Plancherel theorem: Difference between revisions

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{{Short description|Theorem in harmonic analysis}}

In [[mathematics]], the '''Plancherel theorem''' (sometimes called the [[Marc-Antoine Parseval|Parseval]]–Plancherel identity<ref>{{cite book |author1=Cohen-Tannoudji, Claude |author2=Dupont-Roc, Jacques |author3=Grynberg, Gilbert |title=Photons and Atoms : Introduction to Quantum Electrodynamics |year=1997 |url=https://archive.org/details/photonsatomsintr00cohe_398 |url-access=limited |publisher=Wiley |isbn=0-471-18433-0 |page=[https://archive.org/details/photonsatomsintr00cohe_398/page/n39 11]}}</ref>) is a result in [[harmonic analysis]], proven by [[Michel Plancherel]] in 1910. It states that the integral of a function's [[squared modulus]] is equal to the integral of the squared modulus of its [[frequency spectrum]]. That is, if <math>f(x) </math> is ana '''L¹'''function on the real line, and '''L<supmath>2\widehat{f}(\xi)</supmath>''' functionis onits thefrequency realspectrum, line,i.e.then

{{Equation box 1

== Statement ==

In [[mathematics]], the '''Plancherel theorem''' (sometimes called the [[Marc-Antoine Parseval|Parseval]]–Plancherel identity<ref>{{cite book |author1=Cohen-Tannoudji, Claude |author2=Dupont-Roc, Jacques |author3=Grynberg, Gilbert |title=Photons and Atoms : Introduction to Quantum Electrodynamics |year=1997 |url=https://archive.org/details/photonsatomsintr00cohe_398 |url-access=limited |publisher=Wiley |isbn=0-471-18433-0 |page=[https://archive.org/details/photonsatomsintr00cohe_398/page/n39 11]}}</ref>) is a result in [[harmonic analysis]], proven by [[Michel Plancherel]] in 1910. It states that the integral of a function's [[squared modulus]] is equal to the integral of the squared modulus of its [[frequency spectrum]]. That is, if <math>f(x) </math> is an '''L¹''' and '''L²''' function on the real line,i.e.

<math>\int_{-\infty}^{\infty}|f(x)|dx<\infty</math>, <math>\int_{-\infty}^{\infty}|f(x)|^2dx<\infty</math> and <math>\widehat{f}(\xi)</math> is its frequency spectrum, i.e. <math>\hat{f}(\xi)=\int_{-\infty}^{\infty}f(x)\exp(-2\pi i\xi x)dx</math>, then{{Equation box 1

|indent =

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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 map is an isometry with respect to the ''L''² norm. This implies that the Fourier transform map 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.

== Proof ==

'''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>

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>

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

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.

== Extensions ==

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]].

The [[unitary transformation|unitarity]] of the [[Fourier transform]] is often called [[Parseval's theorem]] in science and engineering fields, based on an earlier (but less general) result that was used to prove the unitarity of the [[Fourier series]].

Due to the [[polarization identity]], one can also apply Plancherel's theorem to the [[Lp space|<math>L^2(\mathbb{R})</math>]] [[inner product]] of two functions. That is, if <math>f(x)</math> and <math>g(x)</math> are furthermoretwo <math>L^12(\mathbb{R})</math> functions, thenand <math> \mathcal P</math> denotes the Plancherel transform, then

== Corollary ==

<math display="block">\int_{-\infty}^\infty f(x)\overline{g(x)} \, dx = \int_{-\infty}^\infty (\mathcal P f)(\xi) \overline{(\mathcal P g)(\xi)} \, d\xi,</math>

Due to the [[polarization identity]], one can also apply Plancherel's theorem to the [[Lp space|<math>L^2(\mathbb{R})</math>]] [[inner product]] of two functions. That <math>f(x)</math> and <math>g(x)</math> are furthermore <math>L^1(\mathbb{R})</math> functions, then

and if <math>f(x)</math> and <math>g(x)</math> are furthermore <math>L^1(\mathbb{R})</math> functions, then

<math display="block"> (\mathcal P f)(\xi) = \widehat{f}(\xi) = \int_{-\infty}^\infty f(x) e^{-2\pi i \xi x} \, dx ,</math>

and