Plancherel theorem

Statement

In mathematics, the Plancherel theorem (sometimes called the Parseval–Plancherel identity) 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 $f(x)$ is an L¹ and L² function on the real line,i.e.

$\int _{-\infty }^{\infty }|f(x)|dx<\infty$, $\int _{-\infty }^{\infty }|f(x)|^{2}dx<\infty$ and ${\widehat {f}}(\xi )$ is its frequency spectrum, i.e. ${\hat {f}}(\xi )=\int _{-\infty }^{\infty }f(x)\exp(-2\pi i\xi x)dx$, 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 map is an isometry with respect to the L² norm. This implies that the Fourier transform map 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.

Proof

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 (MCT), 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, $f_{\epsilon }'=f\ast \rho _{\epsilon }'$, $|f_{\epsilon }'|=|f\ast \rho _{\epsilon }'|\leq \|f\|_{L^{2}}\|\rho _{\epsilon }'\|_{L^{2}}$, 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 $\|f\|_{L^{2}}$ 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 $\liminf \limits _{\epsilon \rightarrow 0}|{\hat {f_{\epsilon }}}|=|{\hat {f}}|$, by MCT, we have $\int |{\hat {f}}|^{2}d\xi =\int \liminf \limits _{\epsilon \rightarrow 0}|{\hat {f_{\epsilon }}}|^{2}d\xi =\liminf \limits _{\epsilon \rightarrow 0}\int |{\hat {f_{\epsilon }}}|^{2}d\xi =\liminf \limits _{\epsilon \rightarrow 0}\int |f_{\epsilon }|^{2}dx=\int |f|^{2}dx$

Extensions

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.

The 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.

Corollary

Due to the polarization identity, one can also apply Plancherel's theorem to the $L^{2}(\mathbb {R} )$ inner product of two functions. That $f(x)$ 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

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

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