Motivation for abstract harmonic analysis

I am reading Folland's A Course in Abstract Harmonic Analysis and find this book extremely exciting.

However, it seems Folland does not give many examples to illustrate the motivation behind much of the theory. Thus, I wonder whether there is something showing how these abstract stuff can be applied to solve specific problems and what is the purpose in their mind when they developed this theory.

Thanks!

This question has been open for so long that even someone as lazy as me felt a shame not to close it by providing some understandings. Since KCd pointed to two amazing references On the Evolution of Noncommutative Harmonic Analysis by Gross and Harmonic Analysis as the Exploitation of Symmetry-A Historical Survey by Mackey, I might just summarize what I learnt there. The following follows the first reference.

Classical harmonic analysis begins with Fourier series. It is remarkable that $\{e_n=e^{in\theta}:n\in\mathbb{Z}\}$ are eigenfunctions of $-\Delta$ on $\mathbb{S}^1$, and they form a complete basis for $\mathcal{L}^2(\mathbb{S}^1)$. On the one hand, this gives a satisfactory answer to many important differential equations involving the Laplacian, by convolution with the initial data. On the other hand, it gives a decomposition of $\mathcal{L}^{2}(\mathbb{S}^1)$ into subspaces generated by each of the $e_n$.

This point of view gives the group theoretic nature of Fourier analysis. Note that $e_n$ are precisely the continuous homomorphisms between the two groups $\mathbb{S}^1$ and $\mathbb{C}^*:=\mathbb{C}\backslash\{0\}$, which are the (1-dimensional) irreducible representations of $\mathbb{S}^1$, since $\mathbb{C}^*$ is exactly $GL(\mathbb{C})$. This shows the function space $\mathcal{L}^2(\mathbb{S}^1)$ over one group $\mathbb{S}^1$ is the direct sum of irreducible representations. Also note the collection of irreducible representations also form a group under the rule $e_n \cdot e_m= e_{m+n}$, called the dual group $\hat{\mathbb{S}^1}=\mathbb{Z}$.

This should reminds one of representation theory of finite groups, which became a very elegant theory under the hands of Frobenius, Schur and Burnside not long after Fourier's time. The philosophy is to introduce linear algebraic method into the study of groups. For a group $G$, we study the realization of $G$ inside the symmetry groups of linear spaces, that is, we study homomorphisms $G\xrightarrow{\rho}GL(V)$, where $V$ is a finite dimensional vector space. Note that if we fix a basis of $V$, then each $\rho(g)$ is a matrix $(a_{ij}^{V}(g))$.

If a subspace $W$ of $V$ is invariant under $G$, then $W$ itself gives a representation of $G$. If $V$ has no proper subspaces invariant under $G$, then $V$ is an irreducible representation of $G$. Let $\hat{G}$ be the collection of irreducible representations of $G$, and for each of these vector spaces we fix a basis. Then we have a collection of matrices $\{(a_{ij}^V(g)):V\in\hat{G}, g\in G\}$. Now we have the magic. The function space $C(G)$(since $G$ is finite we might just take all possible functions over $G$) is again a direct sum of irreducible representations, namely, \begin{equation} C(G)=\oplus_{\hat{G}}\operatorname{dim}(V)V \end{equation} (It is actually more than an isomorphism of vector spaces. It is a isomorphism of algebras if we take convolution as multiplication on the left-hand side. ) and the functions $\{a_{ij}^V(g)\}$ form a complete orthogonal basis for $C(G)$ under the inner product \begin{equation} \langle f,h\rangle=\sum_{g\in G}f(g)\overline{h(g)}. \end{equation} Note that this is translation-invariant, that is, you can translate the functions but this does not affect the inner product.

So again this shows a clear similarity between Fourier analysis and representation theory of finite groups. However, this was not appreciated until the beginning of the 20th century. Both group theory and analysis had been revolutionalized. Under the program of of Klein, people are studying geometry through the glasses of group actions. And the group for differential geometry are the Lie groups, which became a focus of people who studied group theory. Meanwhile, people who studied differential equations are also equipped with weapons like functional analysis. Now came Weyl who realized that we might study Lie groups, which on the surface are on the other extreme from finite groups, using almost the same methods used by Frobenius and Schur if one is willing to control the group not by counting, but by topology.

But again he needed an invariant inner product, which is the same thing as an invariant integral. Luckily the existence of such an integral is already proved by Haar on locally compact groups (the uniqueness by von Neumann, and converse by Weil). With this, Peter-Weyl tells us that

The representations of compact groups are direct sums of irreducible ones. And the matrix elements of irreducible representations form a complete basis for the function space.

This is often thought as the beginning of modern harmonic analysis. With his theory, Weyl told us a lot about compact Lie groups.

To go one step further from there, one might like to study locally compact groups. Weil used almost same ideas and constructed a quite satisfactory theory. However, this theory has two major drawbacks when applied to concrete examples. Firstly, now the collection of irreducible representations are often a continuum ($\hat{\mathbb{R}}=\mathbb{R}$) while they are always discrete for compact groups. Correspondingly, representations no longer breaks into direct sums but into direct integrals. Secondly, now we have to consider infinite dimensional representations.

So we still has a very beautiful theory, but the calculations are often technical with locally compact groups. But this theory still has many applications in things like number theory. But also note that it has a close link with quantum mechanics, which involves, by definition, infinite dimensional state spaces. And the uncertainty principles can be modelled on things like Weyl algebras.