Fourier transform of a non-integrable function ...
I am working on Fourier Transform of a non-integrable function. And I am working on the numerical implementation of the FT, using FFT.
Let's call this function F(v). One example is F(v)=-1/(5*i*v)*(1-exp(5*i*v))
The function has a 1/v term in its asymptotic expansion, which corresponds to a jump or Heaviside function in its transform.
Using FFT, I can see the jump, with some overshooting after the jump and undershooting before the jump, along with some ripples.
I want to reduce these artifacts as much as possible, because I am going to integrate this transform against other functions later. Hence obtaining a smooth function without much numerical artifacts is vital. How do I do that?
I have thought about removing the 1/v component from F(v), and then later add it back.
If I could remove 1/v from F(v), then the remaining portion of the function will be of asymptotic order of 1/v^2, and then it is integrable and so its FT will be smooth. And then I manually add the jump/Heaviside function back into its FT.
This scheme sounded good to me. However, I realized that I cannot define a new function G(v)=F(v)-1/v, because 1/v is not defined for v=0. FFT doesn't work on G(0).
I have also thought about "denting" 1/v a little bit at v=0, but without success.
I have also pondered that: using FFT, effectively I have truncated the Fourier integral. Hence the non-integrability is not an issue any more. Instead of trying to remove the 1/v component, I should try make correction about the truncation error. But still, I wasn't able to remove the overshooting, undershooting, and rippling around the jump location...
Thanks a lot!