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| [[Image:2-adic integers with dual colorings.svg|thumb|300px|The [[p-adic integer|2-adic integers]], with selected corresponding characters on [[Prüfer group|their Pontryagin dual group]]]]
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| In mathematics, specifically in [[harmonic analysis]] and the theory of [[topological group]]s, '''Pontryagin duality''' explains the general properties of the [[Fourier transform]] on locally compact groups, such as '''R''', the circle, or [[finite cyclic group]]s. The '''Pontryagin duality theorem''' itself states that locally compact groups identify naturally with their bidual.
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| The subject is named after [[Lev Semenovich Pontryagin]] who laid down the foundations for the theory of locally compact abelian groups and their duality during his early mathematical works in 1934. Pontryagin's treatment relied on the group being [[second-countable space|second-countable]] and either compact or discrete. This was improved to cover the general locally compact abelian groups by [[Egbert van Kampen]] in 1935 and [[André Weil]] in 1940.
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| ==Introduction==
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| Pontryagin duality places in a unified context a number of observations about functions on the real line or on finite abelian groups:
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| * Suitably regular complex-valued [[periodic function]]s on the real line have [[Fourier series]] and these functions can be recovered from their Fourier series;
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| * 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
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| * Complex-valued functions on a finite [[abelian group]] have [[discrete Fourier transform]]s which are functions on the [[#The dual group|dual group]], which is a (non-canonically) isomorphic group. Moreover any function on a finite group can be recovered from its discrete Fourier transform.
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| 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 [[#The dual group|dual group]] of a [[Locally compact space|locally compact]] abelian group.
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| It is analogous to the [[dual vector space]] of a vector space: a finite-dimensional vector space ''V'' and its dual vector space ''V*'' are not naturally isomorphic, but their [[endomorphism]] algebras (matrix algebras) are: End(''V'') ≅ End(''V*''), via the transpose. Similarly, a group ''G'' and its dual group ''G^'' are not in general isomorphic, but their [[group algebra]]s are: ''C''(''G'') ≅ ''C''(''G^'') via the Fourier transform, though one must carefully define these algebras analytically. More categorically, this is not just an isomorphism of endomorphism algebras, but an isomorphism of categories – see [[#Categorical considerations|categorical considerations]].
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| ==Locally compact abelian groups==
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| A [[topological group]] is ''[[locally compact group|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 compact in the topology of ''G''. | |
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| ===Examples===
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| Examples of locally compact [[abelian group|abelian]] groups are:
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| * '''R'''<sup>n</sup>, for ''n'' a positive integer, with vector addition as group operation.
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| * The positive [[real number]]s with multiplication as operation. This group is isomorphic to '''R''', by the exponential map.
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| * Any finite abelian group, with the [[discrete topology]]. By the [[structure theorem for finite abelian groups]], all such groups are products of cyclic groups.
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| * The integers '''Z''' under addition, again with the discrete topology.
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| * The [[circle group]], denoted '''T,''' for [[torus]]. This is the group of complex numbers of [[absolute value|modulus]] 1. '''T''' is isomorphic as a topological group to the [[quotient group]] '''R'''/'''Z''' .
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| * The field '''Q'''<sub>''p''</sub> of [[p-adic number|''p''-adic number]]s under addition, with the usual ''p''-adic topology.
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| ===The dual group===
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| If ''G'' is a locally compact ''abelian'' group, a ''[[Character (mathematics)|character]] of G'' is a [[continuous function (topology)|continuous]] [[group homomorphism]] from ''G'' with values in the [[circle group]] '''T'''. The set of all characters on ''G'' can be made into a locally compact abelian group, called the ''dual group'' of ''G'' and denoted ''G^''. The group operation on the dual group is given by pointwise multiplication of characters, the inverse of a character is its complex conjugate and the [[topology]] on the space of characters is that of [[uniform convergence]] on [[compact set]]s (i.e., the [[compact-open topology]], viewing ''G^'' as a subset of the space of all continuous functions from ''G'' to '''T'''.). This topology in general is not metrizable. However, if the group ''G'' is a [[separable space|separable]] locally compact abelian group, then the dual group is metrizable.
