Pontryagin duality
1 Pontryagin dual
Let G be a locally compact abelian group (http://planetmath.org/TopologicalGroup) and π the 1-torus (http://planetmath.org/NTorus), i.e. the unit circle in β.
Definition - A continuous homomorphism
GβΆπ is called a character
of G. The set of all characters is called the Pontryagin dual of G and is denoted by ΛG.
Under pointwise multiplication ΛG is also an abelian group. Since ΛG is a group of functions we can make it a topological group under the compact-open topology
(topology
of convergence on compact sets).
2 Examples
-
β’
Λβ€β π, via nβ¦zn with zβπ.
-
β’
^πβ β€, via zβ¦zn with nββ€.
-
β’
Λββ β, via tβ¦eist with sββ.
3 Properties
The following are some important of the dual group:
Theorem - Let G be a locally compact abelian group. We have that
-
β’
ΛG is also locally compact.
-
β’
ΛG is second countable if and only if G is second countable.
- β’
-
β’
ΛG is discrete if and only if G is compact.
-
β’
^(βiβJGi)β βiβJ^Gi for any finite set
J. This isomorphism
is natural.
4 Pontryagin duality
Let f:GβΆH be a continuous homomorphism of locally compact abelian groups. We can associate to it a canonical map Λf:ΛHβΆΛG defined by
Λf(Ο)(s):= |
This canonical construction preserves identity mappings and compositions, i.e. the dualization process is a functor:
Theorem - The dualization is a contravariant functor from the category of locally compact abelian groups to itself.
5 Isomorphism with the second dual
Although in general there is not a canonical identification of with its dual , there is a natural isomorphism between and its dualβs dual :
Theorem - The map defined by , where , is a natural isomorphism between and .
6 Applications
The study of dual groups allows one to visualize Fourier series, Fourier transforms 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.
Title | Pontryagin duality |
Canonical name | PontryaginDuality |
Date of creation | 2013-03-22 17:42:42 |
Last modified on | 2013-03-22 17:42:42 |
Owner | asteroid (17536) |
Last modified by | asteroid (17536) |
Numerical id | 7 |
Author | asteroid (17536) |
Entry type | Theorem |
Classification | msc 43A40 |
Classification | msc 22B05 |
Classification | msc 22D35 |
Synonym | Pontrjagin duality |
Synonym | Pontriagin duality |
Related topic | DualityInMathematics |
Defines | Pontryagin dual |
Defines | Pontrjagin dual |
Defines | Pontriagin dual |
Defines | dual of an abelian group |
Defines | character |