I don't exactly know how to ask this question, so forgive any lack of clarity.
I understand that harmonic analysis is an abstract generalization of Fourier analysis. But I am having trouble seeing why one would do harmonic analysis, beyond the heuristic "Fourier analysis is cool, so let's do Fourier analysis but abstract".
I wasn't able to boil down my issue to a single question, so I have boiled it down to the 2 following (similar) questions.
- When studying "pure" harmonic analysis (i.e for its own sake), what are some big goals, in a first course or in current research?
- What are some uses of harmonic analysis in other fields, such as complex analysis, representation theory or really anything else you guys know about?