Editing Fourier transform (section)

== Generalizations ==

=== Fourier–Stieltjes transform on measurable spaces <span class="anchor" id="Fourier-Stieltjes transform"></span>===
{{see also|Bochner–Minlos theorem|Riesz–Markov–Kakutani representation theorem|Fourier series#Fourier-Stieltjes series}}
The Fourier transform of a [[finite measure|finite]] [[Borel measure]] {{mvar|μ}} on {{math|'''R'''<sup>''n''</sup>}} is given by the continuous function:{{sfn|Pinsky|2002|p=256}}
<math display="block">\hat\mu(\xi)=\int_{\mathbb{R}^n} e^{-i 2\pi x \cdot \xi}\,d\mu,</math>
and called the ''Fourier-Stieltjes transform'' due to its connection with the [[Riemann–Stieltjes integral#Application to functional analysis|Riemann-Stieltjes integral]] representation of [[Radon measure|(Radon) measures]].{{sfn|Edwards|1982|pp=53,67,72-73}} If <math>\mu</math> is the [[probability distribution]] of a [[random variable]] <math>X</math> then its Fourier–Stieltjes transform is, by definition, a [[Characteristic function (probability theory)|characteristic function]].<ref>{{harvnb|Katznelson|2004|p=173}} {{pb}} The typical conventions in probability theory take {{math|''e''<sup>''iξx''</sup>}} instead of {{math|''e''<sup>−''i''2π''ξx''</sup>}}.</ref> If, in addition, the probability distribution has a [[probability density function]], this definition is subject to the usual Fourier transform.{{sfn|Billingsley|1995|p=345}} Stated more generally, when <math>\mu</math> is [[Absolute continuity#Absolute continuity of measures|absolutely continuous]] with respect to the Lebesgue measure, i.e., 
<math display="block"> d\mu = f(x)dx,</math>
then
<math display="block">\hat{\mu}(\xi)=\hat{f}(\xi),</math>
and the Fourier-Stieltjes transform reduces to the usual definition of the Fourier transform. That is, the notable difference with the Fourier transform of integrable functions is that the Fourier-Stieltjes transform need not vanish at infinity, i.e., the [[Riemann–Lebesgue lemma]] fails for measures.{{sfn|Katznelson|2004|pp=155,164}}

[[Bochner's theorem]] characterizes which functions may arise as the Fourier–Stieltjes transform of a positive measure on the circle.

One example of a finite Borel measure that is not a function is the [[Dirac measure]].{{sfn|Edwards|1982|p=53}} Its Fourier transform is a constant function (whose value depends on the form of the Fourier transform used).

=== Locally compact abelian groups ===
{{Main|Pontryagin duality}}
The Fourier transform may be generalized to any [[locally compact abelian group]], i.e., an [[abelian group]] that is also a [[locally compact Hausdorff space]] such that the group operation is continuous. If {{mvar|G}} is a locally compact abelian group, it has a translation invariant measure {{mvar|μ}}, called [[Haar measure]]. For a locally compact abelian group {{mvar|G}}, the set of irreducible, i.e. one-dimensional, unitary representations are called its [[character group|characters]]. With its natural group structure and the topology of uniform convergence on compact sets (that is, the topology induced by the [[compact-open topology]] on the space of all continuous functions from <math>G</math> to the [[circle group]]), the set of characters {{mvar|Ĝ}} is itself a locally compact abelian group, called the ''Pontryagin dual'' of {{mvar|G}}. For a function {{mvar|f}} in {{math|''L''<sup>1</sup>(''G'')}}, its Fourier transform is defined by{{sfn|Katznelson|2004}}
<math display="block">\hat{f}(\xi) = \int_G \xi(x)f(x)\,d\mu\quad \text{for any }\xi \in \hat{G}.</math>

The Riemann–Lebesgue lemma holds in this case; {{math|''f̂''(''ξ'')}} is a function vanishing at infinity on {{mvar|Ĝ}}.

The Fourier transform on {{nobr|{{mvar|T}} {{=}} R/Z}} is an example; here {{mvar|T}} is a locally compact abelian group, and the Haar measure {{mvar|μ}} on {{mvar|T}} can be thought of as the Lebesgue measure on [0,1). Consider the representation of {{mvar|T}} on the complex plane {{mvar|C}} that is a 1-dimensional complex vector space. There are a group of representations (which are irreducible since {{mvar|C}} is 1-dim) <math>\{e_{k}: T \rightarrow GL_{1}(C) = C^{*} \mid k \in Z\}</math> where <math>e_{k}(x) = e^{i 2\pi kx}</math> for <math>x\in T</math>.

