Plancherel theorem

In mathematics, the Plancherel theorem (sometimes called the Parseval–Plancherel identity) is a result in harmonic analysis, proven by Michel Plancherel in 1910. It is a generalization of Parseval's theorem; often used in the fields of science and engineering, proving the unitarity of the Fourier transform.

The theorem 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 $f(x)$ is a function on the real line, and ${\widehat {f}}(\xi )$ is its frequency spectrum, then

$\int _{-\infty }^{\infty }|f(x)|^{2}\,dx=\int _{-\infty }^{\infty }|{\widehat {f}}(\xi )|^{2}\,d\xi$

A more precise formulation is that if a function is in both L^p spaces $L^{1}(\mathbb {R} )$ and $L^{2}(\mathbb {R} )$, then its Fourier transform is in $L^{2}(\mathbb {R} )$ and the Fourier transform is an isometry with respect to the L² norm. This implies that the Fourier transform restricted to $L^{1}(\mathbb {R} )\cap L^{2}(\mathbb {R} )$ has a unique extension to a linear isometric map $L^{2}(\mathbb {R} )\mapsto L^{2}(\mathbb {R} )$, sometimes called the Plancherel transform. This isometry is actually a unitary map. In effect, this makes it possible to speak of Fourier transforms of quadratically integrable functions.

Plancherel's theorem remains valid as stated on n-dimensional Euclidean space $\mathbb {R} ^{n}$. The theorem also holds more generally in locally compact abelian groups. 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.

Due to the polarization identity, one can also apply Plancherel's theorem to the $L^{2}(\mathbb {R} )$ inner product of two functions. That is, if $f(x)$ and $g(x)$ are two $L^{2}(\mathbb {R} )$ functions, and ${\mathcal {P}}$ denotes the Plancherel transform, then $$\int _{-\infty }^{\infty }f(x){\overline {g(x)}}\,dx=\int _{-\infty }^{\infty }({\mathcal {P}}f)(\xi ){\overline {({\mathcal {P}}g)(\xi )}}\,d\xi ,$$ and if $f(x)$ and $g(x)$ are furthermore $L^{1}(\mathbb {R} )$ functions, then $$({\mathcal {P}}f)(\xi )={\widehat {f}}(\xi )=\int _{-\infty }^{\infty }f(x)e^{-2\pi i\xi x}\,dx,$$ and $$({\mathcal {P}}g)(\xi )={\widehat {g}}(\xi )=\int _{-\infty }^{\infty }g(x)e^{-2\pi i\xi x}\,dx,$$ so

$\int _{-\infty }^{\infty }f(x){\overline {g(x)}}\,dx=\int _{-\infty }^{\infty }{\widehat {f}}(\xi ){\overline {{\widehat {g}}(\xi )}}\,d\xi .$

Proof

Assumption. $f\in L^{1}\cap L^{2}$, i.e. $\int |f(x)|dx,\int |f(x)|^{2}dx<\infty$

Step 1. The equality holds if f is differentiable and f' is bounded

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

By convolution theorem of Fourier transform, ${\mathcal {F}}(\phi )={\mathcal {F}}(f){\mathcal {F}}(f^{\star })=|{\mathcal {F}}(f)|^{2}=|{\hat {f}}|^{2}$, $\lim \limits _{L\rightarrow \infty }\int _{-L}^{L}|{\hat {f}}(\xi )|^{2}d\xi =\int |{\hat {f}}(\xi )|^{2}d\xi$ by Monotone Convergence Theorem, hence $\int |f(x)|^{2}dx=\int |{\hat {f}}(\xi )|^{2}d\xi$

Step 2. the General Case

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

See also

• Plancherel theorem for spherical functions

References

• Plancherel, Michel (1910), "Contribution à l'étude de la représentation d'une fonction arbitraire par des intégrales définies", Rendiconti del Circolo Matematico di Palermo, 30 (1): 289–335, doi:10.1007/BF03014877, S2CID 122509369.
• Dixmier, J. (1969), Les C*-algèbres et leurs Représentations, Gauthier Villars.
• Yosida, K. (1968), Functional Analysis, Springer Verlag.

External links

• "Plancherel theorem", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Plancherel's Theorem on Mathworld