
Complex Proofs of Real Theorems
American Mathematical Society (Publisher)
Published on 30. March 2012
Book
Paperback/Softback
90 pages
978-0-8218-7559-9 (ISBN)
Description
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.
More details
Series
Language
English
Place of publication
Providence
United States
Target group
Professional and scholarly
Weight
200 gr
ISBN-13
978-0-8218-7559-9 (9780821875599)
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Schweitzer Classification
Persons
Peter D. Lax, Courant Institute, New York, NY, USA
Lawrence Zalcman, Bar-Ilan University, Ramat Gan, Israel
Lawrence Zalcman, Bar-Ilan University, Ramat Gan, Israel
Content
Preface
Early triumphs
Approximation
Operator theory
Harmonic analysis
Banach algebras: The Gleason-Kahane-Zelazko theorem
Complex dynamics: The Fatou-Julia-Baker theorem
The prime number theorem
Coda: Transonic airfoils and SLE
Liouville's theorem in Banach spaces
The Borel-Caratheodory inequality
Phragmen-Lindeloef theorems
Normal families
Early triumphs
Approximation
Operator theory
Harmonic analysis
Banach algebras: The Gleason-Kahane-Zelazko theorem
Complex dynamics: The Fatou-Julia-Baker theorem
The prime number theorem
Coda: Transonic airfoils and SLE
Liouville's theorem in Banach spaces
The Borel-Caratheodory inequality
Phragmen-Lindeloef theorems
Normal families