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One interesting generalization follows Fourier's original analysis for Ordinary Differential Equations on $\mathbb{R}$. It's easiest to break first into ODEs on a half line $[0,\infty)$ for Sturm-Liouville problems $$Lf=-\frac{d^{2}f}{dx^{2}}+q(x)f(x) = g(x),\\ \cos\alpha f(0)+\sin\alpha f'(0) = 0.$$ where ...
I'm hardly qualified to answer this question, but you might find the following references useful. Let's start with the two classical examples of the Fourier transform: the Fourier transforms for $L^2$ functions on the line and the circle. Generalizing the notion of space The line and the circle are both topological abelian groups, and the Fourier ...