I was looking at a question on a physics forum (https://physics.stackexchange.com/questions/45955/splitting-light-into-colors-mathematical-expression-fourier-transforms) and I wanted a more mathematical perspective on it. How would such a convolution of a gaussian path integral as defined in arxiv.org/pdf/1110.2346 with a fourier transform be defined? Is such a thing even mathematically (or physically) meaningful? Please give me your opinions. Thanks in advance!