Pontryagin duality

In mathematics, in particular in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform. It places in a unified context a number of observations about functions on the real line or on finite abelian groups:

Related Topics:
Mathematics - Harmonic analysis - Topological group - Fourier transform

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• Suitably regular complex-valued periodic functions on the real line have Fourier series and that 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.

Related Topics:
Lev Pontryagin - Haar measure - John von Neumann - André Weil - Dual group

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Pontryagin duality: Haar measure ►