Plancherel theorem

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.

\begin{equation} \label{e:1} \lim\limits _ {\omega \rightarrow \infty } \int\limits _ {- \infty } ^ { {+ } \infty } | \widehat{f} ( x) - \widehat{f} _ \omega ( x) | ^ {2} dx = 0. \end{equation}

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 \eqref{e: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).

Comments

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.

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)
[a1]	W. Rudin, "Fourier analysis on groups" , Wiley (1962) Zbl 0107.09603
[a2]	A. Weil, "l'Intégration dans les groupes topologiques et ses applications" , Hermann (1940)
[a3]	L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian)
[a4]	E. Hewitt, K.A. Ross, "Abstract harmonic analysis" , 1–2 , Springer (1979)
[a5]	H. Reiter, "Classical harmonic analysis and locally compact groups" , Oxford Univ. Press (1968)