Complex proofs of real theorems
QA331.7 .L39 2012
- Unknown QA331.7 .L39 2012
- Zalcman, Lawrence Allen.
- Includes bibliographical references.
- Chapter 1. Early triumphs
- 1.1. The Basel problem
- 1.2. The fundamental theorem of algebra
- Chapter 2. Approximation
- 2.1. Completeness of weighted powers
- 2.2. The Müntz approximation theorem
- Chapter 3. Operator theory
- 3.1. The Fuglede-Putnam theorem
- 3.2. Toeplitz operators
- 3.3. A theorem of Beurling
- 3.4. Prediction theory
- 3.5. The Riesz-Thorin convexity theorem
- 3.6. The Hilbert transform
- Chapter 4. Harmonic analysis
- 4.1. Fourier uniqueness via complex variables (d'après D.J. Newman)
- 4.2. A curious functional equation
- 4.3. Uniqueness and nonuniqueness for the Radon transform
- 4.4. The Paley-Wiener theorem
- 4.5. The Titchmarsh convolution theorem
- 4.6. Hardy's theorem
- Chapter 5. Banach algebras: the Gleason-Kahane-Żelazko theorem
- Chapter 6. Complex dynamics: the Fatou-Julia-Baker theorem
- Chapter 7. The prime number theorem
- Coda. Transonic airfoils and SLE
- Appendix A. Liouville's theorem in Banach spaces
- Appendix B. The Borel-Carathéodory inequality
- Appendix C. Phragmén-Lindelöf theorems
- Appendix D. Normal families.
- Publisher's Summary
- Complex Proofs of Real Theorems is an extended meditation on Hadamard's famous dictum, ""The shortest and best way between two truths of the real domain often passes through the imaginary one." Directed at an audience acquainted with analysis at the first year graduate level, it aims at illustrating how complex variables can be used to provide quick and efficient proofs of a wide variety of important results in such areas of analysis as approximation theory, operator theory, harmonic analysis, and complex dynamics. Topics discussed include weighted approximation on the line, Muntz's theorem, Toeplitz operators, Beurling's theorem on the invariant spaces of the shift operator, prediction theory, the Riesz convexity theorem, the Paley-Wiener theorem, the Titchmarsh convolution theorem, the Gleason-Kahane-Zelazko theorem, and the Fatou-Julia-Baker theorem. The discussion begins with the world's shortest proof of the fundamental theorem of algebra and concludes with Newman's almost effortless proof of the prime number theorem. Four brief appendices provide all necessary background in complex analysis beyond the standard first year graduate course. Lovers of analysis and beautiful proofs will read and reread this slim volume with pleasure and profit.
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- Publication date
- Peter D. Lax, Lawrence Zalcman.
- University lecture series ; v. 58