In mathematics, in particular in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform. It places in a unified context a number of observations about functions on the real line or on finite abelian groups: Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Harmonic analysis is the branch of mathematics which studies the representation of functions or signals as the superposition of basic waves. ... In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G × G → G and the inverse operation G → G are continuous maps. ... In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ...

• Suitably regular complex-valued periodic functions on the real line have Fourier series and these functions can be recovered from their Fourier series;

• Suitably regular complex-valued functions on the real line have Fourier transforms that are also functions on the real line and, just as for periodic functions, these functions can be recovered from their Fourier transforms; and

• Complex-valued functions on a finite abelian group have discrete Fourier transforms which are functions on the dual group, which is a (non-canonically) isomorphic group. Moreover any function on a finite group can be recovered from its discrete Fourier transform.

The theory, introduced by Lev Pontryagin and combined with Haar measure introduced by John von Neumann, André Weil and others depends on the theory of the dual group of a locally compact abelian group. In mathematics, a periodic function is a function that repeats its values after some definite period has been added to its independent variable. ... The Fourier series is a mathematical tool used for analyzing periodic functions by decomposing such a function into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. ... In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ... In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ... In mathematics, the discrete Fourier transform (DFT) is a transform for Fourier analysis of finite-domain discrete-time signals. ... Lev Semenovich Pontryagin (Russian: Лев Семёнович Понтрягин) (3 September 1908- 3 May 1988) was a Soviet/Russian mathematician. ... In mathematical analysis, the Haar measure is a way to assign an invariant volume to subsets of locally compact topological groups and subsequently define an integral for functions on those groups. ... John von Neumann (Hungarian Margittai Neumann János Lajos) (born December 28, 1903 in Budapest, Austria-Hungary; died February 8, 1957 in Washington D.C., United States) was a Hungarian-born mathematician and polymath who made contributions to quantum physics, functional analysis, set theory, topology, economics, computer science, numerical analysis... André Weil (May 6, 1906 - August 6, 1998) was one of the great mathematicians of the 20th century. ... In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. ... In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ...

Haar measure

A topological group is locally compact if and only if the identity e of the group has a compact neighborhood. This means that there is some open set V containing e whose closure is relatively compact in the topology of G. One of the most remarkable facts about a locally compact group G is that it carries an essentially unique natural measure, the Haar measure, which allows one to consistently measure the "size" of sufficiently regular subsets of G. "Sufficiently regular subset" here means a Borel set; that is, an element of the σ-algebra generated by the compact sets. More precisely, a right Haar measure on a locally compact group G is a countably additive measure μ defined on the Borel sets of G which is right invariant in the sense that μ(A x) = μ(A) for x an element of and A a Borel subset of G and also satisfies some regularity conditions (spelled out in detail in the article Haar measure). Except for positive scale factors, Haar measures are unique. In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G × G → G and the inverse operation G → G are continuous maps. ... In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G × G → G and the inverse operation G → G are continuous maps. ... In mathematics, a measure is a function that assigns a number, e. ... In mathematical analysis, the Haar measure is a way to assign an invariant volume to subsets of locally compact topological groups and subsequently define an integral for functions on those groups. ... In mathematics, the Borel algebra (or Borel σ-algebra) on a topological space is either of two σ-algebras on a topological space X: The minimal σ-algebra containing the open sets. ... In mathematics, a σ-algebra (or σ-field) X over a set S is a family of subsets of S which is closed under countable set operations; σ-algebras are mainly used in order to define measures on S. The concept is important in mathematical analysis and probability theory. ... Compact as a general noun can refer to: Look up Compact on Wiktionary, the free dictionary a diplomatic contract or covenant among parties, sometimes known as a pact, treaty, or an interstate compact; a British term for a newspaper format; In mathematics, it can refer to various concepts: Mostly commonly... In mathematics, the Borel algebra (or Borel σ-algebra) on a topological space is either of two σ-algebras on a topological space X: The minimal σ-algebra containing the open sets. ... In mathematical analysis, the Haar measure is a way to assign an invariant volume to subsets of locally compact topological groups and subsequently define an integral for functions on those groups. ...

The Haar measure allows us to define the notion of integral for (complex-valued) Borel functions defined on the group. In particular, one may consider various L^p spaces associated to the Haar measure. Specifically, In calculus, the integral of a function is an extension of the concept of a sum. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ...