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| This is analogous to the [[dual space]] in linear algebra: just as for a vector space ''V'' over a field ''K,'' the dual space is Hom(''V'', ''K''), so too is the dual group Hom(''G'', '''T'''). More abstractly, these are both examples of [[representable functor]]s, being represented respectively by ''K'' and '''T.'''
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| A group that is isomorphic (as topological groups) to its dual group is called ''self-dual''. While the reals and '''Z'''/''n'''''Z''' are self-dual, the group and the dual group are not ''naturally'' isomorphic, and should be thought of as two different groups.
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| ====Examples of dual groups====
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| The dual of '''Z''' is isomorphic to the circle group '''T'''.
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| Proof: 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)<sup>''n''</sup>. Moreover, this formula defines a character for any choice of χ(1) in '''T'''. The topology of uniform convergence on compact sets is in this case the topology of [[pointwise convergence]]. This is the topology of the circle group inherited from the complex numbers.
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| The dual of '''T''' is canonically isomorphic with '''Z'''.
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| Proof: A character on '''T''' is of the form ''z'' → ''z''<sup>''n''</sup> 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]].
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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''<sup>iθ''r''</sup>. 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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| Analogously, the ''p''-adic numbers '''Q'''<sub>''p''</sub> are isomorphic to its dual. It follows that the [[adele ring|adele]]s are self-dual.
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| ==The Pontryagin duality theorem==
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| <blockquote>'''Theorem.''' The dual of ''G^'' is canonically isomorphic to ''G'', that is (''G^'')''^'' = ''G'' in a canonical way.</blockquote>
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| [[Canonical form|Canonical]] means that there is a ''naturally'' defined map from ''G'' into (''G^'')''^''; more importantly, the map should be [[functorial]]. The canonical isomorphism is defined as follows:
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| : <math> x \mapsto \{\chi \mapsto \chi(x) \}\mbox{ i.e. } x(\chi):=\chi(x).</math>
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| In other words, each group element ''x'' is identified to the evaluation character on the dual. This is exactly the same as the canonical isomorphism between a [[finite-dimensional vector space]] and its [[double dual]], ''V'' ≅ ''V**''. However, there is also a difference: ''V'' is isomorphic to its dual space ''V*'', although not canonically so, while many groups ''G'' are not isomorphic to their dual groups (for instance, when ''G'' is '''T''' its dual is '''Z''', and '''T''' is not isomorphic to '''Z''' as topological groups). If ''G'' is a '''finite''' abelian group, then ''G'' and ''G^'' are isomorphic, but not canonically. To make precise the statement that there is no canonical isomorphism between finite abelian groups and their dual groups (in general) requires thinking about dualizing not only on groups, but also on maps between the groups, in order to treat dualization as a [[functor]] and prove the identity functor and the dualization functor are not naturally equivalent. Also it should be noted that the duality theorem implies that for any ''G'' (not necessarily finite) the dualization functor is an exact functor.
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| == Pontryagin duality and the Fourier transform ==
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| ===Haar measure===
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| {{main|Haar measure}}
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| One of the most remarkable facts about a locally compact group ''G'' is that it carries an essentially unique natural [[Measure (mathematics)|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 [[sigma-algebra|σ-algebra]] generated by the [[compact set]]s. 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(''Ax'') = μ(''A'') for ''x'' an element of ''G'' and ''A'' a Borel subset of ''G'' and also satisfies some regularity conditions (spelled out in detail in the article on [[Haar measure]]). Except for positive scaling factors, a Haar measure on ''G'' is unique.
