## What is Harmonic Analysis?

Here’s George Mackey’s account of what the *method* of harmonic analysis is (with some notational and terminalogy change, see Harmonic analysis as the exploitation of symmetry–a historical survey)

Let $S$ be a “space” or “set” and let $G$ be a group of automorphism of $S$. Let $[s]x$ denote the transform of $s$ in $S$ by $x$ in $G$ Ordinarily $S$ will have further structure which will be preserved by the transformations of $G$ so that the transformations $s\to [s]x$ are symmetries of $S$. It will be convenient to allow members of $G$ other than the identity $e$ to define the identity map so that some quotient group $G/N$ is the actual transformation group. Now let $\mathfrak{F}$ be some vector space of complex valued functions on $S$ which is $G$ invariant in the sense that $s\to f([s]x)$, the translate of $f$ by $x$, is in $\mathfrak{F}$ whenever $f\in\mathfrak{F}$. Then for each $x\in G$, the mapping $f\mapsto g$ where $g(s)=f([s]x)$ is a linear transformation $V_x:\mathfrak{F}\to\mathfrak{F}$ and $V_{xy}=V_x V_y$ for all $x,y\in G$. The mapping $x\to V_x$ is thus an example of what is called a (linear) representation of the roup $G$. More generally, a (linear) representation of a group $G$ is by definition any homomorphism $x\to W_x$ of $G$ into the group of all automorphisms of some vector space $\mathfrak{H}(W)$. The method I propose to discuss in this article consists (in its simplest form) in attempting to find subspaces $M_\lambda$ of the sapce $\mathfrak{F}$ such that

- $V_x(M_\lambda) = M_\lambda$ for all $x$ and $\lambda$.
- Every element $f\in\mathfrak{F}$ is uniquely a finite or infinite sum $f=\sum f_\lambda$ where each $f_\lambda \in M_\lambda$.
- The subspaces $M_\lambda$ are either not susceptible of further decomposition or are somehow much simpler in structure than $\mathfrak{F}$.

Of course, one must have a topology in $\mathfrak{F}$ in order to make sense of infinite sums. More generally one also considers “continuous direct sums” or direct integrals and vector-valued as well as complex-valued functions. Of course, each $M_\lambda$ is the space of a new representation $V^\lambda$ which is a so-called subrepresentation, and one speaks of the direct sum or direct integral decomposition of $V$. It turns out that the decomposition of functions in $\mathfrak{F}$ into sums and integrals of functions asociated with the components of $V$ is a decomposition that greatly simplifies many problems.

So roughly speaking the method of harmonic analysis is that of ‘divide/classify and conquer’, namely, the study of the decomposition of a representation into direct sum/integral of irreducible representations. Given a symmetry $G$ on a vector-space-like object $V$, $V$ is decomposed into subspaces that behaves in a simple manner under $G$; elements in $V$ then can be decomposed into sums/integrals that behave in a simple manner under $G$.