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Locally Compact Abelian Group
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  • Suitably regular complex-valued functions on the real line have Fourier Transform s 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 Transform s 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 .


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 Set s 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.

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,



u(\chi)

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


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

Theorem The Banach space ''L''1(''G'') is an associative and commutative algebra under convolution.

This algebra is referred to as the ''Group Algebra'' of ''G''. By completeness of ''L''1(''G''), it is a Banach Algebra . The Banach algebra ''L''1(''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'', {''e''''i''}''i'' with the property that
: f \star e_i ightarrow f.

The Fourier transform takes convolution to multiplication, that is:
: \mathcal{F}( f \star g)(\chi) = \mathcal{F}(f)(\chi) \cdot \mathcal{F}(g)(\chi).
In particular, to every group character on ''G'' corresponds a unique ''multiplicative linear functional'' on the group algebra defined by
: f \mapsto \widehat{f}(\chi).
It is an important property of the group algebra that these exhaust the set of non-trivial (that is, not identically zero) multiplicative linear functionals on the group algebra. See section 34 of the ''Loomis'' reference.


PLANCHEREL AND FOURIER INVERSION THEOREMS


As we have stated, the dual group of a locally compact abelian group is a locally compact abelian group in its own right and thus has a Haar measure, or more precisely a whole family of scale-related Haar measures.

Theorem. There is a scaling of Haar measure on the dual group so that the Fourier transform restricted to continuous functions of compact support on ''G'', is an isometric linear map. It has a unique extension to a Unitary Operator

: \mathcal{F}: L^2_\mu(G) ightarrow L^2_
u(\widehat{G})

where
u is the Haar measure on the dual group.

Note that for non-compact locally compact groups ''G'' the space ''L1(G)'' does not contain ''L2(G)'', so one has to resort to some technical trick such as restricting to a dense subspace.

Following the ''Loomis'' reference below, we say that Haar measures on ''G'' and ''G^'' are ''associated'' if and only if the Fourier inversion formula holds. The Unitary character of the Fourier transform implies:




is a morphism into a compact group which is easily shown to satisfy the requisite Universal Property .

See also Almost Periodic Function .


CATEGORICAL CONSIDERATIONS


It is useful to regard the dual group Functorially . In what follows, LCA is the category of locally compact abelian groups and continuous group homomorphisms.
The dual group construction of ''G^'' is a contravariant Functor LCA → LCA. In particular, the iterated functor
''G'' → ''(G^)^'' is ''covariant''.

Theorem. The dual group functor is an Equivalence Of Categories from '''LCA''' to '''LCA'''op.

Theorem. The iterated dual functor is Naturally Isomorphic to the identity functor on '''LCA'''.

This isomorphism is comparable to the double dual of finite-dimensional Vector Space s (a special case, for real and complex vector spaces).

The duality interchanges the subcategories of discrete groups and Compact groups. If R is a Ring and G is a left R- Module , the dual group G^ will become a right R-module; in this way we can also see that discrete left R-modules will be Pontryagin dual to compact right R-modules. The ring End(G) of Endomorphism s in LCA is changed by duality into its Opposite ring (change the multiplication to the other order). For example if G is an infinite cyclic discrete group, G^ is a circle group: the former has End(G) = '''Z''' so this is true also of the latter.


NON-COMMUTATIVE THEORY


Such a theory cannot exist in the same form for non-commutative groups G, since in that case the appropriate dual object G^ of isomorphism classes of representations cannot only contain one-dimensional representations, and will fail to be a group. The generalisation that has been found useful in Category Theory is called Tannaka-Krein Duality ; but this diverges from the connection with Harmonic Analysis , which needs to tackle the question of the '' Plancherel measure'' on G^.



HISTORY


The foundations for the theory of locally compact abelian groups and their duality were laid down by Lev Semenovich Pontryagin in 1934. His treatment relied on the group being Second-countable and either compact or discrete. This was improved to cover the general locally compact abelian groups by E.R. Van Kampen in 1935 and André Weil in 1953.


REFERENCES


The following books (available in most university libraries) have chapters on locally compact abelian groups, duality and Fourier transform. The Dixmier reference (also available in English translation) has material on non-commutative harmonic analysis.