If *G* is an *abelian* locally compact group, we define a *character of G* to be a continuous group homomorphism φ : *G* `->` *S*^{1}. The set of all characters on *G* is another locally compact abelian group, the **dual group** G' of *G*.

In detail, the **dual group** is defined as follows:

If `G` is an abelian locally compact group, two such characters can be multiplied to form a new character, and with the trivial character `x → 1` as the identity and the topology of uniform convergence on compact sets, the set of all characters on `G` is a locally compact abelian group, called the *dual group* `G'` of `G`.

*Note:* Here `S _{1}` is the circle group, which can be realised as the complex numbers of modulus 1 or the quotient group

This duality is a symmetric relationship, since the dual group of a dual group is the original group.

The dual group is introduced as the underlying space for an abstract version of the Fourier transform. In this context, functions on the group `G` (e.g. functions in `L ^{1}(G)` or

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 L^{2}(*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 L^{2}(*G*) and *x* is an element of *G*, we define the function *xf* by (*xf*)(*y*) = *f*(*yx*) for all *y* in *G*.

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

For example, a character on the infinite cyclic group of integers **Z** is determined by its value φ(1), since φ(n) = (φ(1))^{n} gives its values on all other elements of **Z**. Moreover, this formula defines a character for any choice of φ(1) in `S _{1}` and the topology of uniform convergence on compacta (appearing here as pointwise convergence) is the natural topology of

Conversely, a character φ on `S _{1}` is of the form

The other "classical group" example, the group of real numbers **R**, is its own dual. The characters on **R** are of the form `φ _{y}: x → e^{ixy}`.

With these dualities, the version of the Fourier transform to be introduced next coincides with the classical Fourier transform on **R**, and the exponential form of the Fourier series on **Z**.

The most natural Fourier transform generalization is then given by the operator

defined by- (
*Ff*)(φ) = ∫*f*(*x*)φ(*x*) d*x*

*F*(*f***g*) =*Ff*·*Fg*

In the case of *G* = **R**^{n}, we have *G*' = **R**^{n} and we recover the ordinary continuous Fourier transform; in the case *G* = *S*^{1}, 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.

In detail, the dual group construction of G^{^} is a contravariant functor **LCA** -> **LCA**^{op} allowing us to identify the category **LCA** of locally compact abelian topological groups with its own opposite category. We have G^{^^} isomorphic to G, in a natural way that is comparable to the double dual of finite-dimensional vector spaces (a special case, for real and complex vector spaces).

The duality interchanges the subcategories of discrete groups and compact groups. If R is a ring (mathematics) and G is a left R-module, the dual group G^ will become a right R-module; in this way we can also see that discrete left R-modules will be Pontryagin dual to compact right R-modules. The ring End(G) of endomorphisms in **LCA** is changed by duality into its opposite ring (change the multiplication to the other order). For example if G is an infinite cyclic discrete group, G^ is a circle group: the former has End(G) = **Z** so this is true also of the latter.

One use made of Pontryagin duality is to give a general definition of an almost-periodic function on a non-compact group G in **LCA**. For that, we define the *Bohr compactification* B(G) of G as H^{^}, where H is as a group G^{^}, but given the discrete topology. Since H -> G^{^} is continuous and a homomorphism, the dual morphism G -> B(G) is defined, and realizes G as a subgroup of a compact group. The restriction to G of continuous functions on B(G) gives a class of *almost-periodic* functions; one can imagine them as analogous to the restrictions to a copy of **R** wound round a torus.

Such a theory cannot exist in the same form for non-commututive groups G, since in that case the appropriate dual object G^{^} of isomorphism classes of representations cannot only contain one-dimensional representations, and will fail to be a group. The generalisation that has been found useful in category theory is called Tannaka-Krein duality; but this diverges from the connection with harmonic analysis, which needs to tackle the question of the *Plancherel measure* on G^{^}.

The foundations for the theory of Locally compact abelian groups and their duality was laid down by Lev Semenovich Pontryagin in the 1934. His treatment, which relied on the group being Second countable and either compact or discrete. This was improved to cover the general locally compact abelian groups by E.R. van Kampen in 1935 and André Weil in 1953.