Parseval's identity

Parseval's equation (after Marc- Antoine Parseval ), also known as seclusion relation, from the area of functional analysis is the general form of the Pythagorean theorem for inner product spaces. At the same time it is important for Orthogonalzerlegungen in these spaces, in particular for the generalized Fourier transform.

Formulation

There were a pre-Hilbert space and orthonormal system given - ie all elements of are orthogonal to each other and also have the norm. is exactly then a complete orthonormal system ( orthonormal basis ) of if for all the Parseval equation

Is satisfied. Here, denotes the inner product and the associated norm of.

Is an incomplete orthonormal system, after all, is still the Bessel inequality.

Applications

The equation has the physical evidence that the energy of a signal in the pulse chamber looking identical to the energy of the signal in the spatial domain is.

Another formulation of the equation is the statement that the L2 - norm of a function equal to - or is norm of the coefficients of the Fourier series of this function. The generalization of Parseval's equation to the Fourier transform is the set of Plancherel.

Special case of the Fourier series

If the Fourier coefficients of the (real) is the Fourier series development of the periodic function, that is

Then the following equation holds

This identity is a special case of the general Parseval equation described above, if the orthonormal system, the trigonometric functions, n = 1,2, ..., takes the inner product

Set of Plancherel

The Parseval equation for the Fourier series corresponding to an identity of the Fourier transform, which is commonly referred to as a set of Plancherel:

If the Fourier transform of is, does the equation

The Fourier transform is thus an isometry in Hilbert space L2. This equation is very similar to the Parseval but it does not follow from this, since the Fourier transformation is not assigned orthogonal.