Let G be a locally compact abelian group (lcag). Thus G is an abelian group with a topology making the group operations continuous functions from G and G2 to G, and under which any point has a neighborhood contained in a compact set. (Since translation by a group element is continuous, it is enough to find such a neighborhood of the origin of G.) Examples are any finite abelian group (with the discrete topology), the direct sum of finitely many lcag's, and the groups Z, T, R, C (which we also know as the direct sum of two R's). The p-adic integers Zp and rationals Qp are also lcag's for each prime p. An example of a topological abelian group that is not locally compact is an infinite-dimensional Hilbert space.

The Pontrjagin dual G^ of a lcag G is the group of continuous homomorphisms from G to the unit circle C1. For instance, the duals of Z/nZ, Z, T, R are isomorphic with Z/nZ, T, Z, R respectively. For each g in G, the map taking any g^ in G^ to g^(g) is a homomorphism from G^ to C1. If G^ is given the weakest topology making all those homomorphisms continuous, it turns out that G^ becomes an lcag itself, and its Pontrjagin dual becomes canonically identified with G. Note that on the other hand there is in general no canonical identification between G and G^ even when these two groups are isomorphic as lcag's (e.g. for Z/nZ or R).

Further properties of Pontrjagin duality: the dual of a finite direct sum is naturally identified with the direct sum of the duals; a continuous homomorphism f from G to H yields a continuous homomorphism f^ from H^ to G^; this dual homomorphism f^ is injective if f is surjective, and vice versa; and f^^=f. (These results can all be checked either directly or with the assistance of the theorem G^^=G.) In particular if G is a closed subgroup of H then G^ is the quotient of H^ by the annihilator of G in H^, i.e. the subgroup consisting of homomorphisms taking every element of G to the identity in C1. Conversely the dual of a quotient H/G is the annihiliator of G in H^, provided G is a closed subgroup of H. (Check that this works for the subgroups Z of R, and nZ of Z.) The dual of a compact group is discrete, and vice versa (hard to prove, but at least check it the examples we've seen).

Except for that last sentence, the properties of Pontrjagin duality may remind you of duality of vector spaces. Indeed Pontrjagin duality is a generalization of vector-space duality, at least for vector spaces over R and Z/pZ. For instance, if V is a real vector space then V* may be identified with V^ by regarding the functional v* as the homomorphism sending any v in V to eiv*(v). For Z/pZ we instead raise a fixed p-th root of unity to the power v*(v).

So, what does all this theory have to do with Fourier? In each case we can express ``any'' function f on G as a ``linear combination'' of its homomorphisms to C1, with the coefficients of the homomorphisms g^ being given by a ``Fourier transform'' f^(g^) of f. The ``linear combination'' may be a finite sum, infinite sum, or integral over G^ (actually it's always an ``integral'' for a sufficiently general notion of integration), and except in the finite case the sense in which this linear combination is known coincide with f depends on what kind of function f is. In each case, though, f^(g^) may be written as an inner product (f,g^), and f^^ is a function on G proportional to f(-g).