# 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

is satisfied. Here denotes the inner product and the associated standard of .

If the orthonormal system is incomplete, Bessel's inequality still applies .

## 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 .

### Special case of the Fourier series

If the Fourier coefficients are the (real) Fourier series expansion of the -periodic real-valued function , that is

- ,

then the equation applies

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

- .

### 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

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