Integrability requirements for FT to whole complex plane

Stream: new members

[Nicholas Wilson]

for a function f : ℝ → ℝ to have a convergent Fourier transform on the whole complex plane the conditionf is in L^1(ℝ) and f = O(exp(-x^2) is sufficient. How do I state those conditions and how would I use that to prove

Integrable fun x => (↑(f x) - ↑(f (-x))) * Complex.xxx (s * ↑x)

(xxx=sin, cos)
?