I'm asking this question because I've been told by some people that Fourier analysis is "just representation theory of $S^1$."
I've been introduced to the idea that Fourier analysis is related to representation theory. Specifically, when considering the representations of a finite abelian group $A$, these representations are all $1$-dimensional, hence correspond to characters $A \to \mathbb{R}/\mathbb{Z} \cong S^1 \subseteq \mathbb{C}$. On the other side, finite Fourier analysis is, in a simplistic sense, the study of characters of finite abelian groups. Classical Fourier analysis is, then, the study of continuous characters of locally compact abelian groups like $\mathbb{R}$ (classical Fourier transform) or $S^1$ (Fourier series). However, in the case of Fourier analysis, we have something beyond characters/representations: We have the Fourier series / transform. In the finite case, this is a sum which looks like $\frac{1}{n} \sum_{0 \le r < n} \omega^r \rho(r)$ for some character $\rho$, and in the infinite case, we have the standard Fourier series and integrals (or, more generally, the abstract Fourier transform). So it seems like there is something more you're studying in Fourier analysis, beyond the representation theory of abelian groups. To phrase this as a question (or two):
(1) What is the general Fourier transform which applies to abelian and non-abelian groups?
(2) What is the category of group representations we consider (and attempt to classify) in Fourier analysis? That is, it seems like Fourier analysis is more than just the special case of representation theory for abelian groups. It seems like Fourier analysis is trying to do more than classify the category of representations of a locally compact abelian group $G$ on vector spaces over some fixed field. Am I right? Or can everything we do in Fourier analysis (including the Fourier transform) be seen as one piece in the general goal of classifying representations?
Let me illustrate this in another way. The basic result of Fourier series is that every function in $L^2(S^1)$ has a Fourier series, or in other words that $L^2$ decomposes as a (Hilbert space) direct sum of one dimensional subspaces corresponding to $e^{2 \pi i n x}$ for $n \in \mathbb{Z}$. If we encode this in a purely representation-theoretic fact, this says that $L^2(S^1)$ decomposes into a direct sum of the representations corresponding to the unitary characters of $S^1$ (which correspond to $\mathbb{Z}$). But this fact is not why Fourier analysis is interesting (at least in the sense of $L^2$-convergence; I'm not even worrying about pointwise convergence). Fourier analysis states furthermore an explicit formula for the function in $L^2$ giving this representation. Though I guess by knowing the character corresponding to the representation would tell you what the function is.
So is Fourier analysis merely similar to representation theory, or is it none other than the abelian case of representation theory?
(Aside: This leads into a more general question of mine about the use of representation theory as a generalization of modular forms. My question is the following: I understand that a classical Hecke eigenform (of some level $N$) can be viewed as an element of $L^2(GL_2(\mathbb{Q})\ GL_2(\mathbb{A}_{\mathbb{Q}})$ which corresponds to a subrepresentation. But what I don't get is why the representation tells you everything you would have wanted to know about the classical modular form. A representation is nothing more than a vector space with an action of a group! So how does this encode the information about the modular form?)
Best Answer
I would like to elaborate slightly on my comment. First of all, Fourier analysis has a very broad meaning. Fourier introduced it as a means to study the heat equation, and it certainly remains a major tool in the study of PDE. I'm not sure that people who use it in this way think of it in a particularly representation-theoretic manner.
Also, when one thinks of the Fourier transform as interchanging position space and frequency space, or (as in quantum mechanics) position space and momentum space, I don't think that a representation theoretic view-point necessarily need play much of a role.
So, when one thinks about Fourier analysis from the point of view of group representation theory, this is just one part of Fourier analysis, perhaps the most foundational part, and it is probably most important when one wants to understand how to extend the basic statements regarding Fourier transforms or Fourier series from functions on $\mathbb R$ or $S^1$ to functions on other (locally compact, say) groups.
