Thanks for all your comments. It's definitely time to read a good book now.
My original problem is a convolution of two complex functions given as samples over quite different intervals with different n. The imaginary part of one of these functions is Lorentz-shaped. I thought it might be good to resample them in the frequency domain, then multiply and transform back. For the resampling I have to make sure the two resulting frequency axises are equivalent/physically meaningful. Also, of course, I wanted to understand what the NumPy/SciPy routines do and how to use them correctly. Now I'll try to just resample in the time domain, transform without looking at the result, then blindly multiply and transform back. This seems to work, but I'll have to find a different testcase so I can make sure the results are trustworthy.