Exploring Fourier Transforms and Integrability for Different Functions

[shoplifter]

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))?

 

[jbunniii]

Is this homework? If so, what have you done so far?

 

[shoplifter]

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).

sry :(

 

[shoplifter]

any help please? :(