# 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

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

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.