پولینوم‌های عمود بر دایره واحد. جلد اول: نظریه کلاسیک

This two-part volume gives a comprehensive overview of the theory of probability measures on the unit circle, viewed especially in terms of the orthogonal polynomials defined by those measures. A major theme involves the connections between the Verblunsky coefficients (the coefficients of the recurrence equation for the orthogonal polynomials) and the measures, an analog of the spectral theory of one-dimensional Schrödinger operators. Among the topics discussed along the way are the asymptotics of Toeplitz determinants (Szegő's theorems), limit theorems for the density of the zeros of orthogonal polynomials, matrix representations for multiplication by z (CMV matrices), periodic Verblunsky coefficients from the point of view of meromorphic functions on hyperelliptic surfaces, and connections between the theories of orthogonal polynomials on the unit circle and on the real line. Readership: Graduate students and research mathematicians interested in analysis. Cover 1 S Title 2 Orthogonal Polynomials on the Unit Circle, Part 1 : Classical Theory 4 © 2005 by the American Mathematical Society 5 ISBN 0-8218-3446-0 (part 1) 5 ISBN 0-8218-3675-7 (part 2) 5 QA404.5 .S45 2004 515'.55-dc22 5 LCCN 2004046219 5 Dedication 6 Contents 8 Preface to Part 1 12 Notation 18 CHAPTER 1 The Basics 28 1.1. Introduction 28 1.2. Orthogonal Polynomials on the Real Line 38 1.3. Caratheodory and Schur Functions 52 1.4. An Introduction to Operator and Spectral Theory 67 1.5. Verblunsky Coefficients and the Szego Recurrence 82 1.6. Examples of OPUC 98 1.7. Zeros and the First Proof of Verblunsky's Theorem 117 CHAPTER 2 Szego's Theorem 136 2.1. Toeplitz Determinants and Verblunsky Coefficients 136 2.2. Extremal Properties, the Christoffel Functions, and the Christoffel-Darboux Formula 144 2.3. Entropy Semicontinuity and the First Proof of Szego's Theorem 163 2.4. The Szego Function 170 2.5. Szego's Theorem Using the Poisson Kernel 178 2.6. Khrushchev's Proof of Szego's Theorem 183 2.7. Consequences of Szego's Theorem 186 2.8. A Higher-Order Szego Theorem 199 2.9. The Relative Szego Function 205 2.10. Totik's Workshop 211 2.11. Riesz Products and Khrushchev's Workshop 216 2.12. The Workshop of Denisov and Kupin 224 2.13. Matrix-Valued Measures 233 CHAPTER 3 Tools for Geronimus' Theorem 244 3.1. Verblunsky's Viewpoint: Proofs of Verblunsky's and Geronimus' Theorems 244 3.2. Second Kind Polynomials 249 3.3. KW Pairs 266 3.4. Coefficient Stripping and Associated Polynomials 272 CHAPTER 4 Matrix Representations 278 4.1. The GGT Representation 278 4.2. The CMV Representation 289 4.3. Spectral Consequences of the CMV Representation 301 4.4. The Resolvent of the CMV Matrix 314 4.5. Rank Two Perturbations and Decoupling of CMV Matrices 320 CHAPTER 5 Baxter's Theorem 328 5.1. Wiener-Hopf Factorization and the Inverses of Finite Toeplitz Matrices 328 5.2. Baxter's Proof 340 CHAPTER 6 The Strong Szego Theorem 346 6.1. The lbragimov and Golinskii-Ibragimov Theorems 346 6.2. The Borodin-Okounkov Formula 360 6.3. Representations of U(n) and the Bump-Diaconis Proof 373 6.4. Toeplitz Determinants as the Statistical Mechanics of Coulomb Gases and Johansson's Proof 379 6.5. The Combinatorial Approach and Kac's Proof 395 6.6. A Second Look at Ibragimov's Theorem 403 CHAPTER 7 Verblunsky Coefficients With Rapid Decay 408 7.1. The Rate of Exponential Decay and a Theorem of Nevai-Totik 408 7.2. Detailed Asymptotics of the Verblunsky Coefficients 414 CHAPTER 8 The Density of Zeros 418 8.1. The Density of Zeros Measure via Potential Theory 418 8.2. The Density of Zeros Measure via the CMV Matrix 430 8.3. Rotation Numbers 437 8.4. A Gallery of Zeros 439 Bibliography 452 Author Index 484 Subject Index 490 Back Cover 494 This two-part book is a comprehensive overview of the theory of probability measures on the unit circle, viewed especially in terms of the orthogonal polynomials defined by those measures. A major theme involves the connections between the Verblunsky coefficients (the coefficients of the recurrence equation for the orthogonal polynomials) and the measures, an analog of the spectral theory of one-dimensional Schrödinger operators. Among the topics discussed along the way are the asymptotics of Toeplitz determinants (Szegő's theorems), limit theorems for the density of the zeros of orthogonal polynomials, matrix representations for multiplication by $z$ (CMV matrices), periodic Verblunsky coefficients from the point of view of meromorphic functions on hyperelliptic surfaces, and connections between the theories of orthogonal polynomials on the unit circle and on the real line. This two-part book is a comprehensive overview of the theory of probability measures on the unit circle, viewed especially in terms of the orthogonal polynomials defined by those measures. A major theme involves the connections between the Verblunsky coefficients (the coefficients of the recurrence equation for the orthogonal polynomials) and the measures, an analog of the spectral theory of one-dimensional Schrödinger operators. Among the topics discussed along the way are the asymptotics of Toeplitz determinants (Szegö's theorems), limit theorems for the density of the zeros of orthogonal polynomials, matrix representations for multiplication by z (CMV matrices), periodic Verblunsky coefficients from the point of view of meromorphic functions on hyperelliptic surfaces, and connections between the theories of orthogonal polynomials on the unit circle and on the real line Chapter 1 The Basics Chapter 2 Szegő's theorem Chapter 3 Tools for Geronimus' theorem Chapter 4 Matrix representations Chapter 5 Baxter's theorem Chapter 6 The strong Szegő theorem Chapter 7 Verblunsky coefficients with rapid decay Chapter 8 The density of zeros Bibliography Author index Subject index