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| The Haar measure on ''G'' allows us to define the notion of [[integral]] for ([[complex number|complex]]-valued) Borel functions defined on the group. In particular, one may consider various ''L<sup>p</sup>'' spaces associated to the Haar measure. Specifically,
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| :<math> L^p_\mu(G) = \left\{f: G \rightarrow \mathbf{C}\;\left|\; \int_G |f(x)|^p\, d \mu(x) < \infty \right.\right\}. </math>
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| Note that, since any two Haar measures on ''G'' are equal up to a scaling factor, this ''L<sup>p</sup>''-space is independent of the choice
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| of Haar measure and thus perhaps could be written as ''L<sup>p</sup>(G)''. However, the ''L<sup>p</sup>''-norm on this space
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| depends on the choice of Haar measure, so if one wants to talk about isometries it is important to keep track of the Haar measure being used.
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| === Fourier transform and Fourier inversion formula for ''L''<sup>1</sup>-functions ===
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| The dual group of a locally compact abelian group is used as the underlying space for an abstract version of the [[Fourier transform]]. If a function is in ''L''<sup>1</sup>(''G''), then the Fourier transform is the function <math>\widehat f</math> on ''G^'' defined by
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| :<math> \widehat f(\chi) = \int_G f(x) \overline{\chi(x)}\;d\mu(x),</math>
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| where the integral is relative to [[Haar measure]] μ on ''G''. This is also denoted <math>({\mathcal F}f)(\chi)</math>. Note the Fourier transform depends on the
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| choice of Haar measure.
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| It is not too difficult to show that the Fourier transform of an ''L''<sup>1</sup> function on ''G'' is a bounded continuous function on ''G^'' which vanishes at infinity. The Fourier inversion formula for ''L<sup>1</sup>''-functions says that for each Haar measure μ on ''G'' there is a unique Haar measure ν on ''G^'' such that whenever ''f'' is in ''L''<sup>1</sup>(''G'') and its Fourier transform <math>\widehat f</math> is in ''L''<sup>1</sup>(''G^''), we have
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| :<math> f(x) = \int_{\widehat{G}} \widehat f(\chi)\chi(x)\;d\nu(\chi) </math>
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| for μ-almost all ''x'' in ''G''. If ''f'' is continuous then this identity holds for all ''x''. (The ''inverse Fourier transform'' of an integrable function on ''G^'' is given by | |
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| :<math> \check{g} (x) = \int_{\widehat{G}} g(\chi) \chi(x)\;d\nu(\chi),</math>
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| where the integral is relative to the Haar measure ν on the dual group ''G^''.) The measure ν on ''G^'' that appears in the Fourier inversion formula is called the [[dual measure]] to μ and may be denoted <math>\widehat{\mu}</math>.
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| The various Fourier transforms can be classified in terms of their domain and transform domain (the group and dual group) as follows:
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| {|
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| ! Transform !! Original domain !! Transform domain
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| | Fourier transform || '''R''' || '''R'''
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| | [[Fourier series]] || '''T''' || '''Z'''
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| |-
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| | [[Discrete-time Fourier transform]] (DTFT) || '''Z''' || '''T'''
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| | [[Discrete Fourier transform]] (DFT) || '''Z'''/(''n'') || '''Z'''/(''n'')
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| |}
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| As an example, suppose ''G'' = '''R'''<sup>''n''</sup>, so we can think about ''G^'' as '''R'''<sup>''n''</sup> by the pairing <math>({\mathbf v},{\mathbf w}) \mapsto e^{i{\mathbf v}\cdot {\mathbf w}}</math>. If we use for μ Lebesgue measure on Euclidean space, we obtain the ordinary [[Fourier transform]] on '''R'''<sup>''n''</sup> and the [[dual measure]] needed for the Fourier inversion formula is <math>\widehat{\mu} = 1/(2\pi)^n\mu</math>. If we want to get a Fourier inversion formula with the same measure on both sides (that is, since we can think about '''R'''<sup>''n''</sup> as its own dual space we can ask for <math>\widehat{\mu}</math> to equal μ) then we need to use
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| :<math> \mu = (2 \pi)^{-n/2} \times \mbox{Lebesgue measure}</math>
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| :<math> \widehat{\mu} = (2 \pi)^{-n/2} \times \mbox{Lebesgue measure}.</math>
