Editing Fourier transform (section)

=== 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.