Complex Proofs of Real Theorems
About this Title
Peter D. Lax, Courant Institute, New York, NY and Lawrence Zalcman, Bar-Ilan University, Ramat Gan, Israel
Publication: University Lecture Series
Publication Year 2012: Volume 58
ISBNs: 978-0-8218-7559-9 (print); 978-0-8218-8489-8 (online)
MathSciNet review: MR2827550
MSC: Primary 30-02; Secondary 11-02, 26-02, 46-01
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, Müntz'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–Żelazko 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.
Graduate students and research mathematicians interested in analysis.
Table of Contents
- Chapter 1. Early triumphs
- Chapter 2. Approximation
- Chapter 3. Operator theory
- Chapter 4. Harmonic analysis
- 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