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| However, if we change the way we identify '''R'''<sup>''n''</sup> with its dual group, by using the pairing <math>({\mathbf v},{\mathbf w}) \mapsto e^{2\pi{i}{\mathbf v}\cdot {\mathbf w}}</math>, then Lebesgue measure on '''R'''<sup>''n''</sup> is equal to its own [[dual measure]]. This convention minimizes the number of factors of 2π that show up in various places when computing Fourier transforms or inverse Fourier transforms on Euclidean space. (In effect it limits the 2π only to the exponent rather than as some messy factor outside the integral sign.) Note that the choice of how to identify '''R'''<sup>''n''</sup> with its dual group affects the meaning of the term *self-dual function*, which is a function on '''R'''<sup>''n''</sup> equal to its own Fourier transform: using the classical pairing <math>({\mathbf v},{\mathbf w}) \mapsto e^{i{\mathbf v}\cdot {\mathbf w}}</math> the function <math>e^{-x^2/2}</math> is self-dual, but using the (cleaner) pairing <math>({\mathbf v},{\mathbf w}) \mapsto e^{2\pi{i}{\mathbf v}\cdot {\mathbf w}}</math> makes <math>e^{-\pi x^2}</math> self-dual instead.
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| === The group algebra ===
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| 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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| :<math> [f \star g](x) = \int_G f(x - y) g(y)\, d \mu(y).</math>
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| <blockquote>'''Theorem.''' The Banach space ''L''<sup>1</sup>(''G'') is an associative and commutative algebra under convolution.</blockquote>
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| This algebra is referred to as the ''Group Algebra'' of ''G''. By the [[Fubini's theorem|Fubini-Tonelli theorem]], the convolution is submultiplicative with respect to the ''L''<sup>1</sup> norm, making ''L''<sup>1</sup>(''G'') a [[Banach algebra]]. The Banach algebra ''L''<sup>1</sup>(''G'') has a multiplicative identity element if and only if ''G'' is a discrete group, namely the function that is 1 at the identity and zero elsewhere. In general, however, it has an [[approximate identity]] which is a net (or generalized sequence) indexed on a directed set ''I'', {''e''<sub>''i''</sub>}<sub>''i''</sub> with the property that
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| :<math> f \star e_i \rightarrow f. </math>
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| The Fourier transform takes convolution to multiplication, i.e. it is a homomorphism of abelian Banach algebras from ''L''<sup>1</sup>(''G'') to ''C''<sub>0</sub>(''G^'') (of norm ≤ 1):
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| :<math> \mathcal{F}( f \star g)(\chi) = \mathcal{F}(f)(\chi) \cdot \mathcal{F}(g)(\chi).</math>
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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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| :<math> f \mapsto \widehat{f}(\chi).</math>
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| 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. This means the Fourier transform is a special case of the [[Gelfand transform]].
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| === Plancherel and ''L''<sup>2</sup> Fourier inversion theorems ===
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| 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.
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| <blockquote>'''Theorem.''' Choose a Haar measure μ on ''G'' and let ν be the dual measure on ''G^'' as defined above. If ''f'' is a continuous complex-valued function of compact support on ''G'', its Fourier transform <math>\widehat{f}</math> is in ''L''<sup>2</sup>(''G^'') and
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| : <math> \int_G |f(x)|^2 \ d \mu(x) = \int_{\widehat{G}} |\widehat{f}(\chi)|^2 \ d \nu(\chi).</math>
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| In particular, the Fourier transform is an ''L''<sup>2</sup> isometry from the complex-valued continuous functions of compact support on ''G'' to the ''L''<sup>2</sup>-functions on ''G^'' (using the ''L''<sup>2</sup>-norm with respect to μ for functions on ''G'' and the ''L''<sup>2</sup>-norm with respect to ν for functions on ''G^'').</blockquote>
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| Since the complex-valued continuous functions of compact support on ''G'' are ''L''<sup>2</sup>-dense, there is a unique extension of the Fourier transform from that space to a [[unitary operator]]
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| : <math> \mathcal{F}: L^2_\mu(G) \rightarrow L^2_\nu(\widehat{G}).</math>
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| and we have the formula
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| : <math> \int_G |f(x)|^2 \ d \mu(x) = \int_{\widehat{G}} |\widehat{f}(\chi)|^2 \ d \nu(\chi) </math>
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| for all ''f'' in ''L''<sup>2</sup>(''G'').
