prove that f(x) = (1 + |x|)^{-a} is the fourier transform of some integrable function on R, when a > 1. what happens when 0 < a <= 1? how about the function f(x) = 1/(log(|x|^2 + 2))?
Is this homework? If so, what have you done so far?
#3
shoplifter
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0
yes, and i realize that we want the integral to converge when we take the inverse transform. so in order to do that, i'm guessing the denominator has to have a power > 1, which is why we have that condition on a. so it will fail the second time, i guess. but i can't formalize my argument (and i'm clueless about the third one).