Pontryagin dual
In mathematics, specifically in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform on locally compact groups, such as , the circle, or finite cyclic groups. The Pontryagin duality theorem itself states that locally compact abelian groups identify naturally with their bidual.
The subject is named after Lev Semenovich Pontryagin who laid down the foundations for the theory of locally compact abelian groups and their duality during his early mathematical works in 1934. Pontryagin's 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 Egbert van Kampen in 1935 and André Weil in 1940.
Pontryagin duality places in a unified context a number of observations about functions on the real line or on finite abelian groups:
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.
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