The character of such representation, that is the trace of <math>e_{k}(x)</math> for each <math>x\in T</math> and <math>k\in Z</math>, is <math>e^{i 2\pi kx}</math> itself. In the case of representation of finite group, the character table of the group {{mvar|G}} are rows of vectors such that each row is the character of one irreducible representation of {{mvar|G}}, and these vectors form an orthonormal basis of the space of class functions that map from {{mvar|G}} to {{mvar|C}} by Schur's lemma. Now the group {{mvar|T}} is no longer finite but still compact, and it preserves the orthonormality of character table. Each row of the table is the function <math>e_{k}(x)</math> of <math>x\in T,</math> and the inner product between two class functions (all functions being class functions since {{mvar|T}} is abelian) <math>f,g \in L^{2}(T, d\mu)</math> is defined as <math display="inline">\langle f, g \rangle = \frac{1}{|T|}\int_{[0,1)}f(y)\overline{g}(y)d\mu(y)</math> with the normalizing factor <math>|T|=1</math>. The sequence <math>\{e_{k}\mid k\in Z\}</math> is an orthonormal basis of the space of class functions <math>L^{2}(T,d\mu)</math>.

For any representation {{mvar|V}} of a finite group {{mvar|G}}, <math>\chi_{v}</math> can be expressed as the span <math display="inline">\sum_{i} \left\langle \chi_{v},\chi_{v_{i}} \right\rangle \chi_{v_{i}}</math> (<math>V_{i}</math> are the irreps of {{mvar|G}}), such that <math display="inline">\left\langle \chi_{v}, \chi_{v_{i}} \right\rangle = \frac{1}{|G|}\sum_{g\in G}\chi_{v}(g)\overline{\chi}_{v_{i}}(g)</math>. Similarly for <math>G = T</math> and <math>f\in L^{2}(T, d\mu)</math>, <math display="inline">f(x) = \sum_{k\in Z}\hat{f}(k)e_{k}</math>. The Pontriagin dual <math>\hat{T}</math> is <math>\{e_{k}\}(k\in Z)</math> and for <math>f \in L^{2}(T, d\mu)</math>, <math display="inline">\hat{f}(k) = \frac{1}{|T|}\int_{[0,1)}f(y)e^{-i 2\pi ky}dy</math> is its Fourier transform for <math>e_{k} \in \hat{T}</math>.

=== Gelfand transform ===
{{Main|Gelfand representation}}
The Fourier transform is also a special case of [[Gelfand transform]]. In this particular context, it is closely related to the Pontryagin duality map defined above.

Given an abelian [[locally compact space|locally compact]] [[Hausdorff space|Hausdorff]] [[topological group]] {{mvar|G}}, as before we consider space {{math|''L''<sup>1</sup>(''G'')}}, defined using a Haar measure. With convolution as multiplication, {{math|''L''<sup>1</sup>(''G'')}} is an abelian [[Banach algebra]]. It also has an [[Involution (mathematics)|involution]] * given by
<math display="block">f^*(g) = \overline{f\left(g^{-1}\right)}.</math>

Taking the completion with respect to the largest possibly {{math|''C''*}}-norm gives its enveloping {{math|''C''*}}-algebra, called the group {{math|''C''*}}-algebra {{math|''C''*(''G'')}} of {{mvar|G}}. (Any {{math|''C''*}}-norm on {{math|''L''<sup>1</sup>(''G'')}} is bounded by the {{math|''L''<sup>1</sup>}} norm, therefore their supremum exists.)

Given any abelian {{math|''C''*}}-algebra {{mvar|A}}, the Gelfand transform gives an isomorphism between {{mvar|A}} and {{math|''C''<sub>0</sub>(''A''^)}}, where {{math|''A''^}} is the multiplicative linear functionals, i.e. one-dimensional representations, on {{mvar|A}} with the weak-* topology. The map is simply given by
<math display="block">a \mapsto \bigl( \varphi \mapsto \varphi(a) \bigr)</math>
It turns out that the multiplicative linear functionals of {{math|''C''*(''G'')}}, after suitable identification, are exactly the characters of {{mvar|G}}, and the Gelfand transform, when restricted to the dense subset {{math|''L''<sup>1</sup>(''G'')}} is the Fourier–Pontryagin transform.

