Plancherel定理

此式也可以用L²范数范数写为 ||f(x)||_(L²)=||f(ξ)||_(L²)

更精确的表述是，如果一个函数同时在 Lp 空间 L¹(R) 和 L²(R) 中，那么它的傅里叶变换也在 L²(R) 中，且傅里叶变换是关于 L² 范数的等距映射。这意味着， L¹(R)∩ L²(R) 上的傅里叶变换可以唯一地扩张为一个 （公式略） 的等距同构，后者有时称为普朗歇尔变换。这个等距同构实际上是一个幺正映射。这使得我们可以对全体平方可积函数进行傅里叶变换。

普朗歇尔定理在 n 维欧几里德空间 R ⁿ 上仍然有效。更一般地，该定理对局部紧阿贝尔群也成立。对于满足某些技术上的假定的非交换局部紧群，还有另一个版本的普朗歇尔定理。这是非交换调和分析的主题。

傅里叶变换的幺正性在科学和工程领域通常被称为帕塞瓦尔定理，该定理基于一个用于证明傅里叶级数幺正性的早期结果（但不那么具有一般性）。

借助极化恒等式，我们还可以将普朗歇尔定理用于计算 L²(R) 中两个函数的内积。也就是说，设 f(x) 和 g(x) 是两个 L²(R) 中的函数，而 P 表示普朗歇尔变换，则 ∫ _(-∞)^(∞)f(x)g(x) dx=∫ _(-∞)^(∞)(Pf)(ξ)(Pg)(ξ) dξ, 若 f(x) 和 g(x) 还是 L¹(R) 函数，那么还有 (Pf)(ξ)=f(ξ)=∫ _(-∞)^(∞)f(x)e^(-2π iξ x) dx, 和 (Pg)(ξ)=g(ξ)=∫ _(-∞)^(∞)g(x)e^(-2π iξ x) dx, 于是有

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 f(ξ) is its frequency spectrum, then

Formal definition

The Fourier transform of an L1 function f on the real line R is defined as the Lebesgue integral f(ξ)=∫ _(R)f(x)e^(-2π ixξ)dx.

f belongs to both L¹ and L², then the Plancherel theorem states that f also belongs to L², and the Fourier transform is an isometry with respect to the L2 norm, which is to say that ∫ _(-∞)^(∞)|f(x)|² dx=∫ _(-∞)^(∞)|f(ξ)|² dξ

This implies that the Fourier transform restricted to L¹(R)∩ L²(R) has a unique extension to a linear isometric map （公式略）, sometimes called the Plancherel transform. This isometry is actually a unitary map. In effect, this makes it possible to speak of Fourier transforms of L2 functions.

A proof of the theorem is available from Rudin (1987, Chapter 9). The basic idea is to prove it for Gaussian distributions, 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 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 L²(R) inner product of two functions. That is, if f(x) and g(x) are two L²(R) functions, and P denotes the Plancherel transform, then ∫ _(-∞)^(∞)f(x)g(x) dx=∫ _(-∞)^(∞)(Pf)(ξ)(Pg)(ξ) dξ,

and if f(x) and g(x) are furthermore L¹(R) functions, then (Pf)(ξ)=f(ξ)=∫ _(-∞)^(∞)f(x)e^(-2π iξ x) dx,

and (Pg)(ξ)=g(ξ)=∫ _(-∞)^(∞)g(x)e^(-2π iξ x) dx,

so

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 Pontryagin dual group G of characters on G. Given a Haar measure on G, the Fourier transform of a function in L¹(G) f(χ)=∫ _(G)χ (g)f(g) dg

for χ a character on G.

The Plancherel theorem states that there is a Haar measure on G, the dual measure such that ‖f‖_(G)²=‖f‖_(G)²

for all f∈ L¹∩ L² (and the Fourier transform is also in L²).

The theorem also holds in many non-abelian locally compact groups, except that the set of irreducible unitary representations G may not be a group. For example, when G is a finite group, 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=‖f‖_(G)²

‖ ‖f‖_(G)²=1/(|G|)∑ _(g∈ G)|f(g)|², ‖f‖_(G)²=∑ _(ρ ∈ G)(dim ρ)²|f(ρ)|².

More generally, when f is not a class function, the norm is ‖f‖_(G)²=∑ _(ρ ∈ G)dim ρ tr (f(ρ)^(*)f(ρ))

so the Plancherel measure weights each representation by its dimension.

In full generality, a Plancherel theorem is ‖f‖_(G)²=∫ _(G)‖f(ρ)‖_(HS)²dμ (ρ)

where the norm is the Hilbert–Schmidt norm of the operator f(ρ)=∫ _(G)f(g)ρ (g)^(*) dg

and the measure μ, if one exists, is called the Plancherel measure.