Parseval's equation

The Parseval equation (after Marc-Antoine Parseval ), also known as the closure relation , from the field of functional analysis is the general form of the Pythagorean theorem for interior product spaces . At the same time, it is important for orthogonal decompositions in these spaces, especially for the generalized Fourier transform .

formulation

Let there be a Prähilbert space and an orthonormal system - i.e. H. all elements of are mutually orthogonal and also have the norm . is a complete orthonormal system ( orthonormal basis ) of if and only if Parseval's equation for all ${\displaystyle V}$ ${\displaystyle S\subset V}$ ${\displaystyle S}$ ${\displaystyle 1}$ ${\displaystyle S}$ ${\displaystyle V}$ ${\displaystyle v\in V}$

${\displaystyle \|v\|^{2}=\langle v,v\rangle =\sum _{s\in S}|\langle v,s\rangle |^{2}}$

is satisfied. Here denotes the inner product and the associated standard of . ${\displaystyle \langle \cdot ,\cdot \rangle }$ ${\displaystyle \|\cdot \|}$ ${\displaystyle V}$

If the orthonormal system is incomplete, Bessel's inequality still applies . ${\displaystyle S}$

Applications

The equation has the physical statement that the energy of a signal in momentum space is identical to the energy of the signal in space .

Another formulation of the equation is the statement that the L ² norm of a function is equal to the or norm of the coefficients of the Fourier series of this function. The generalization of Parseval's equation to the Fourier transform is Plancherel's theorem . ${\displaystyle \ell ^{2}}$ ${\displaystyle L^{2}(\mathbb {Z} )}$

Special case of the Fourier series

→ Main article : Parseval's theorem

If the Fourier coefficients are the (real) Fourier series expansion of the -periodic real-valued function , that is ${\displaystyle a_{k},b_{k}}$ ${\displaystyle 2\pi }$ ${\displaystyle f}$

${\displaystyle f(x)\sim {\frac {a_{0}}{2}}+\sum _{k=1}^{\infty }\left(a_{k}\,\cos(kx)+b_{k}\,\sin(kx)\right)}$,

then the equation applies

${\displaystyle {\frac {1}{\pi }}\int _{-\pi }^{\pi }f(x)^{2}\,\mathrm {d} x={\frac {a_{0}^{2}}{2}}+\sum _{k=1}^{\infty }(a_{k}^{2}+b_{k}^{2}).}$

This identity is a special case of the general Parseval equation described above, if the trigonometric functions as an orthonormal system , takes, with the dot product ${\displaystyle {\tfrac {1}{\sqrt {2}}},\,\cos(nx),\,\sin(nx)}$ ${\displaystyle n=1,2,\dotsc }$

${\displaystyle \langle f,\,g\rangle ={\frac {1}{\pi }}\int _{-\pi }^{\pi }f(x)g(x)\;dx}$.

Plancherel's theorem

→ Main article : Plancherel's theorem

The Parseval equation for the Fourier series corresponds to an identity of the Fourier transform , which is commonly referred to as Plancherel's theorem:

If the Fourier transform is of, then the equation holds ${\displaystyle {\hat {f}}(\xi )}$ ${\displaystyle f(x)}$

${\displaystyle \int _{-\infty }^{\infty }|f(x)|^{2}\mathrm {d} x=\int _{-\infty }^{\infty }|{\hat {f}}(\xi )|^{2}\mathrm {d} \xi }$

The Fourier transformation is therefore an isometry in the Hilbert space L ² . This equation is very similar to Parseval's, but it does not follow from it, since the Fourier transform is not associated with an orthogonal system.

literature

• Dirk Werner : functional analysis, Springer-Verlag, Berlin, ISBN 978-3-540-72533-6