Examples of locally compact abelian groups are: In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ...

• Rⁿ, for n a positive integer, with vector addition as group operation.

• The positive real numbers with multiplication as operation. This group is clearly seen to be isomorphic to R. In fact, the exponential mapping implements that isomorphism.

• Any finite abelian group, with the discrete topology. By the structure theorem for finite abelian groups, all such groups are products of cyclic groups.

• The integers Z under addition, again with the discrete topology.

• The circle group, denoted T. This is the group of complex numbers of modulus 1. T is isomorphic as a topological group to the quotient group R/Z .

• The field Q_p of p-adic numbers under addition, with the usual p-adic topology.

In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ... In topology and related fields of mathematics, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are isolated from each other in a certain sense. ... In topology and related fields of mathematics, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are isolated from each other in a certain sense. ... In mathematics, the circle group, denoted by T (or in blackboard bold by ), is the multiplicative group of all complex numbers with absolute value 1. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... Mathematical meanings Especially in British/European usage, the modulus of a number is its absolute value. ... In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is intuitively a group that collapses the normal subgroup N to the identity element. ... The p-adic number systems were first described by Kurt Hensel in 1897. ...

The dual group

If G is a locally compact abelian group, a character of G is a continuous group homomorphism from G with values in the circle group T. It can be shown that the set of all characters on G is itself a locally compact abelian group, called the dual group of G. The group operation on the dual group is given by pointwise multiplication of characters, the inverse of a character is its complex conjugate and the topology on the space of characters is that of uniform convergence on compact sets (i.e., the compact-open topology). This topology in general is not metrizable. However, if the group G is a separable locally compact abelian group, then the dual group is metrizable. The dual group of an abelian group G is denoted G^. In topology and related areas of mathematics a continuous function is a morphism between topological spaces. ... Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G -> H such that for all u and v in G it holds that h(u * v) = h(u) · h(v) From this property, one can deduce that h maps the identity element... A Möbius strip, a surface with only one side and one edge; such shapes are an object of study in topology. ... In the mathematical field of analysis, uniform convergence is a type of convergence stronger than pointwise convergence. ... Compact as a general noun can refer to: Look up Compact on Wiktionary, the free dictionary a diplomatic contract or covenant among parties, sometimes known as a pact, treaty, or an interstate compact; a British term for a newspaper format; In mathematics, it can refer to various concepts: Mostly commonly... In mathematics, the compact-open topology is a topology defined on the set of continuous maps between two topological spaces. ...

Theorem The dual of G^ is canonically isomorphic to G, that is (G^)^ = G in a canonical way.

Canonical means that there is naturally defined map from G into (G^)^; more importantly, the map should be functorial. The precise formulation of this idea involves the concept of natural transformation. This fact is important; for instance, any finite abelian group is isomorphic to its dual, but the isomorphism is not canonical. The canonical isomorphism is defined as follows: Canonical is an adjective derived from canon. ... For functors in computer science, see the function object article. ... In category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i. ...

In other words, each group element x is identified to the evaluation character on the dual.

Examples

A character on the infinite cyclic group of integers Z under addition is determined by its value at the generator 1. Thus for any character χ on Z, χ(n)=χ(1)ⁿ. Moreover, this formula defines a character for any choice of χ(1) in T. Thus it follows easily that algebraically the dual of Z is isomorphic to the circle group T. The topology of uniform convergence on compact sets is in this case the topology of pointwise convergence. It is also easily shown that this is the topology of the circle group inherited from the complex numbers. In mathematics, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that every element of the group is a power of a. ... Suppose { fn } is a sequence of functions sharing the same domain in common (for the moment, we defer making precise the nature of the values of these functions, but the reader may take them to be real numbers if that makes anyone feel good). ...

Hence the dual group of Z is canonically isomorphic with T.

Conversely, a character on T is of the form z → zⁿ for n an integer. Since T is compact, the topology on the dual group is that of uniform convergence, which turns out to be the discrete topology. As a consequence of this, the dual of T is canonically isomorphic with Z. In topology and related fields of mathematics, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are isolated from each other in a certain sense. ...

