What do you mean, not much intuition for Fourier transforms? Sure, sometimes you get weird results and need to work with fairly generalized versions of functions, and a bit of background in functional analysis is helpful, but a strong intuition about the Fourier transform is both extremely useful, and not that hard to develop.

For starters, I'd recommend the course notes to Stanford's EE261 (https://see.stanford.edu/materials/lsoftaee261/book-fall-07....) - well written, very funny, a nice level of rigour, etc...