Complex proofs of real theorems
 Responsibility
 Peter D. Lax, Lawrence Zalcman.
 Language
 English.
 Imprint
 Providence, R.I. : American Mathematical Society, c2012.
 Physical description
 xi, 90 p. ; 26 cm.
 Series
 University lecture series (Providence, R.I.) ; 58.
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Creators/Contributors
 Author/Creator
 Lax, Peter D.
 Contributor
 Zalcman, Lawrence Allen.
Contents/Summary
 Bibliography
 Includes bibliographical references.
 Contents

 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 FugledePutnam theorem
 3.2. Toeplitz operators
 3.3. A theorem of Beurling
 3.4. Prediction theory
 3.5. The RieszThorin 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 PaleyWiener theorem
 4.5. The Titchmarsh convolution theorem
 4.6. Hardy's theorem
 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.
 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 PaleyWiener theorem, the Titchmarsh convolution theorem, the GleasonKahaneZelazko theorem, and the FatouJuliaBaker 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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Subjects
Bibliographic information
 Publication date
 2012
 Series
 University lecture series ; v. 58
 ISBN
 9780821875599 (alk. paper)
 0821875590 (alk. paper)