=== Compact non-abelian groups ===
The Fourier transform can also be defined for functions on a non-abelian group, provided that the group is [[compact space|compact]]. Removing the assumption that the underlying group is abelian, irreducible unitary representations need not always be one-dimensional. This means the Fourier transform on a non-abelian group takes values as Hilbert space operators.<ref>{{harvnb|Hewitt|Ross|1970|loc=Chapter 8}}</ref> The Fourier transform on compact groups is a major tool in [[representation theory]]<ref>{{harvnb|Knapp|2001}}</ref> and [[non-commutative harmonic analysis]].

Let {{mvar|G}} be a compact [[Hausdorff space|Hausdorff]] [[topological group]]. Let {{math|Σ}} denote the collection of all isomorphism classes of finite-dimensional irreducible [[unitary representation]]s, along with a definite choice of representation {{math|''U''{{isup|(''σ'')}}}} on the [[Hilbert space]] {{math|''H<sub>σ</sub>''}} of finite dimension {{math|''d<sub>σ</sub>''}} for each {{math|''σ'' ∈ Σ}}. If {{mvar|μ}} is a finite [[Borel measure]] on {{mvar|G}}, then the Fourier–Stieltjes transform of {{mvar|μ}} is the operator on {{math|''H<sub>σ</sub>''}} defined by
<math display="block">\left\langle \hat{\mu}\xi,\eta\right\rangle_{H_\sigma} = \int_G \left\langle \overline{U}^{(\sigma)}_g\xi,\eta\right\rangle\,d\mu(g)</math>
where {{math|{{overline|''U''}}{{isup|(''σ'')}}}} is the complex-conjugate representation of {{math|''U''<sup>(''σ'')</sup>}} acting on {{math|''H<sub>σ</sub>''}}. If {{mvar|μ}} is [[absolutely continuous]] with respect to the [[Haar measure|left-invariant probability measure]] {{mvar|λ}} on {{mvar|G}}, [[Radon–Nikodym theorem|represented]] as
<math display="block">d\mu = f \, d\lambda</math>
for some {{math|''f'' ∈ [[Lp space|''L''<sup>1</sup>(''λ'')]]}}, one identifies the Fourier transform of {{mvar|f}} with the Fourier–Stieltjes transform of {{mvar|μ}}.

The mapping
<math display="block">\mu\mapsto\hat{\mu}</math>
defines an isomorphism between the [[Banach space]] {{math|''M''(''G'')}} of finite Borel measures (see [[rca space]]) and a closed subspace of the Banach space {{math|'''C'''<sub>∞</sub>(Σ)}} consisting of all sequences {{math|''E'' {{=}} (''E<sub>σ</sub>'')}} indexed by {{math|Σ}} of (bounded) linear operators {{math|''E<sub>σ</sub>'' : ''H<sub>σ</sub>'' → ''H<sub>σ</sub>''}} for which the norm
<math display="block">\|E\| = \sup_{\sigma\in\Sigma}\left\|E_\sigma\right\|</math>
is finite. The "[[convolution theorem]]" asserts that, furthermore, this isomorphism of Banach spaces is in fact an isometric isomorphism of [[C*-algebra]]s into a subspace of {{math|'''C'''<sub>∞</sub>(Σ)}}. Multiplication on {{math|''M''(''G'')}} is given by [[convolution]] of measures and the involution * defined by
<math display="block">f^*(g) = \overline{f\left(g^{-1}\right)},</math>
and {{math|'''C'''<sub>∞</sub>(Σ)}} has a natural {{math|''C''*}}-algebra structure as Hilbert space operators.

The [[Peter–Weyl theorem]] holds, and a version of the Fourier inversion formula ([[Plancherel's theorem]]) follows: if {{math|''f'' ∈ ''L''<sup>2</sup>(''G'')}}, then
<math display="block">f(g) = \sum_{\sigma\in\Sigma} d_\sigma \operatorname{tr}\left(\hat{f}(\sigma)U^{(\sigma)}_g\right)</math>
where the summation is understood as convergent in the {{math|''L''<sup>2</sup>}} sense.

The generalization of the Fourier transform to the noncommutative situation has also in part contributed to the development of [[noncommutative geometry]].{{Citation needed|date=May 2009}} In this context, a categorical generalization of the Fourier transform to noncommutative groups is [[Tannaka–Krein duality]], which replaces the group of characters with the category of representations. However, this loses the connection with harmonic functions.