Harmonic analysis

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Harmonic analysis is the branch of mathematics which studies analysis on groups. The core motivating idea is the Fourier transform, which can be generalized to a transform of functions defined on locally compact groups. Fourier analysis is concerned with finding the "frequencies" present in a given function, and these frequencies are also called "harmonics", hence the name harmonic analysis.

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 modulus 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 left invariant in the sense that |gA| = |A| for every g in G, and it is finite for compact subsets A. This measure allows to define the notion of integral for functions defined on G, and one may consider the Hilbert space L2(G) of all square integrable functions on 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(g)φ(g) dg

for every f in L2(G) and φ in G'. F is an isometric isomorphism of Hilbert spaces.

In the case of G = Rn, we have G' = Rn and we recover the ordinary 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 Zn (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.