I'm trying to use Mathematica to do a convolution of two gaussians with a rectangular function but I am getting very strange results.
Gaussian[x0_, \[Sigma]_] :=
1/((2 \[Pi])^(1/2) \[Sigma]) Exp[-((x - x0)^2/(2 \[Sigma]^2))]
\[Rho] = Gaussian[-1/2, \[Sigma]] +
Gaussian[1/2, \[Sigma]] /. \[Sigma] -> 1/10
F = FourierTransform[\[Rho], x, h]
In fourier terms we can do the same thing using a convolution with an averaging filter
Kernel = 1/PixelSize UnitStep[PixelSize/2 + h]
UnitStep[PixelSize/2 - h]
Filter = 2 (\[Pi]/2)^(1/2) InverseFourierTransform[Kernel, h, x]
Blurred = FullSimplify[InverseFourierTransform[F, h, x]*Filter]
The problem is here. The result of the fourier transform doesn't make any sense and it completely different of the non filtered \[Rho]
BlurredF = FourierTransform[Blurred, x, h]
The result is complitely different from the expected two blurred gaussians.
Where did I go wrong?