There are many problems which have implications on P vs. NP and other complexity classes.  Supposing that we're interested in Fourier transforms and/or multiplication algorithms, do faster Fourier transforms carry any implications for P vs. NP or other complexity classes?  For example, I seem to recall seeing something regarding faster matrix multiplication as it relates to P vs. NP.I'm really trying, although I may be somewhat vague, to determine if any Fourier-related transforms have P vs. NP implications, even if we have to trace implications through several problems or complexity classes.
What would faster Fourier Transform(FFT?) and/or multiplication algorithms imply?
complexity classes;p vs np;fourier analysis;intuition;fft