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:

Examples

A character on the infinite cyclic group of integers Z under addition is determined by its value at the generator 1. Thus for any character χ on Z, χ(n)=χ(1)n. Moreover, this formula defines a character for any choice of χ(1) in T. Thus it follows easily that algebraically the dual of Z is isomorphic to the circle group T. The topology of uniform convergence on compact sets is in this case the topology of pointwise convergence. It is also easily shown that this is the topology of the circle group inherited from the complex numbers.

Related Topics:
Infinite cyclic group - Pointwise convergence

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Hence the dual group of Z is canonically isomorphic with T.

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Conversely, a character on T is of the form z → zn for n an integer. Since T is compact, the topology on the dual group is that of uniform convergence, which turns out to be the discrete topology. As a consequence of this, the dual of T is canonically isomorphic with Z.

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The group of real numbers R, is isomorphic to its own dual; the characters on R are of the form r → e i θ r. With these dualities, the version of the Fourier transform to be introduced next coincides with the classical Fourier transform on R.

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