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Difficult convolution problem: can I ask for a solution in terms of elliptic integrals?

POSTED BY: Patrick McMullen
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Same solution as Henrik's one, a little more simplified :

f[t_] := If[-1 <= t <= 1, 2*(2/Pi)*Sqrt[1 - t^2], 0];
g[z_] := FourierTransform[f[x], x, z];
Carré[u_] := g[u] g[u];
FullSimplify@
 FunctionExpand[
  InverseFourierTransform[Carré[\[Omega]], \[Omega], t], -2 <= t <= 2]

giving :

(8 Sqrt[2] ((4 + t^2) EllipticE[1 - t^2/4] - 
   2 t^2 EllipticK[1 - t^2/4]))/(3 \[Pi]^(5/2))
POSTED BY: Claude Mante

Thank you -- this is helpful.
POSTED BY: Patrick McMullen

Strange that MMA has problems doing it ...

Another option would be to use the convolution theorem:

f[t_] := HeavisidePi[t/2] Sqrt[1 - t^2]
ft = FourierTransform[f[t], t, k];
convol = InverseFourierTransform[ft^2, k, t] // FullSimplify

enter image description here

Plotting this strange function took me quite some time - it seems not easy to calculate the respective values.
POSTED BY: Henrik Schachner
POSTED BY: Mariusz Iwaniuk
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