Contents
Preface ix
Chapter 1. Early Triumphs 1
1.1. The Basel Problem 1
1.2. The Fundamental Theorem of Algebra 3
Chapter 2. Approximation 5
2.1. Completeness of Weighted Powers 5
2.2. The Müntz Approximation Theorem 7
Chapter 3. Operator Theory 13
3.1. The Fuglede-Putnam Theorem 13
3.2. Toeplitz Operators 14
3.3. A Theorem of Beurling 22
3.4. Prediction Theory 28
3.5. The Riesz-Thorin Convexity Theorem 34
3.6. The Hilbert Transform 40
Chapter 4. Harmonic Analysis 45
4.1. Fourier Uniqueness via Complex Variables (d’après D.J. Newman) 45
4.2. A Curious Functional Equation 46
4.3. Uniqueness and Nonuniqueness for the Radon Transform 49
4.4. The Paley-Wiener Theorem 54
4.5. The Titchmarsh Convolution Theorem 57
4.6. Hardy’s Theorem 58
Chapter 5. Banach Algebras: The Gleason-Kahane-Żelazko Theorem 63
Chapter 6. Complex Dynamics: The Fatou-Julia-Baker Theorem 67
Chapter 7. The Prime Number Theorem 71
Coda: Transonic Airfoils and SLE 77
Appendix A. Liouville’s Theorem in Banach Spaces 81
Appendix B. The Borel-Carathéodory Inequality 83
Appendix C. Phragmén-Lindelöf Theorems 85
Appendix D. Normal Families 87
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