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==Locally compact groups==
There is also a Plancherel theorem for the Fourier transform in [[locally compact group]]s. In the case of an [[abelian group]] <math>G</math>, there is a [[Pontrjagin dual]] group <math>\widehat G</math> of characters on <math>G</math>. Given a [[Haar measure]] on <math>G</math>, the Fourier transform of a function in <math>L^1(G)</math> is
<math display="block">\hat f(\chi) = \int_G \overline{\chi(g)}f(g)\,dg</math>
for <math>\chi</math> a character on <math>G</math>.
The Plancherel theorem states that there is a Haar measure on <math>\widehat G</math>, the ''dual measure'' such that
<math display="block">\|f\|_G^2 = \|\hat f\|_{\widehat G}^2</math>
for all <math>f\in L^1\cap L^2</math> (and the Fourier transform is also in <math>L^2</math>).
The theorem also holds in many non-abelian locally compact groups, except that the set of irreducible unitary representations <math>\widehat G</math> may not be a group. For example, when <math>G</math> is a finite group, <math>\widehat G</math> is the set of irreducible characters. From basic [[character theory]], if <math>f</math> is a [[class function]], we have the Parseval formula
<math display="block">\|f\|_G^2 = \|\hat f\|_{\widehat G}^2</math>
<math display="block">\|f\|_G^2 = \frac{1}{|G|}\sum_{g\in G} |f(g)|^2, \quad \|\hat f\|_{\widehat G}^2 = \sum_{\rho\in\widehat G} (\dim\rho)^2|\hat f(\rho)|^2.</math>
More generally, when <math>f</math> is not a class function, the norm is
<math display="block">\|\hat f\|_{\widehat G}^2 = \sum_{\rho\in\widehat G} \dim\rho\,\operatorname{tr}(\hat f(\rho)^*\hat f(\rho))</math>
so the [[Plancherel measure]] weights each representation by its dimension.
In full generality, a Plancherel theorem is
<math display="block">\|f\|^2_G = \int_{\hat G} \|\hat f(\rho)\|_{HS}^2d\mu(g)</math>
where the norm is the [[Hilbert-Schmidt norm]] of the operator
<math display="block">\hat f(\rho) = \int_G f(g)\rho(g)^*\,dg</math>
and the measure <math>\mu</math>, if one exists, is called the Plancherel measure.
==See also==
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