PARSEVAL

Parseval refers to the work of Marc-Antoine Parseval des Besages, a French mathematician known for his contributions to the theory of trigonometric series. In mathematics, the name is associated with Parseval's theorem (or Parseval's identity), a fundamental result in harmonic analysis that expresses how energy or L2 norm is preserved under Fourier expansions and related transforms. The theorem is a statement about square-integrable functions and, more generally, about inner products in Hilbert spaces.

In the context of Fourier series, if a function f is 2π-periodic and square-integrable on a interval such as [-π, π], its Fourier series has coefficients a0, a1, b1, a2, b2, and so on. Parseval's identity for Fourier series states that (a0^2)/2 + Σ_{n=1}^∞ (a_n^2 + b_n^2) = (1/π) ∫_{-π}^{π} f(x)^2 dx. This equality expresses the equality of the total energy contained in the time (or spatial) domain and in the frequency domain, as represented by the Fourier coefficients.

For Fourier transforms, Parseval's theorem generalizes the energy conservation to functions in L2(R) and their transforms. With a given normalization of the transform, the L2 norm of a function equals (up to a constant depending on convention) the L2 norm of its transform: ∫ |f(x)|^2 dx = C ∫ |F(ξ)|^2 dξ. This result underpins many practical uses in signal processing, physics, and engineering, where energy or power can be analyzed equally in time and frequency domains.

Beyond Fourier analysis, Parseval's identity is a special case of the preservation of inner products under orthonormal expansions, and it extends to finite-dimensional settings as energy preservation under orthogonal or unitary transforms.