The group of real numbers R, is isomorphic to its own dual; the characters on R are of the form r → e ^{i θ r}. With these dualities, the version of the Fourier transform to be introduced next coincides with the classical Fourier transform on R. In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ...

Fourier transform

The dual group of a locally compact abelian group is introduced as the underlying space for an abstract version of the Fourier transform. If a function is in L¹(G), then the Fourier transform is the function on G^ such that In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ...

where the integral is relative to Haar measure μ on G. It is not too difficult to show that the Fourier transform of an L¹ function on G is a bounded continuous function on G^ which vanishes at infinity. Similarly, the inverse Fourier transform of an integrable function on G^ is given by In mathematical analysis, the Haar measure is a way to assign an invariant volume to subsets of locally compact topological groups and subsequently define an integral for functions on those groups. ...

where the integral is relative to the Haar measure ν on the dual group G^.

The group algebra

The space of integrable functions on a locally compact abelian group G is an algebra, where multiplication is convolution: if f, g are integrable functions then the convolution of f and g is defined as Algebra is a branch of mathematics concerning the study of structure, relation and quantity. ...

Theorem The Banach space L¹(G) is an associative and commutative algebra under convolution.

This algebra is referred to as the Group Algebra of G. By completeness of L¹(G), it is a Banach algebra. The Banach algebra L¹(G) does not have a multiplicative identity element unless G is a discrete group. In general, however, it has an approximate identity which is a net (or generalized sequence) indexed on a directed set I, {e_i}_i with the property that In functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space. ... In functional analysis, a right approximate identity in a Banach algebra A is a net (or a sequence) such that for every element of , the net (or sequence) has limit . ...

The Fourier transform takes convolution to multiplication, that is:

In particular, to every group character on G corresponds a unique multiplicative linear functional on the group algebra defined by

It is an important property of the group algebra that these exhaust the set of non-trivial (that is, not identically zero) multiplicative linear functionals on the group algebra. See section 34 of the Loomis reference.

Plancherel and Fourier inversion theorems

As we have stated, the dual group of a locally compact abelian group is a locally compact abelian group in its own right and thus has a Haar measure, or more precisely a whole family of scale-related Haar measures.

Theorem. There is a scaling of Haar measure on the dual group so that the Fourier transform restricted to continuous functions of compact support on G, is an isometric linear map. It has a unique extension to a unitary operator In functional analysis, a unitary operator is a bounded linear operator U on a Hilbert space satisfying U*U=UU*=I where I is the identity operator. ...

where ν is the Haar measure on the dual group.

Note that for non-compact locally compact groups G the space L¹(G) does not contain L²(G), so one has to resort to some technical trick such as restricting to a dense subspace.

Following the Loomis reference below, we say that Haar measures on G and G^ are associated if and only if the Fourier inversion formula holds. The unitary character of the Fourier transform implies: In functional analysis, a unitary operator is a bounded linear operator U on a Hilbert space satisfying U*U=UU*=I where I is the identity operator. ...

for every continuous complex-valued function of compact support on G.

It is the unitary extension of the Fourier transform which we consider to be the Fourier transform on the space of square integrable functions. The dual group also has an inverse Fourier transform in its own right; it can be characterized as the inverse (or adjoint, since it is unitary) of the Fourier transform. This is the content of the Fourier inversion formula which follows.

Theorem. The adjoint of the Fourier transform restricted to continuous functions of compact support is the inverse Fourier transform

where the measures on G and G^ are associated.

In the case of G = Rⁿ, we have G′ = Rⁿ and we recover the ordinary Fourier transform on the Rⁿ by taking In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ...

In the case G = T, 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. The Fourier series is a mathematical tool used for analyzing periodic functions by decomposing such a function into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. ...

If G is a finite group, we recover the discrete Fourier transform. Note that this case is very easy to prove directly. In mathematics, the discrete Fourier transform (DFT) is a transform for Fourier analysis of finite-domain discrete-time signals. ...

Bohr compactification and almost-periodicity

One important application of Pontryagin duality is the following characterization of compact abelian topological groups:

Theorem. A locally compact abelian group G is compact if and only if the dual group G^{^} is discrete. Conversely, G is discrete if and only if G^{^} is compact. It has been suggested that this article or section be merged with Logical biconditional. ...

