I am currently taking classes in Combinatorics and Abstract Harmonic Analysis, which deal with the Fourier Analysis of functions on (mostly Hausdorff) locally compact topological (mostly abelian) groups. However, I cannot find sufficient motivation for this study, i.e., why just (Hausdorff) locally compact (abelian) groups? Why not any other group, even a ring, or some devilish algebraic structure?
It'd be helpful if the folks at MSE could highlight the said motivations, possibly in terms of examples of useful and pivotal connections with other areas of mathematics, interesting results, conjectures, etc. Thank you!
This post is somewhat helpful but not thoroughly satisfying. Among other things, it doesn't seem to talk about why other algebraic structures are not used, for instance.