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 R, the circle or finite cyclic groups. The Pontryagin duality theorem itself states that locally compact 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.
Read more about Pontryagin Duality: Introduction, Locally Compact Abelian Groups, The Pontryagin Duality Theorem, Bohr Compactification and Almost-periodicity, Categorical Considerations