CONTENTS xi
Volume 1: Hardy, Hankel and Toeplitz
Part A. An Invitation to Hardy Classes 1
Chapter 1. Invariant Subspaces of
L2(/x)
7
1.1. Basic Definitions 7
1.2. Doubly Invariant Subspaces 8
1.3. Simply Invariant Subspaces, the Case fi = m 9
1.4. Inner Functions. A Uniqueness Theorem 10
1.5. Invariant Subspaces of L2(/i): the General Case 10
1.6. Exercises and Further Results 13
1.7. Notes and Remarks 17
Chapter 2. First Applications 21
2.1. Straightforward Corollaries 21
2.2. The Problem of Weighted Polynomial Approximation 22
2.3. A Probabilistic Interpretation 23
2.4. The Inner-Outer Factorization 23
2.5. Arithmetic of Inner Functions 24
2.6. A Characterization of Outer Functions 25
2.7. Szego Infimum and the Riesz Brothers' Theorem 25
2.8. Exercises and Further Results 27
2.9. Notes and Remarks 28
Chapter 3.
Hp
Classes. Canonical Factorization 31
3.1. The Main Definition 31
3.2. Straightforward Properties 32
3.3. A Digression on Convolutions and Fourier series 32
3.4. Identifying fl*(0) and H? 34
3.5. Jensen's Formula and Jensen's Inequality 35
3.6. The Boundary Uniqueness Theorem 36
3.7. Blaschke Products 37
3.8. Nontangential Boundary Limits 39
3.9. The Riesz-Smirnov Canonical Factorization 41
3.10. Approximation by inner functions and Blaschke products 44
3.11. Vector valued
#p-spaces
and the Fatou theorem 46
3.12. Exercises and Further Results 54
3.13. Notes and Remarks 57
Chapter 4. Szego Infimum, and Generalized Phragmen-Lindelof Principle 65
4.1. Szego Infimum and Weighted Polynomial Approximation 65
4.2. How to Recognize an Outer Function 67
4.3. Locally Outer Functions 68
4.4. The Smirnov Class D 72
4.5. A Conformally Invariant Framework 72
4.6. The Generalized Phragmen-Lindelof Principle 73
4.7. Classical Examples 74
4.8. Exercises and Further Results 75
4.9. Notes and Remarks 87
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