I've been taught (in the context of Sturm-Liouville problems) that Fourier series can be explained using inner products and the idea of projection onto eigenfunctions in a Hilbert space. In those cases, the eigenvalues are infinite, but discrete. I'm now taking a quantum mechanics course, and reading the 3rd chapter of Griffiths' Quantum book piqued my interest about the idea of continuous sets of eigenvalues and their corresponding eigenfunctions which may not be normalizable in the usual sense. Being in electrical engineering, the Fourier Transform is something that's relatively familiar to me, and it seems very much related to this idea. So my question is, is it possible to interpret the Fourier transform in terms of projection onto a set of eigenfunctions corresponding to a continuous set of eigenvalues (similar to Fourier series)? If so, can anyone point me to some resources explaining how this is done which would be understandable to a student who isn't majoring in math (but is willing to learn)? As an idea of what I've done so far, I've been trying to read up on Rigged Hilbert Spaces and distribution theory, but I've mostly been getting bogged down in unfamiliar mathematics and I haven't been able to make a solid connection between those ideas and the Physics/Electrical Engineering I'm trying to understand better. Thanks for any help.