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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. ==Introduction== Pontryagin duality places in a unified context a number of observations about functions on the real line or on finite abelian groups: * Suitably regular complex-valued periodic functions on the real line have Fourier series and these functions can be recovered from their Fourier series; * Suitably regular complex-valued functions on the real line have Fourier transforms that are also functions on the real line and, just as for periodic functions, these functions can be recovered from their Fourier transforms; and * Complex-valued functions on a finite abelian group have discrete Fourier transforms which are functions on the dual group, which is a (non-canonically) isomorphic group. Moreover, any function on a finite group can be recovered from its discrete Fourier transform. 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. It is analogous to the dual vector space of a vector space: a finite-dimensional vector space ''V'' and its dual vector space ''V *'' are not naturally isomorphic, but their endomorphism algebras (matrix algebras) are: End(''V'') ≅ End(''V *''), via the transpose. Similarly, a group ''G'' and its dual group ''G^'' are not in general isomorphic, but their group algebras are: ''C''(''G'') ≅ ''C''(''G^'') via the Fourier transform, though one must carefully define these algebras analytically. More categorically, this is not just an isomorphism of endomorphism algebras, but an isomorphism of categories – see categorical considerations. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Pontryagin duality」の詳細全文を読む スポンサード リンク
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