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单词 PontryaginDuality
释义

Pontryagin duality


1 Pontryagin dual

Let G be a locally compact abelian groupMathworldPlanetmath (http://planetmath.org/TopologicalGroup) and 𝕋 the 1-torus (http://planetmath.org/NTorus), i.e. the unit circle in .

Definition - A continuousPlanetmathPlanetmath homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath G𝕋 is called a characterPlanetmathPlanetmath of G. The set of all characters is called the Pontryagin dual of G and is denoted by G^.

Under pointwise multiplication G^ is also an abelian group. Since G^ is a group of functions we can make it a topological groupMathworldPlanetmath under the compact-open topologyMathworldPlanetmath (topologyMathworldPlanetmath of convergence on compact sets).

2 Examples

  • ^𝕋, via nzn with z𝕋.

  • 𝕋^, via zzn with n.

  • ^, via teist with s.

3 Properties

The following are some important of the dual group:

Theorem - Let G be a locally compact abelian group. We have that

  • G^ is also locally compact.

  • G^ is second countable if and only if G is second countable.

  • G^ is compact if and only if G is discrete.

  • G^ is discrete if and only if G is compact.

  • (iJGi)^iJGi^ for any finite setMathworldPlanetmath J. This isomorphismMathworldPlanetmathPlanetmath is natural.

4 Pontryagin duality

Let f:GH be a continuous homomorphism of locally compact abelian groups. We can associate to it a canonical map f^:H^G^ defined by

f^(ϕ)(s):=ϕ(f(s)),ϕH^,sG

This canonical construction preserves identity mappings and compositions, i.e. the dualization process ^ is a functorMathworldPlanetmath:

Theorem - The dualization ^:𝐋𝐜𝐀𝐋𝐜𝐀 is a contravariant functor from the categoryMathworldPlanetmath of locally compact abelian groups to itself.

5 Isomorphism with the second dual

Although in general there is not a canonical identification of G with its dual G^, there is a natural isomorphism between G and its dual’s dual G^^:

Theorem - The map GG^^ defined by ss^^, where s^^(ϕ):=ϕ(s), is a natural isomorphism between G and G^^.

6 Applications

The study of dual groups allows one to visualize Fourier series, Fourier transformsMathworldPlanetmath and discrete Fourier transforms from a more abstract and unified view-point, providing the for a general definition of Fourier transform. Thus, dual groups and Pontryagin duality are the of the of abstract abelian harmonic analysis.

TitlePontryagin duality
Canonical namePontryaginDuality
Date of creation2013-03-22 17:42:42
Last modified on2013-03-22 17:42:42
Ownerasteroid (17536)
Last modified byasteroid (17536)
Numerical id7
Authorasteroid (17536)
Entry typeTheorem
Classificationmsc 43A40
Classificationmsc 22B05
Classificationmsc 22D35
SynonymPontrjagin duality
SynonymPontriagin duality
Related topicDualityInMathematics
DefinesPontryagin dual
DefinesPontrjagin dual
DefinesPontriagin dual
Definesdual of an abelian group
Definescharacter
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