift-Li-Tomei Integration: Calculation of the Lax Unitaries 404 6.8 Complete Integrability of Periodic Toda Flow and Isospectral Tori 408 6.9 Independence of Toda Flows and Trace Gradients 413 6.10 Flows for OPUC 416 Chapter 7. Right Limits 418 7.1 Overview 418 7.2 The Essential Spectrum 419 7.3 The Last-Simon Theorem on A.C. Spectrum 426 7.4 Remling's Theorem on A.C. Spectrum 431 7.5 Purely Reflectionless Jacobi Matrices on Finite Gap Sets 452 7.6 The Denisov-Rakhmanov-Remling Theorem 454 Chapter 8. Szego and Killip-Simon Theorems for Periodic OPRL 456 8.1 Overview 456 8.2 The Magic Formula 457 8.3 The Determinant of the Matrix Weight 460 8.4 A Shohat-Nevai Theorem for Periodic Jacobi Matrices 463 8.5 Controlling the L2 Approach to the Isospectral Torus 465 8.6 A Killip-Simon Theorem for Periodic Jacobi Matrices 473 8.7 Sum Rules for Periodic OPUC 475 Chapter 9. Szego's Theorem for Finite Gap OPRL 477 9.1 Overview 477 9.2 Fractional Linear Transformations 478 9.3 Mobius Transformations 496 9.4 Fuchsian Groups 505 9.5 Covering Maps for Multiconnected Regions 518 9.6 The Fuchsian Group of a Finite Gap Set 525 9.7 Blaschke Products and Green's Functions 540 9.8 Continuity of the Covering Map 556 9.9 Step-by-Step Sum Rules for Finite Gap Jacobi Matrices 562 9.10 The Szego-Shohat-Nevai Theorem for Finite Gap Jacobi Matrices 564 9.11 Theta Functions and Abel's Theorem 570 9.12 Jost Functions and the Jost Isomorphism 576 9.13 Szego Asymptotics 583 Chapter 10. A.C. Spectrum for Bethe-Cayley Trees 591 10.1 Overview 591 10.2 The Free Hamiltonian and Radially Symmetric Potentials 594 10.3 Coefficient Stripping for Trees 597 10.4 A Step-by-Step Sum Rule for Trees 600 10.5 The Global l2 Theorem 601 10.6 The Local l2 Theorem 603 Bibliography 607 Author Index 641 Subject Index 647

"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