Plancherel theorem: Difference between revisions

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== Formal definition ==

A more precise formulation is that if a function is in both [[Lp space|''L''^''p'' spaces]] <math>L^1(\mathbb{R})</math> and <math>L^2(\mathbb{R})</math>, then its Fourier transform is in <math>L^2(\mathbb{R})</math> and the Fourier transform is an [[isometry]] with respect to the ''L''² norm. This implies that the Fourier transform restricted to <math>L^1(\mathbb{R}) \cap L^2(\mathbb{R})</math> has a unique extension to a [[Linear isometry|linear isometric map]] <math>L^2(\mathbb{R}) \mapsto L^2(\mathbb{R})</math>, sometimes called the Plancherel transform. This isometry is actually a [[unitary operator|unitary]] map. In effect, this makes it possible to speak of Fourier transforms of [[quadratically integrable function]]s.

The [[Fourier transform]] of an [[Lp space|''L''^''1'']] function <math>f</math> on the [[real line]] <math>\mathbb R</math> is defined as the [[Lebesgue integral]]

<math display="block">\hat f(\xi) = \int_{\mathbb R} f(x)e^{-2\pi i x\xi}dx.</math>

If <math>f</math> belongs to both <math>L^1</math> and <math>L^2</math>, then the Plancherel theorem states that <math>\hat f</math> also belongs to <math>L^2</math>, and the Fourier transform is an [[isometry]] with respect to the ''L''² norm, which is to say that

<math display="block">\int_{-\infty}^\infty |f(x)|^2 \, dx = \int_{-\infty}^\infty |\widehat{f}(\xi)|^2 \, d\xi</math>

A more precise formulation is that if a function is in both [[Lp space|''L''^''p'' spaces]] <math>L^1(\mathbb{R})</math> and <math>L^2(\mathbb{R})</math>, then its Fourier transform is in <math>L^2(\mathbb{R})</math> and the Fourier transform is an [[isometry]] with respect to the ''L''² norm. This implies that the Fourier transform restricted to <math>L^1(\mathbb{R}) \cap L^2(\mathbb{R})</math> has a unique extension to a [[Linear isometry|linear isometric map]] <math>L^2(\mathbb{R}) \mapsto L^2(\mathbb{R})</math>, sometimes called the Plancherel transform. This isometry is actually a [[unitary operator|unitary]] map. In effect, this makes it possible to speak of Fourier transforms of [[quadratically integrable function]]s.

A proof of the theorem is available from ''Rudin (1987, Chapter 9)''. The basic idea is to prove it for [[Gaussian distribution]]s, and then use density. But a standard Gaussian is transformed to itself under the Fourier transformation, and the theorem is trivial in that case. Finally, the standard transformation properties of the Fourier transform then imply Plancherel for all Gaussians.

Plancherel's theorem remains valid as stated on ''n''-dimensional [[Euclidean space]] <math>\mathbb{R}^n</math>. The theorem also holds more generally in [[locally compact abelian group]]s. There is also a version of the Plancherel theorem which makes sense for non-commutative locally compact groups satisfying certain technical assumptions. This is the subject of [[non-commutative harmonic analysis]].

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==Locally Proofcompact 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 [[Pontryagin 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

'''Assumption.''' <math>f\in L^1 \cap L^2</math>, i.e. <math>\int |f(x)|dx, \int |f(x)|^2 dx<\infty</math>

<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>.

