Complex Proofs of Real Theorems (University Lecture Series) (University Lecture Series, 58)

The International Society for Analysis, its Applications and Computation (ISAAC) has held its international congresses biennially since 1997. This proceedings volume reports on the progress in analysis, applications and computation in recent years as covered and discussed at the 7th ISAAC Congress. This volume includes papers on partial differential equations, function spaces, operator theory, integral transforms and equations, potential theory, complex analysis and generalizations, stochastic analysis, inverse problems, homogenization, continuum mechanics, mathematical biology and medicine. With over 500 participants from almost 60 countries attending the congress, the book comprises a broad selection of contributions in different topics 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 "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." -- Publisher 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, MÃ1⁄4ntz'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- elazko 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 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. 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 Peter D. Lax, Lawrence Zalcman. Includes Bibliographical References.