We will read Complex Proofs of Real Theorems by Lax and Zalcman
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–Zelazko theorem, and the Fatou–Julia–Baker theorem.
https://moodle2.uni-potsdam.de/course/view.php?id=42548
