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:

Fourier transform

The dual group of an locally compact abelian group is introduced as the underlying space for an abstract version of the Fourier transform. If a function is in L1(G), then the Fourier transform is the function on G^ such that

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: widehat f(chi) = int_G f(x) overline{chi(x)};dmu(x)

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where the integral is relative to Haar measure μ on G. It is not too difficult to show that the Fourier transform of an L1 function on G is a bounded continuous function on G^ which vanishes at infinity. Similarly, the inverse Fourier transform of a integrable function on G^ is given by

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: check{g} (x) = int_{widehat{G}} g(chi) chi(x);d u(chi)

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where the integral is relative to the Haar measure ν on the dual group G^.

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