'''Step 1. The equality holds if ''f'' is differentiable and ''f'<nowiki/>'' is bounded'''

Let <math>f^{\star}(y)=\bar{f}(-y), \phi(x)=(f\ast f^{\star})(x)=\int f(x-y) f^{\star}(y)dy=\int f(x-y)\bar{f}(-y)dy=\int f(x+t)\bar{f}(t)dt</math>, then <math>|\frac{\partial [f(x+t)\bar{f}(t)]}{\partial x}|=|f'(x+t)\bar{f}(t)|\leq C|f(t)|</math>, and the [[Dominated convergence theorem|Dominated Convergence Theorem]] implies the interchangibility of differentiation and integration, thus <math>\phi '(x)=\int f'(x+t)\bar{f}(t)dt</math>, <math>\phi</math> is differentiable, hence by [[Fourier inversion theorem]], <math>\int|f(x)|^2 dx=\phi (0)=\lim\limits_{L\rightarrow \infty}\int_{-L}^{L} \mathcal{F}(\phi)(\xi)exp(2\pi i\cdot 0\cdot \xi)d\xi=\lim\limits_{L\rightarrow \infty}\int_{-L}^{L} \mathcal{F}(\phi)(\xi)d\xi</math>

The Plancherel theorem states that there is a Haar measure on <math>\widehat G</math>, the ''dual measure'' such that

By [[convolution theorem]] of Fourier transform, <math>\mathcal{F}(\phi)=\mathcal{F}(f)\mathcal{F}(f^{\star})=|\mathcal{F}(f)|^2=|\hat{f}|^2</math>, <math>\lim\limits_{L\rightarrow \infty}\int_{-L}^{L} |\hat{f}(\xi)|^2 d\xi=\int |\hat{f}(\xi)|^2 d\xi</math> by [[Monotone convergence theorem|Monotone Convergence Theorem]], hence <math>\int |f(x)|^2 dx=\int |\hat{f}(\xi)|^2 d\xi</math>

<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

'''Step 2. the General Case'''

<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

Let <math>\rho _\epsilon</math> be a family of [[Mollifier|mollifiers]], <math>f_\epsilon=f \ast \rho_\epsilon</math>, then for each ε, <math>f_\epsilon'=f\ast \rho_\epsilon'</math>, <math>|f_\epsilon'|=|f\ast \rho_\epsilon'|\leq \|f\|_{L^2}\|\rho_\epsilon'\|_{L^2}</math> by [[Hölder's inequality]], hence <math>f_\epsilon</math> is differentiable and has a bounded derivative. By '''Step 1''', <math>\int |f_\epsilon(x)|^2 dx=\int |\hat{f_\epsilon }(\xi)|^2 d\xi</math>. By the property of mollification, the left hand side converges to <math>\|f\|_{L^2}^2</math> as <math>\epsilon\rightarrow 0</math>, and by [[convolution theorem]], <math>|\hat{f_\epsilon }|=|\hat{f}||\hat{\rho_\epsilon }|\rightarrow |\hat{f}| \text{ as }\epsilon\rightarrow 0 </math>, hence by [[Fatou's lemma|Fatou' lemma]], we have <math>\int |\hat{f}|^2 d\xi \leq \liminf \limits_{\epsilon\rightarrow 0}\int |\hat{f_\epsilon}|^2 d\xi = \liminf \limits_{\epsilon\rightarrow 0} \int |f_\epsilon|^2 dx =\int |f|^2 dx </math>, thus <math>|\hat{f}|^2 </math> is integrable. Thus the right hand side converges to <math>\|\hat{f}\|_{L^2}^2</math> as <math>\epsilon\rightarrow 0</math> by [[Dominated convergence theorem|Dominated Convergence Theorem]]. Q.E.D.

<math display="block">\|f\|^2_G = \int_{\hat G} \|\hat f(\rho)\|_{HS}^2d\mu(\rho)</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==

* [[Carleson's theorem]]

*[[Plancherel theorem for spherical functions]]

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* {{citation|first=J.|last=Dixmier|authorlink=Jacques Dixmier|title=Les C*-algèbres et leurs Représentations|publisher=Gauthier Villars|year=1969}}.

* {{citation|first=K.|last=Yosida|authorlink=Kōsaku Yosida|title=Functional Analysis|publisher=Springer Verlag|year=1968}}.

* {{citation|first=Walter|last=Rudin|authorlink=Walter Rudin|year=1987|title=Real and Complex Analysis|publisher=McGraw-Hill Book Company|chapter=9 Fourier Transforms|edition=3}}.

==External links==