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| Note that for non-compact locally compact groups ''G'' the space ''L<sup>1</sup>(G)'' does not contain ''L<sup>2</sup>(G)'', so the Fourier transform of general ''L''<sup>2</sup>-functions on ''G'' is *not* given by any kind of integration formula (or really any explicit formula). To define the ''L''<sup>2</sup> Fourier transform one has to resort to some technical trick such as starting on a dense subspace like the continuous functions with compact support and then extend the isometry by continuity to the whole space. This unitary extension of the Fourier transform is what we mean by the Fourier transform on the space of square integrable functions.
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| The dual group also has an inverse Fourier transform in its own right; it can be characterized as the inverse (or adjoint, since it is unitary) of the ''L''<sup>2</sup> Fourier transform. This is the content of the ''L''<sup>2</sup> Fourier inversion formula which follows.
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| <blockquote>'''Theorem.''' The adjoint of the Fourier transform restricted to continuous functions of compact support is the inverse Fourier transform
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| : <math> L^2_\nu(\widehat{G}) \rightarrow L^2_\mu(G) </math>
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| where ν is the dual measure to μ.</blockquote>
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| In the case ''G'' = '''T''', the dual group ''G''^ is naturally isomorphic to the group of integers '''Z''' and the Fourier transform specializes to the computation of coefficients of [[Fourier series]] of periodic functions.
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| If ''G'' is a finite group, we recover the [[discrete Fourier transform]]. Note that this case is very easy to prove directly. | |
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| ==Bohr compactification and almost-periodicity==<!-- This section is linked from [[Discrete space]] -->
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| One important application of Pontryagin duality is the following characterization of compact abelian topological groups:
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| <blockquote>'''Theorem'''. A locally compact ''abelian'' group ''G'' is compact [[if and only if]] the dual group ''G^'' is discrete. Conversely,
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| ''G'' is discrete if and only if ''G^'' is compact.</blockquote>
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| That ''G'' being compact implies ''G^'' is discrete or that ''G'' being discrete implies that ''G^'' is compact is an elementary consequence of the definition of the compact-open topology on ''G^'' and does not need Pontryagin duality. One uses Pontryagin duality to prove the converses.
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| The [[Bohr compactification]] is defined for any topological group ''G'', regardless of whether ''G'' is locally compact or abelian. One use made of Pontryagin duality between compact abelian groups and discrete abelian groups is to characterize the Bohr compactification of an arbitrary abelian ''locally compact'' topological group. The ''Bohr compactification'' ''B(G)'' of ''G'' is ''H^'', where ''H'' has the group structure ''G^'', but given the [[discrete topology]]. Since the [[inclusion map]]
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| : <math> \iota: H \rightarrow \widehat{G} </math> | |
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| is continuous and a homomorphism, the dual morphism
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| : <math> G \sim \widehat{\widehat{G}} {\rightarrow} \widehat{H} </math>
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|
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| is a morphism into a compact group which is easily shown to satisfy the requisite [[universal property]].
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| See also [[almost periodic function]].
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| ==Categorical considerations ==
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| It is useful to regard the dual group [[functor|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''', represented (in the sense of [[representable functor]]s) by the circle group '''T,''' as ''G^'' = Hom(''G'', '''T'''). In particular, the iterated functor ''G'' → ''(G^)^'' is ''covariant''.
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| '''Theorem'''. The dual group functor is an [[equivalence of categories]] from '''LCA''' to '''LCA'''<sup>op</sup>.
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| '''Theorem'''. The iterated dual functor is [[natural transformation|naturally isomorphic]] to the identity functor on '''LCA'''.
