Calculating the Convolution Using DFT (FFT)

I have the following convolution as part of a numerical simulation.

$$T(r)=\int \mathrm{d}^3r_2\, p(r_2)f(r_2)\alpha(r-r_2)\, .$$

My problem is that the analytical expressions for $$f$$ and $$p$$ do exist but, I have the expression for $$\alpha$$ only in the Fourier domain in the form of $$\alpha(k)$$. I planned to evaluate in the following way:

1. Construct a grid using the mesh grid of $$100\times100\times100$$ using meshgrid and linspace in numpy
ran = linspace(-1,1,N_r)
x,y,z = meshgrid(ran,ran,ran) #position space

1. Construct the components xf, yf, zf in the Fourier domain from x, y, z
xf = fftn(x)
yf = fftn(y)
zf = fftn(z)

1. Find the Fourier transform of $$f(r)\times p(r)$$ using fftn in numpy
2. Multiply it with $$\alpha(k)$$
3. Taking the inverse Fourier transform using ifftn in numpy.

I am not very sure that the above method works and I actually failed to verify it properly. I tried using scipy.ndimage.convolve to compare the results with the inverse Fourier transform of the product in the Fourier domain. Is it correct as to what I am doing with the code? And is there a way where I can verify that a method is working with a simpler example?

Trying to verify:

I have tried the following to test the theory. Seems like it does not work. I expect that the result res_1 and res_2 to be the same. I also used the function real to truncate the tiny imaginary part which results from the fftn and ifftn functions.

x = linspace(-1,1,10)
xf = fftn(x)

def f(x):
return x**2+x**3*sin(x)

def g(k):
return k**2+k**3/(3-k**2)

g_k = g(xf)
g_x = real(ifftn(g_k))

res_1 = img_con(g_x,f(x))

res_2 = real(ifftn(g(xf)*fftn(f(x))))

print(res_1)
print(res_2)


Am I doing something which is wrong?