Harmonic analysis

Harmonic analysis is the branch of mathematics which studies the representation of functions or signals as the superposition of basic waves. It investigates and generalizes the notions of Fourier series and Fourier transforms. The basic waves are called "harmonics", hence the name "harmonic analysis."

The classical Fourier transform is still an interesting area of research. For instance, if we impose some requirements on a function f, we can attempt to translate these requirements in terms of the Fourier transform of f. For example, if a function is compactly supported, then its Fourier transform may not also be compactly supported; this is a very elementary form of an Uncertainty Principle in a Harmonic Analysis setting (there are more sophisticated examples of this.)

Fourier series can be conveniently studied in the context of Hilbert spaces, which provides a connection between harmonic analysis and functional analysis.

One of the more modern branches of Harmonic Analysis, having its roots in the mid-twentieth century, is analysis on topological groups. The core motivating idea are the various Fourier transforms, which can be generalized to a transform of functions defined on locally compact groups.

Examples of abelian locally compact groups are: Euclidean space with vector addition as operation, the positive real numbers with multiplication as operation, the group S¹ of all complex numbers of absolute value 1, with complex multiplication as operation, and every finite abelian group.

The most important feature of a locally compact group G is that it carries an essentially unique natural measure, the Haar measure, which allows to consistently measure the "size" |A| of subsets A of G. This measure is right invariant in the sense that |Ax| = |A| for every x in G, and it is finite for compact subsets A. This measure allows to define the notion of integral for (complex-valued) functions defined on G, and one may then consider the Hilbert space L²(G) of all square-integrable functions on G. The group G acts on this Hilbert space as a group of isometric automorphisms via right shift: if f is a function in L²(G) and x is an element of G, we define the function xf by (xf)(y) = f(yx) for all y in G.

If G is an abelian locally compact group, we define a character of G to be a continuous group homomorphism φ : G -> S¹. Two such characters can be multiplied to form a new character, and this operation turns the set of all characters on G into a locally compact abelian group, the dual group G' of G. The most natural Fourier transform generalization is then given by the operator

F : L²(G) -> L²(G')

defined by

(Ff)(φ) = ∫ f(x)φ(x) dx

for every f in L²(G) and φ in G'. F is an isometric isomorphism of Hilbert spaces. The convolution f*g of two elements f, g in L²(G) can be defined by

<math>(f*g)(t)=\int f(x)g(t-x)\,dx</math>

(this is a function in L¹(G)) and the convolution theorem

F(f*g) = Ff · Fg

relating the Fourier transform of the convolution and the product of the two Fourier transforms remains valid.

In the case of G = Rⁿ, we have G' = Rⁿ and we recover the ordinary continuous Fourier transform; in the case G = S¹, the dual group G' is naturally isomorphic to the group of integers Z and the above operator F specializes to the computation of coefficients of Fourier series of periodic functions; if G is the finite cyclic group Z_n (see modular arithmetic), which coincides which its own dual group, we recover the discrete Fourier transform.

Harmonic analysis studies the properties of this transform and attempts to extend it to different settings, for instance to the case of non-abelian Lie groups.

The Peter-Weyl Theorem[?] explains how one may get harmonics by choosing one irreducible representation out of each equivalence class of representations. This choice of harmonics enjoys some of the useful properties of the classical Fourier transform in terms of carrying convolutions to pointwise products, or otherwise showing a certain understanding of the underlying group structure.