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:

The group algebra

The space of integrable functions on a locally compact abelian group G is an algebra, where multiplication is convolution: if f, g are integrable functions then the convolution of f and g is defined as

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: (x) = int_G f(x - y) g(y), d mu(y).

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Theorem The Banach space L1(G) is an associative and commutative algebra under convolution.

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This algebra is referred to as the Group Algebra of G. By completeness of L1(G), it is a Banach algebra. The Banach algebra L1(G) does not have a multiplicative identity element unless G is a discrete group. In general, however, it has an approximate identity which is a net (or generalized sequence) indexed on a directed set I, {ei}i with the property that

Related Topics:
Banach algebra - Approximate identity

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: f star e_i ightarrow f.

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The Fourier transform takes convolution to multiplication, that is:

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: mathcal{F}( f star g)(chi) = mathcal{F}(f)(chi) cdot mathcal{F}(g)(chi).

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In particular, to every group character on G corresponds a unique multiplicative linear functional on the group algebra defined by

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: f mapsto widehat{f}(chi).

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It is an important property of the group algebra that these exhaust the set of non-trivial multiplicative linear functionals on the group algebra. See section 34 of the Loomis reference.

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