Complex proofs of real theorems /
Complex analysis has been pushed to the margins of mathematics over the past half century say mathematicians Lax (New York U.) and Zalcman (Bar-Ilan U.). They present an overview for anyone who enjoys analysis and reading pretty proofs, and are familiar with basic functional analysis and some elemen...
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Format: | Book |
Language: | English |
Published: |
Providence, R.I. :
American Mathematical Society,
[2012], ©2012
Providence, R.I. : c2012 Providence, R.I. : [2012] |
Series: | University lecture series (Providence, R.I.) ;
58 University lecture series ; v. 58 |
Subjects: |
Table of 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 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'après 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-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
- 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 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'après 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-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
- ch. 1 Early triumphs. 1.1. The Basel problem ; 1.2. The fundamental theorem of algebra
- ch. 2. Approximation. 2.1. Completeness of weighted powers ; 2.2. The Müntz approximation theorem
- ch. 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
- ch. 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 Paley-Wiener theorem ; 4.5. The Titchmarsh convolution theorem ; 4.6. Hardy's theorem
- ch. 5. Banach algebras : the Gleason-Kahane-Żelazko theorem
- ch. 6. Complex dynamics : the Fatou-Julia-Baker theorem
- ch. 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.
- 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