Let $G$ be a locally compact group. To the best of my understanding, harmonic analysis has three legs that all work perfectly in the case that $G$ is in addition compact and abelian, but have generalizations in other cases:
- The notion of a "dual" $\hat{G}$ (itself a topological group if $G$ is abelian, but generalizations do not require that) in such a way that $G$ is naturally isomorphic to $\hat{\hat{G}}$ and such that one has an isometry $L^2(G)\rightarrow L^2(\hat{G})$.
- Decomposing $L^2(G)$ as a direct integral (direct sum in the case that $G$ is compact) of irreducible unitary representations.
- Coming up with a simultaneous eigenbasis (w.r.t the $G$ action) for $L^2(G)$ (in the case that $G$ is compact).
Perhaps this is because I don't have any engineering background, but it's difficult for me to tell what each (or even any!) of these legs of harmonic analysis are actually good for.
Question
For each of the above "legs of harmonic analysis" separately: why is such a result desirable, and what can it be naturally applied towards in order to solve? What makes these results interesting other than for their own sake?
Of course I know about Langlands, and the great benefits to the world of number theory of ideas coming from representation theory. But I'm looking for much more basic applications than that...