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.
Let $\beta_n$ denote the flag $h$-vector (as defined in EC1, Section
3.13) of the partition lattice $\Pi_n$ (EC1, Example 3.10.4). Then
$$ \mathrm{mag}_{n,{n-1\choose 2}-j} = \sum_S \beta_n(S), $$
where $S$ ranges over all subsets of $\lbrace 1,2,\dots,n-2\rbrace$ whose
elements sum to $j$. An explicit formula for $\beta_n(S)$ is given by
$$ \beta_n(S) = \sum_{T\subseteq S} (-1)^{|S-T|}
\alpha_n(T), $$
where if the elements of $T$ are $t_1<\cdots < t_k$, then
$$ \alpha_n(T) = S(n,n-t_1)S(n-t_1,n-t_2)
S(n-t_2,n-t_3)\cdots S(n-t_{k-1},n-t_k). $$
Here $S(m,j)$ denotes a Stirling number of the second kind.
Addendum. A combinatorial description of the mag numbers is
somewhat complicated. Consider all ways to start with the $n$ sets
$\lbrace 1 \rbrace,\dots, \lbrace n \rbrace$. At each step we take two
of our sets and replace them by their union. After $n-1$ steps we will
have the single set $\lbrace 1,2,\dots,n \rbrace$. An example for
$n=6$ is (writing a set like $\lbrace 2,3,5\rbrace$ as 235)
1-2-3-4-5-6, 1-2-36-4-5, 14-36-2-5, 14-356-2, 14356-2, 123456. At the
$i$th step suppose we take the union of two sets $S$ and $T$. Let
$a_i$ be the least integer $j$ such that $j$ belongs to one of the
sets $S$ or $T$, and some number smaller than $j$ belongs to the other
set. For the example above we get $(a_1,\dots,a_5)=(6,4,5,3,2)$. If
$\nu$ denotes this merging process, then let $f(\nu) = \sum i$,
summed over all $i$ for which $a_i>a_{i+1}$. For the above example,
$f(\nu) = 1+3+4=8$. (The number $f(\nu)$ is called the major index
of the sequence $(a_1,\dots,a_{n-1})$.) Then
$\mathrm{mag}_{n,{n-1\choose 2}-j}$ is the number
of merging processes $\nu$ for which $f(\nu)=j$. This might look
completely contrived to the uninitiated, but it is very natural within
the theory of flag $h$-vectors.
Best Answer
I would suggest "Handbook of Formulas and Tables for Signal Processing", by Alexander D. Poularikas.
This link should help you.
This is the amazon link.