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| This isomorphism is analogous to the [[double dual]] of [[finite-dimensional vector space]]s (a special case, for real and complex vector spaces).
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| The duality interchanges the subcategories of discrete groups and [[compact group]]s. If ''R'' is a [[ring (mathematics)|ring]] and ''G'' is a left ''R''-[[module (mathematics)|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#Constructing new rings from given ones|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.
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| ==Generalizations==
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| ===Non-commutative theory===
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| 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 ''[[Michel Plancherel|Plancherel]] measure'' on ''G^''.
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| There are analogues of duality theory for noncommutative groups, some of which are formulated in the language of [[C-star algebra|C*-algebras]].
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| === Others ===
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| When ''G'' is a Hausdorff abelian topological group, the group ''G^'' with the compact-open topology is a Hausdorff abelian topological group and the natural mapping from ''G'' to its double-dual ''G^^'' makes sense. If this mapping is an isomorphism, we say that ''G'' satisfies Pontryagin duality. This has been extended in a number directions beyond the case that ''G'' is locally compact.
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| * S.Kaplan, in "Extensions of the Pontryagin duality" ("part I: infinite products", Duke Math. J. 15 (1948) 649–658, and "part II: direct and inverse limits", same journal, 17 (1950), 419–435) showed that arbitrary products and countable inverse limits of locally compact (Hausdorff) abelian groups satisfy Pontryagin duality. Note that an infinite product of locally compact non-compact spaces is not locally compact.
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| * Later, in 1975, R.Venkataraman ("Extensions of Pontryagin Duality", Math. Z. 143, 105-112) showed, among other facts, that every open subgroup of an abelian topological group which satisfies Pontryagin duality itself satisfies Pontryagin duality.
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| * More recently, S. Ardanza-Trevijano and M.J. Chasco have extended the results of Kaplan mentioned above. They showed, in "The Pontryagin duality of sequential limits of topological Abelian groups", Journal of Pure and Applied Algebra 202 (2005), 11–21, that direct and inverse limits of sequences of abelian groups satisfying Pontryagin duality also satisfy Pontryagin duality if the groups are metrizable or ''k''<sub>ω</sub>-spaces but not necessarily locally compact, provided some extra conditions are satisfied by the sequences.
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| However, there is a fundamental aspect that changes if we want to consider Pontryagin duality beyond the locally compact case. In E. Martin-Peinador, ''A reflexible admissible topological group must be locally compact'', Proc. Amer. Math. Soc. 123 (1995), 3563-3566, it is proved that if ''G'' is a Hausdorff abelian topological group that satisfies Pontryagin duality and the natural evaluation pairing from ''G'' × ''G^'' to '''T''', where (''x'', χ) goes to χ(''x''), is continuous, then ''G'' is locally compact. Thus any non-locally compact example of Pontryagin duality is a group where the natural evaluation pairing of ''G'' and ''G^'' is not continuous.
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| ==See also==
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| * [[Peter–Weyl theorem]]
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| * [[Cartier Duality]]
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| * [[Stereotype space]]
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| ==References==
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| The following books 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.
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| * Jacques Dixmier, ''Les C*-algèbres et leurs Représentations'', Gauthier-Villars,1969.
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| * Lynn H. Loomis, ''An Introduction to Abstract Harmonic Analysis'', D. van Nostrand Co, 1953
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| * [[Walter Rudin]], ''Fourier Analysis on Groups'', 1962
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| * Hans Reiter, Classical Harmonic Analysis and Locally Compact Groups, 1968 (2nd ed produced by [[Jan D. Stegeman]], 2000).
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| * Hewitt and Ross, ''Abstract Harmonic Analysis, vol 1'', 1963.
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| [[Category:Topological groups]]
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| [[Category:Harmonic analysis]]
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| [[Category:Duality theories]]
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| [[Category:Theorems in analysis]]
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| [[Category:Fourier analysis]]
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| [[ja:ポントリャーギン双対]]
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