The Bohr compactification is defined for any topological group G, regardless of whether G is locally compact or abelian. One use made of Pontryagin duality between compact abelian groups and discrete abelian groups is to characterize the Bohr compactification of an arbitrary abelian locally compact topological group. The Bohr compactification B(G) of G is H^{^}, where H has the group structure G^{^}, but given the discrete topology. Since the inclusion map In mathematics, the Bohr compactification of a topological group G is a compact Hausdorff topological group H that may be canonically associated to G. Its importance lies in the reduction of the theory of uniformly almost periodic functions on G to the theory of continuous functions on H. The concept... In topology and related fields of mathematics, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are isolated from each other in a certain sense. ... In mathematics, inclusion is a partial order on sets. ...

is continuous and a homomorphism, the dual morphism

is a morphism into a compact group which is easily shown to satisfy the requisite universal property. In various branches of mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique morphism under certain conditions. ...

See also almost periodic function. In mathematics, almost periodicity is a property of dynamical systems that appear to retrace their paths through phase space, but not exactly. ...

Categorical considerations

It is useful to regard the dual group functorially. In what follows, LCA is the category of locally compact abelian groups and continuous group homomorphisms. The dual group construction of G^{^} is a contravariant functor LCA → LCA. In particular, the iterated functor G → (G^{^})^{^} is covariant. Did somebody just say functor? In category theory, a functor is a special type of mapping between categories. ... Did somebody just say functor? In category theory, a functor is a special type of mapping between categories. ...

Theorem. The dual group is a category isomorphism from LCA to LCA^op. In category theory, two categories C and D are isomorphic if there exist functors F : C → D and G : D → C which are mutually inverse to each other, i. ...

Theorem. The iterated dual functor is naturally isomorphic to the identity functor on LCA. In category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i. ...

This isomorphism is comparable to the double dual of finite-dimensional vector spaces (a special case, for real and complex vector spaces). In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ...

The duality interchanges the subcategories of discrete groups and compact groups. If R is a ring 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. Compact as a general noun can refer to: Look up Compact on Wiktionary, the free dictionary a diplomatic contract or covenant among parties, sometimes known as a pact, treaty, or an interstate compact; a British term for a newspaper format; In mathematics, it can refer to various concepts: Mostly commonly... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalars to lie in a field, the scalars may lie in an arbitrary ring. ... In mathematics, an endomorphism is a morphism (or homomorphism) from a mathematical object to itself. ... In category theory, an abstract branch of mathematics, the dual of a category C is the category formed by reversing all the morphisms of C. That is, we take Cop to be the category with objects that are those of C, but with the morphisms from X to Y in...

Non-commutative theory

Such a theory cannot exist in the same form for non-commutative 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^{^}. In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ... Tannaka-Krein duality theory concerns the interaction with a group and the category of its representations. ... Harmonic analysis is the branch of mathematics which studies the representation of functions or signals as the superposition of basic waves. ... Michel Plancherel (16 January 1885 - 4 March 1967) was a Swiss mathematician. ...

There are analogues of duality theory for noncommutative groups, some of which are formulated in the language of C*- algebras. C*-algebras are an important area of research in functional analysis. ...

History

The foundations for the theory of locally compact abelian groups and their duality was laid down by Lev Semenovich Pontryagin in 1934. His treatment 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. Lev Semenovich Pontryagin. ... In topology, a second-countable space is a topological space satisfying the second axiom of countability. Specifically, a space is said to be second-countable if its topology has a countable base. ... André Weil (May 6, 1906 - August 6, 1998) was one of the great mathematicians of the 20th century. ...

References

The following books (available in most university libraries) have chapters on locally compact abelian groups, duality and Fourier transform. The Dixmier reference (also available in English translation) has material on non-commutative harmonic analysis.

• Jacques Dixmier, Les C*-algèbres et leurs Représentations, Gauthier-Villars,1969.
• Lynn H. Loomis, An Introduction to Abstract Harmonic Analysis, D. van Nostrand Co, 1953
• Walter Rudin, Fourier Analysis on Groups, 1962
• Hans Reiter, Classical Harmonic Analysis and Locally Compact Groups, 1968 (2nd ed produced by Jan D. Stegeman, 2000).
• Hewitt and Ross, Abstract Harmonic Analysis, vol 1, 1963.