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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:

  1. 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 $L^2(G)\rightarrow L^2(\hat{G})$.
  2. Decomposing $L^2(G)$ as a direct integral (direct sum in the case that $G$ is compact) of irreducible unitary representations.
  3. 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...

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  • $\begingroup$ What does $\hat{\hat G}$ mean in the situations where $\hat G$ is not a topological group? $\endgroup$
    – LSpice
    Dec 3, 2020 at 3:24
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    $\begingroup$ To try and get a better idea of what you are looking for: do you feel that for each of the above, you have seen results and applications in the abelian case which answer your questions? If not, then that would be a natural issue to address first $\endgroup$
    – Yemon Choi
    Dec 3, 2020 at 3:29
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    $\begingroup$ @JochenGlueck I must confess I interpreted the question as mainly being about the non-abelian setting, because in the abelian setting I had the impression that the Fourier transform's utlity would be familiar from things typically seen in UG courses (ODE, PDE, probability theory ...) $\endgroup$
    – Yemon Choi
    Dec 3, 2020 at 3:33
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    $\begingroup$ @AndrewNC: On compact intervals yes. If we want to solve the heat equation $\dot u = \Delta u$ on $\mathbb{R}^d$, there is no eigenbasis, though (in the literal sense of a Hilbert space ONB); but we can apply that the Fourier transform from $L^2(\mathbb{R}^d)$ to $L^2(\mathbb{R}^d)$ intertwines the Laplace operator with the mulitplication by $-x^2$. This is one way to compute the heat kernel of $\dot u = \Delta u$. $\endgroup$ Dec 3, 2020 at 3:38
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    $\begingroup$ BTW character theory of finite non-abelian groups seems like an "application" of 2) to me, if when you ask "what can it be naturally applied towards in order to solve?" you are including applications to other areas of "pure" mathematics $\endgroup$
    – Yemon Choi
    Dec 3, 2020 at 3:43

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