Softcover ISBN:  9780821875599 
Product Code:  ULECT/58 
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Electronic ISBN:  9780821884898 
Product Code:  ULECT/58.E 
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Book DetailsUniversity Lecture SeriesVolume: 58; 2012; 90 ppMSC: Primary 30; 41; 47; 42; 46; 26; 11; 60;
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.ReadershipGraduate students and research mathematicians interested in analysis.

Table of Contents

Chapters

Chapter 1. Early triumphs

Chapter 2. Approximation

Chapter 3. Operator theory

Chapter 4. Harmonic analysis

Chapter 5. Banach algebras: The GleasonKahaneŻelazko theorem

Chapter 6. Complex dynamics: The FatouJuliaBaker theorem

Chapter 7. The prime number theorem

Coda: Transonic airfoils and SLE

Appendix A. Liouville’s theorem in Banach spaces

Appendix B. The BorelCarathéodory inequality

Appendix C. PhragménLindelöf theorems

Appendix D. Normal families


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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.

Chapters

Chapter 1. Early triumphs

Chapter 2. Approximation

Chapter 3. Operator theory

Chapter 4. Harmonic analysis

Chapter 5. Banach algebras: The GleasonKahaneŻelazko theorem

Chapter 6. Complex dynamics: The FatouJuliaBaker theorem

Chapter 7. The prime number theorem

Coda: Transonic airfoils and SLE

Appendix A. Liouville’s theorem in Banach spaces

Appendix B. The BorelCarathéodory inequality

Appendix C. PhragménLindelöf theorems

Appendix D. Normal families