Pontryagin duality

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In mathematics, particularly harmonic analysis and topological group theory, the Pontriaguin duality explains the general properties of the Fourier transform. It puts into a unified context a number of observations about functions on the real line or on finite abelian groups, eg.

The conveniently regular periodical functions in the real straight have Fourier series and these functions can be recovered from your Fourier series;
The conveniently regular complex-valued functions in the real line have Fourier transformation that are also functions in the real line and, as well as periodic functions, these functions can be recovered from its Fourier transformation; and

complex-valued functions in a finite abelian group have discreet Fourier transformation that are functions in the dual group, which is an isomorph group (not canonically). Even more any function in a finite group can be recovered from its transformation of Fourier discreet.

The theory, introduced by Lev Pontriaguin and combined with the Haar measure introduced by John von Neumann, André Weil and others depends on the dual group theory of a locally compact abelian group.

Haar's 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 a certain open set V containing e that 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 de Haar measure, which allows the "size" of sufficiently regular subsets of G. In this sense, the de Haar measure is a function of "area" or "volume" generalized defined on subsets of G. More precisely, a right Haar measure on a locally compact group G is a countably additive measure:

${displaystyle Amapsto mu (A)quad Asubseteq G}$ a set of Borel

defined on the Borel sets of G that is right invariant in the sense that

${displaystyle mu (Ax)=mu (A)quad {mbox{para }}xin G}$

is finite for compact subsets A and nonzero and positive for open sets. Except for positive scale factors, de Haar measures are unique. Note that it is impossible to define a countably additive right invariant measure on all subsets ' ' of G if the axiom of choice is assumed. See non-measurable set. Note that one can similarly define the left Haar measure. Haar's left and right measures are related by the modular function.

The Haar measure allows defining the notion of Integral for Borelian functions taking complex values defined in the group. In particular, one can consider several L^p spaces associated with the de Haar measure. Specifically,

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Examples of locally abelian groups are:

RⁿStop. n a positive integer, with the addition of vectors as a group operation.

The positive real numbers with multiplication as an operation. This group is clearly seen isomorph to R. In fact, the exponential function implements that isomorphism.

Any finite abelian group. By the structure theorem for finite abelian groups, all these groups are cyclic group products.

The integers Z under the addition.

The group of the unitary circumference, denoted T (i.e., T1 single-dimensional bull). This is the group of complex module numbers 1. T isomorph as the topological group quotient to R/Z.

The dual group

If G is a locally compact abelian group, we define a character of G as a continuous homomorphism of group φ: G → T. The set of all characters in G is another locally compact abelian group, called the dual group of G and denoted as G^ . In more detail, the dual group is defined as follows: If G is an abelian locally compact group, two such characters can be multiplied point by point to form a new character, and the trivial character x → 1 is the identity of G^. The topology of G^ is that of uniform convergence on compacts. It can be shown that the group G^ with the topology thus defined is a locally compact abelian group. Note: Here T is the group of the unit circle, which can be seen as the complex numbers of module 1 or the quotient group R/Z as you see fit. This duality, like most, is an involutionary function, since the dual group of a dual group is the original group. The dual group is presented as the underlying space for an abstract version of the Fourier transform. In this context, functions on the group G (e.g. functions on L¹(G) or L²(G)) become functions with domain on the dual group G^. This is implemented via the integral

{displaystyle {hat {f}}(phi)=int _{G}f(x)phi (-x);dx}

where the integral uses the Haar measure.

Fourier transform in general

The generalization of the most natural Fourier transformation is given, then, by the operator ${displaystyle F:L^{2}(G)mapsto L^{2}(G^{wedge })}$ defined by

(Ff(φ) = ∫ f(x)φ(x) dx

for each f in L²(G) and φ in G^. F is an isometric isomorphism between Hilbert spaces. The f*g of the convolution of two elements f, g in L²(G) can be defined

{displaystyle (f*g)(t)=int f(x)g(t-x),dx}

(this is a function on L²(G) and the convolution theorem F(f*g) = Ff·Fg relating the Fourier transform of the convolution to the product of the two Fourier transforms remains valid. In the case of G = Rⁿ, we have G^ = Rⁿ and recovering the ordinary continuous Fourier transformation, in the case G = S¹, the dual group G^ is naturally isomorphic to the group of integers Z and the aforementioned operator F reduces to the computation of coefficients of the Fourier series of periodic functions; if G is the finite cyclic group Z_n (see modular arithmetic), which coincides with its own dual group, we recover the discrete Fourier transformation.

Examples

For example, a character in the infinite cyclic group of integers Z is determined by its value φ(1), since φ(n) = (φ(1))ⁿ gives its values in the rest of the Z elements. Furthermore, this formula defines a character for any choice of φ(1) in S¹ and the topology of uniform convergence on compacts (appearing here as point-to-point convergence) is the natural topology of Yes¹. Therefore, the dual group of Z is identified by S¹. Conversely, is a character in S¹ of the form z |-> zⁿ for n ∈ Z. Since S¹ 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 S¹ is identified with Z. The other example of a "classical group", the group of real numbers R, is its own dual. Characters in R are of the form φ_y: x |-> e^ixy. With these dualities, the version of the Fourier transform to be introduced later coincides with the Fourier transform in R, and the exponential form of the Fourier series in Z.

The abstract point of view

More precisely, the dual construction of the group G^ of G is a contravariant functor (.)^: LCA - > LCA^op allowing us to identify the LCA category 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 vector spaces and complex). The duality interchanges the subcategories of discrete groups and compact groups. If R is a ring and G is a left-module R, the dual group G^ will become a right R-module; in this way we can also see that the discrete left R-modules will be Pontriaguin dual of the compact right R-modules. The End(G) ring of endomorphisms in LCA is changed by the duality in its opposite ring (changes the multiplication to the opposite order). For example, if G is an infinite cyclic discrete group, then G^ is a circle group: the first has End(G) = Z therefore also End(G^) = Z.

Bohr compaction and quasi-periodicity

One use made of Pontriaguin duality is to give a general definition of a quasi-periodic function on a non-compact group G in LCA. For this, we define the Bohr compactification B(G) of G as H^, where H is like 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 quasi-periodic functions; they can be thought of as analogous to the constraints on a copy of R coiled around a torus.

The non-commutative theory

Such a theory cannot exist in the same form for noncommutative groups G, since in that case the proper dual object G^ of the isomorphism classes of representations cannot contain only one-dimensional representations, and cannot be a group. The generalization that has been found useful in category theory is called the Tannaka-Krein duality; but this diverges from the connection to harmonic analysis, which needs to address the question of the Plancherel measure in G^.

History

The foundations of the theory of locally compact abelian groups and their duality were laid by Lev Pontriaguin in 1934. His treatment was based on groups that were second-countable and compact or discrete. This was improved to cover locally compact abelian groups in general by E.R. van Kampen in 1935 and André Weil in 1953.

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