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.

A proof of the theorem is available from Rudin (1987, Chapter 9).

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

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

Locally compact groups

There is also a Plancherel theorem for the Fourier transform in locally compact groups. In the case of an abelian group $G$, there is a Pontrjagin dual group ${\widehat {G}}$ of characters on $G$. Given a Haar measure on $G$, the Fourier transform of a function in $L^{1}(G)$ is $${\hat {f}}(\chi )=\int _{G}{\overline {\chi (g)}}f(g)\,dg$$ for $\chi$ a character on $G$.

The Plancherel theorem states that there is a Haar measure on ${\widehat {G}}$, the dual measure such that $$\|f\|_{G}^{2}=\|{\hat {f}}\|_{\widehat {G}}^{2}$$ for all $f\in L^{1}\cap L^{2}$ (and the Fourier transform is also in $L^{2}$).

The theorem also holds in many non-abelian locally compact groups, except that the set of irreducible unitary representations ${\widehat {G}}$ may not be a group. For example, when $G$ is a finite group, ${\widehat {G}}$ is the set of irreducible characters. From basic character theory, if $f$ is a class function, we have the Parseval formula $$\|f\|_{G}^{2}=\|{\hat {f}}\|_{\widehat {G}}^{2}$$ $$\|f\|_{G}^{2}={\frac {1}{|G|}}\sum _{g\in G}|f(g)|^{2},\quad \|{\hat {f}}\|_{\widehat {G}}^{2}=\sum _{\rho \in {\widehat {G}}}(\dim \rho )^{2}|{\hat {f}}(\rho )|^{2}.$$ More generally, when $f$ is not a class function, the norm is $$\|{\hat {f}}\|_{\widehat {G}}^{2}=\sum _{\rho \in {\widehat {G}}}\dim \rho \,\operatorname {tr} ({\hat {f}}(\rho )^{*}{\hat {f}}(\rho ))$$ so the Plancherel measure weights each representation by its dimension.

In full generality, a Plancherel theorem is $$\|f\|_{G}^{2}=\int _{\hat {G}}\|{\hat {f}}(\rho )\|_{HS}^{2}d\mu (g)$$ where the norm is the Hilbert-Schmidt norm of the operator $${\hat {f}}(\rho )=\int _{G}f(g)\rho (g)^{*}\,dg$$ and the measure $\mu$, if one exists, is called the Plancherel measure.

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.
• Rudin, Walter (1987), "9 Fourier Transforms", Real and Complex Analysis (3 ed.), McGraw-Hill Book Company.

External links

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