Home > Pontryagin duality
Topological groups Harmonic analysis TheoremsIn 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:
- 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.
1 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 which 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 measureMeasure can mean: To perform a measurement. In mathematics, a measure is a way to assign non-negative real numbers to subsets of a given set, in order to "measure their sizes or probabilities". See measure (mathematics) for a treatment of the concept., the Haar measure, which allows to consistently measure the "size" of sufficiently regular subsets of G. Sufficiently regular, here means Borel set, that is an element of the σ-algebraIn mathematics, a sigma;-algebra (or sigma;-field X over a set S is a family of subsets of S which is closed under countable set operations; σ-algebras are mainly used in order to define measures on S''. The concept is important in mathematical anal generated by the compactSeveral specialized usages of the terms compact and compactness exist. Multiple definitions of the term "compact" are found in mathematics: The most common usage relates to topology, where one considers compact spaces . This article also includes the clos 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.
Haar measure allows 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 Haar measure. Specifically,
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Examples of abelian locally compact groups are:
- 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. By the structure theorem for finite abelian groups, all such groups are products of cyclic groups.
- The positive integers Z under addition.
