In mathematics, in particular in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general features of the Fourier transform. It places in context the observations that periodic functions have Fourier series, functions on the real line transforms that are also functions of one continuous variable, and functions on a finite abelian group can be expressed (as in fast Fourier transform techniques) as transforms of functions on an isomorphic group but with some bit-reversal. The theory, introduced by Lev Pontryagin and combined with Haar measure by André Weil, depends on the theory of the dual group and is its expression in the language of category theory; that is, it also includes the behaviour of frequency domains with respect to group homomorphisms.
If G is an abelian locally compact group, we define a character of G to be a continuous group homomorphism φ : G -> S1. 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 S1 is the circle group, which can be realised as the complex numbers of modulus 1 or the quotient group R/Z as convenient.
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 L1(G) or L2(G)) are transformed into functions whose domain is the dual group G'. This is implemented via the integral
The dual group
where the integral uses Haar measure.
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.
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.
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 S1 and the topology of uniform convergence on compacta (appearing here as pointwise convergence) is the natural topology of S1. Hence the dual group of Z is identified with S1.
Conversely, a character φ on S1 is of the form z → zn for n ∈ Z. Since S1 is compact, the topology on the dual group is that if uniform convergence, which turns out to be the discrete topology. As a consequence of this, the dual of S1 is identified with Z.
The other "classical group" example, the group of real numbers R, is its own dual. The characters on R are of the form φy: x → eixy.
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
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 Zn (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 -> LCAop 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.Haar measure
Examples
Fourier transforms in general
defined by
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
(this is a function in L1(G)) and the convolution theorem
relating the Fourier transform of the convolution and the product of the two Fourier transforms remains valid.Abstract point of view
Bohr compactification and almost-periodicity
Non-commutative theory
History