As I noted in my comment, the basic question is: how to decompose the regular representation of $G$ on the Hilbert space $L^2(G)$. When $G$ is locally compact abelian, this has a very satisfactory answer in terms of the Pontrjagin dual group $\widehat{G}$, as described in Dick Palais's answer: one has a Fourier transform relating $L^2(G)$ and $L^2(\widehat{G})$. A useful point to note is that $G$ is discrete/compact if and only if $\widehat{G}$ is compact/discrete. So $L^2(G)$ is always described as the Hilbert space direct integral of the characters of $G$ (which are the points of $\widehat{G}$) with respect to the Haar measure on $\widehat{G}$, but when $G$ is compact, so that $\widehat{G}$ is discrete, this just becomes a Hilbert space direct sum, which is more straightforward (thus the series of Fourier series are easier than the integrals of Fourier transforms).
I will now elide Dick Palais's distinction between the Fourier case and the more general context of harmonic analysis, and move on to the non-abelian case. As Dick Palais also notes, when $G$ is compact, the Peter--Weyl theorem nicely generalizes the theory of Fourier series; one again describes $L^2(G)$ as a Hilbert space direct sum, not of characters, but of finite dimensional representations, each appearing with multiplicity equal to its degree (i.e. its dimension). Note that the set over which one sums now is still discrete, but is not a group. And there is less homogeneity in the description: different irreducibles have different dimensions, and so contribute in different amounts (i.e. with different multiplicities) to the direct sum.
When G is locally compact but neither compact nor abelian, the theory becomes more complex. One would like to describe $L^2(G)$ as a Hilbert space direct integral of matrix coefficients of irreducible unitary representations, and for this, one has to find the correct measure (the so-called Plancherel measure) on the set $\widehat{G}$ of irreducible unitary representations. Since $\widehat{G}$ is now just a set, a priori there is no natural measure to choose (unlike in the abelian case, when $\widehat{G}$ is a locally compact group, and so has its Haar measure), and in general, as far as I understand, one doesn't have such a direct integral decomposition of $L^2(G)$ in a reasonable sense.
But in certain situations (when $G$ is of "Type I") there is such a decomposition, for a uniquely determined measure, so-called Plancherel measure, on $\widehat{G}$. But this measure is not explicitly given. Basic examples of Type I locally compact groups are semi-simple real Lie groups, and also semi-simple $p$-adic Lie groups.
The major part of Harish-Chandra's work was devoted to explicitly describing the Plancherel measure for semi-simple real Lie groups. The most difficult part of the question is the existence of atoms (i.e. point masses) for the measure; these are irreducible unitary representations of $G$ that embed as subrepresentations of $L^2(G)$, and are known as "discrete series" representations. Harish-Chandra's description of the discrete series for all semi-simple real Lie groups is one of the major triumphs of 20th century representation theory (indeed, 20th century mathematics!).
For $p$-adic groups, Harish-Chandra reduced the problem to the determination of the discrete series, but the question of explicitly describing the discrete series in that case remains open.
One important thing that Harish-Chandra proved was that not all points of $\widehat{G}$ (when $G$ is a real or $p$-adic semisimple Lie group) are in the support of Plancherel measure; only those which satisfy the technical condition of being "tempered". (So this is another difference from the abelian case, where Haar measure is supported uniformly over all of $\widehat{G}$.) Thus in explicitly describing Plancherel measure, and hence giving an explicit form of Fourier analysis for any real semi-simple Lie group, he didn't have to classify all unitary representations of $G$.
Indeed, the classification of all such reps. (i.e. the explicit description of $\widehat{G}$) remains an open problem for real semi-simple Lie groups (and even more so for $p$-adic semi-simple Lie groups, where even the discrete series are not yet classified).
This should give you some sense of the relationship between Fourier analysis in its representation-theoretic interpretation (i.e. the explicit description of $L^2(G)$ in terms of irreducibles) and the general classification of irreducible unitary representations of $G$. They are related questions, but are certainly not the same, and one can fully understand one without understanding the other.