Parseval's identity - Alchetron, The Free Social Encyclopedia

In mathematical analysis, Parseval's identity, named after Marc-Antoine Parseval, is a fundamental result on the summability of the Fourier series of a function. Geometrically, it is the Pythagorean theorem for inner-product spaces.

Informally, the identity asserts that the sum of the squares of the Fourier coefficients of a function is equal to the integral of the square of the function,

∑ n = − ∞ ∞ | c n | 2 = 1 2 π ∫ − π π | f ( x ) | 2 d x ,

where the Fourier coefficients c_n of ƒ are given by

c n = 1 2 π ∫ − π π f ( x ) e − i n x d x .

More formally, the result holds as stated provided ƒ is square-integrable or, more generally, in L²[−π,π]. A similar result is the Plancherel theorem, which asserts that the integral of the square of the Fourier transform of a function is equal to the integral of the square of the function itself. In one-dimension, for ƒ ∈ L²(R),

∫ − ∞ ∞ | f ^ ( ξ ) | 2 d ξ = ∫ − ∞ ∞ | f ( x ) | 2 d x .

Generalization of the Pythagorean theorem

The identity is related to the Pythagorean theorem in the more general setting of a separable Hilbert space as follows. Suppose that H is a Hilbert space with inner product 〈•,•〉. Let (e_n) be an orthonormal basis of H; i.e., the linear span of the e_n is dense in H, and the e_n are mutually orthonormal:

⟨ e m , e n ⟩ = { 1 if m = n 0 if m ≠ n .

Then Parseval's identity asserts that for every x ∈ H,

∑ n | ⟨ x , e n ⟩ | 2 = ∥ x ∥ 2 .

This is directly analogous to the Pythagorean theorem, which asserts that the sum of the squares of the components of a vector in an orthonormal basis is equal to the squared length of the vector. One can recover the Fourier series version of Parseval's identity by letting H be the Hilbert space L²[−π,π], and setting e_n = e^−inx for n ∈ Z.

More generally, Parseval's identity holds in any inner-product space, not just separable Hilbert spaces. Thus suppose that H is an inner-product space. Let B be an orthonormal basis of H; i.e., an orthonormal set which is total in the sense that the linear span of B is dense in H. Then

∥ x ∥ 2 = ⟨ x , x ⟩ = ∑ v ∈ B | ⟨ x , v ⟩ | 2 .

The assumption that B is total is necessary for the validity of the identity. If B is not total, then the equality in Parseval's identity must be replaced by ≥, yielding Bessel's inequality. This general form of Parseval's identity can be proved using the Riesz–Fischer theorem.

References

Parseval's identity Wikipedia

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