Parseval's equation

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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 .


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 .


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.