# Plancherel theorem

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For any square-summable function $f \in L _ {2} (- \infty , + \infty )$ the integral

$$\widehat{f} _ \omega ( x) = \frac{1}{\sqrt {2 \pi } } \int\limits _ {- \omega } ^ { {+ } \omega } f( y) e ^ {-} ixy dy$$

converges in $L _ {2}$ to some function $\widehat{f} \in L _ {2}$ as $\omega \rightarrow \infty$, i.e.

$$\tag{1 } \lim\limits _ {\omega \rightarrow \infty } \int\limits _ {- \infty } ^ { {+ } \infty } | \widehat{f} ( x) - \widehat{f} _ \omega ( x) | ^ {2} dx = 0.$$

Here the function $f$ itself is representable as the limit in $L _ {2}$ of the integrals

$$f _ \eta ( x) = \frac{1}{\sqrt {2 \pi } } \int\limits _ {- \eta } ^ { {+ } \eta } \widehat{f} ( y) e ^ {ixy} dy,\ \eta > 0,$$

as $\eta \rightarrow \infty$, i.e.

$$\lim\limits _ {\eta \rightarrow \infty } \int\limits _ {- \infty } ^ \infty | f( x) - f _ \eta ( x) | ^ {2} dx = 0.$$

Also, the following relation holds:

$$\int\limits _ {- \infty } ^ { {+ } \infty } | f( x) | ^ {2} dx = \int\limits _ {- \infty } ^ { {+ } \infty } | \widehat{f} ( \lambda ) | ^ {2} d \lambda$$

(the Parseval–Plancherel formula).

The function

$$\widehat{f} ( x) = \lim\limits _ {\omega \rightarrow \infty } \frac{1}{\sqrt {2 \pi } } \int\limits _ {- \omega } ^ { {+ } \omega } f( y) e ^ {-} iyx dy,$$

where the limit is understood in the sense of convergence in $L _ {2}$( as in (1)), is called the Fourier transform of $f$; it is sometimes denoted by the symbolic formula:

$$\tag{2 } \widehat{f} ( x) = \frac{1}{\sqrt {2 \pi } } \int\limits _ {- \infty } ^ { {+ } \infty } f( y) e ^ {-} ixy dy ,$$

where the integral in (2) must be understood in the sense of the principal value at $\infty$ in the metric of $L _ {2}$. One similarly interprets the equation

$$\tag{3 } f( x) = \frac{1}{\sqrt {2 \pi } } \int\limits _ {- \infty } ^ { {+ } \infty } \widetilde{f} ( y) e ^ {ixy} dy.$$

For functions $f \in L _ {2}$, the integrals (2) and (3) exist in the sense of the principal value for almost all $x$.

The functions $f$ and $\widehat{f}$ also satisfy the following equations for almost-all $x$:

$$\widehat{f} ( x) = \frac{d}{dx} \left \{ \frac{1}{\sqrt {2 \pi } } \int\limits _ {- \infty } ^ { {+ } \infty } f( y) \frac{e ^ {-} ixy - 1 }{-} iy dy \right \} ,$$

$$f( x) = \frac{d}{dx} \left \{ \frac{1}{\sqrt {2 \pi } } \int\limits _ {- \infty } ^ { {+ } \infty } \widehat{f} ( y) \frac{e ^ {ixy} - 1 }{iy} dy \right \} .$$

If Fourier transformation is denoted by ${\mathcal F}$ and if ${\mathcal F} ^ {-} 1$ denotes the inverse, then Plancherel's theorem can be rephrased as follows: ${\mathcal F}$ and ${\mathcal F} ^ {-} 1$ are mutually-inverse unitary operators on $L _ {2}$( cf. Unitary operator).

The theorem was established by M. Plancherel (1910).

#### References

 [1] A. Zygmund, "Trigonometric series" , 2 , Cambridge Univ. Press (1988) [2] E.C. Titchmarsh, "Introduction to the theory of Fourier integrals" , Oxford Univ. Press (1948) [3] S. Bochner, "Lectures on Fourier integrals" , Princeton Univ. Press (1959) (Translated from German)

The heart of Plancherel's theorem is the assertion that if $f \in L _ {1} ( \mathbf R ) \cap L _ {2} ( \mathbf R )$, then: a) $\widehat{f} \in L _ {2} ( \mathbf R )$, where $\widehat{f} ( y)$ is defined by (2) for $y \in \mathbf R$; b) $\| \widehat{f} \| _ {2} = \| f \| _ {2}$; and c) the set of all such $\widehat{f}$ is dense in $L _ {2} ( \mathbf R )$. Then one extends this mapping $f \rightarrow \widehat{f}$ to a unitary mapping ${\mathcal F}$ of $L _ {2} ( \mathbf R )$ onto itself which satisfies $( {\mathcal F} ^ {-} 1 f ) ( y) = ( {\mathcal F} f )(- y)$ for almost every $y \in \mathbf R$. There are generalizations of Plancherel's theorem in which $\mathbf R$ is replaced by $\mathbf R ^ {n}$ or by any locally compact Abelian group. Cf. also Harmonic analysis, abstract.