《實(shí)定理的復(fù)證明》是對(duì)Hadamard的格言“實(shí)域中兩個(gè)真理之間的最好和最短路程是通過復(fù)域”的延伸思考。面向熟悉研究生一年級(jí)水平分析學(xué)的受眾，此書的目的在于解釋復(fù)變量是如何對(duì)分析的一些領(lǐng)域中的許多類重要結(jié)果提供了快速而高效的證明, 這些領(lǐng)域包括諸如近似理論、算子理論、調(diào)和分析和復(fù)動(dòng)力系統(tǒng)。
Preface
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 Muntz 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'apres 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-Zelazko 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-Caratheodory Inequality
Appendix C． Phragmen-Lindelof Theorems
Appendix D． Normal Families
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