Math 135 Fall 2024 102324 Paley Wiener Theorem
Автор: Winston Ou
Загружено: 2025-02-17
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Review: if the Fourier transform has exponential decay, then f can be extended to a holomorphic function on a strip (dependent on the decay exponent).
Statement of Paley-Wiener theorem: the Fourier transform is supported within [-M,M] iff f has an entire extension whose growth is exponential (with exponent M). Forward direction: use holomorphic Fourier inversion to define a (holomorphic) extension. Converse direction: Step I: Use Phragmén-Lindelöf to show that in fact we have the slightly better bound on the growth (note: there's a typo at 17:38 when I write that the Fourier transform's growth should be bounded: it's the function's growth, not its Fourier transform). Step II: "Find an epsilon of room/leeway" - create functions whose Fourier transforms converge to that of the original function and that in fact satisfy stronger bounds that allow us to obtain the vanishing of their Fourier transforms outside of the interval. Step III: Show that the perturbed functions' Fourier transforms actually do vanish outside of the interval.
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