Fourier transform for spectroscopy spectra?
As Mad Scientist said, many spectroscopy methods are already based on Fourier transform of a time-dependent observable. (This is not true of the basic spectrophotometry measurements, where you directly measure the transmittance as a function of frequency.) NMR is a method based on Fourier transform, as is FTIR.

Regarding NMR signal processing, it's a very wide field, which includes interesting research ranging from academic interest to software development. So, it's hard to answer your question other than saying “yes, of course, further processing of NMR signals is sometime possible”. One thing you have to realize is: while you have probably learnt proton NMR during your studies, this is only one NMR technique, and the simplest of them all. There are plenty of different NMR techniques, depending on the type of excitation and sequence of RF pulses, many of them being multidimensional. Each method can use a variety of signal processing techniques.