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:

Haar measure

A topological group is locally compact if and only if the identity e of the group has a compact neighborhood. This means that there is some open set V containing e whose closure is relatively compact in the topology of G. One of the most remarkable facts about a locally compact group G is that it carries an essentially unique natural measure, the Haar measure, which allows one to consistently measure the "size" of sufficiently regular subsets of G. "Sufficiently regular subset" here means a Borel set; that is, an element of the σ-algebra generated by the compact sets. More precisely, a right Haar measure on a locally compact group G is a countably additive measure μ defined on the Borel sets of G which is right invariant in the sense that μ(A x) = μ(A) for x an element of and A a Borel subset of G and also satisfies some regularity conditions (spelled out in detail in the article Haar measure). Except for positive scale factors, Haar measures are unique.

Related Topics:
Topological group - Measure - Haar measure - Borel set - σ-algebra - Compact

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The Haar measure allows us to define the notion of integral for (complex-valued) Borel functions defined on the group. In particular, one may consider various Lp spaces associated to the Haar measure. Specifically,

Related Topics:
Integral - Complex

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: L^p_mu(G) = {f: G ightarrow mathbb{C}: int_G |f(x)|^p, d mu(x) < infty }

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Examples of locally compact abelian groups are:

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• Rⁿ, for n a positive integer, with vector addition as group operation.
• The positive real numbers with multiplication as operation. This group is clearly seen to be isomorphic to R. In fact, the exponential mapping implements that isomorphism.
• Any finite abelian group, with the discrete topology. By the structure theorem for finite abelian groups, all such groups are products of cyclic groups.
• The positive integers Z under addition, again with the discrete topology.
• The circle group, denoted T. This is the group of complex numbers of modulus 1. T is isomorphic as a topological group to the quotient group R/Z .
• The field Q_p of p-adic numbers under addition, with the usual p-adic topology.