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Oct 31, 2016 at 16:47 comment added Christian Remling And conversely, if the FT has exponential decay, then $g$ is holomorphic on a strip of that width.
Oct 31, 2016 at 15:47 comment added Willie Wong How representative are the examples you gave? Taking the analytic continuation you can evaluate the Fourier transforms along the line $x - i \epsilon$ and get exponential decay. But this of course require your analytic continuation to be to a strip of a fixed width, which I don't know if is what you meant.
Oct 31, 2016 at 14:53 history asked Fedor Petrov CC BY-SA 3.0