Abstract Harmonic Analysis

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

  1. $V_x(M_\lambda) = M_\lambda$ for all $x$ and $\lambda$.
  2. Every element $f\in\mathfrak{F}$ is uniquely a finite or infinite sum $f=\sum f_\lambda$ where each $f_\lambda \in M_\lambda$.
  3. 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 $G$.

Locally compact groups and group $C^\ast$-algebras.