Sorry if this question is a bit vague
I took a course on PDEs and learned or reviewed a lot of math revolving around Fourier transforms, convolutions, distributions, Gaussian functions, etc, all simply in the context of solving PDEs. I later took another course, this time on image processing (which was moreover in the neuroscience department rather than mathematics) and was quite surprised to see the same mathematics pop up: we used Fourier transforms, convolutions, the Dirac delta distribution, and Gaussian functions, but now all entirely in the context of transforming signals/images and highlighting different features of them.
It seems strange to me that the same mathematics and so much of it should pop up in two very different subjects - at least on the surface level there doesn't seem to be much connection at all between PDEs and signal processing. I was wondering if anyone has some sort of explanation on how they are somehow connected, or why the same mathematics pops up.