Parsevals

Parseval's theorem, named after the French mathematician Marc-Antoine Parseval, is a fundamental result in Fourier analysis. It relates the energy of a function to the energy of its frequency components and expresses the equivalence of norms between a function and its Fourier representation. The theorem has continuous and discrete variants and is widely used in signal processing, physics, and mathematics.

In the setting of Fourier series for a real-valued 2π-periodic function f, with coefficients a0, a_n, b_n defined by a0 = (1/π) ∫_{-π}^{π} f(x) dx, a_n = (1/π) ∫_{-π}^{π} f(x) cos(nx) dx, and b_n = (1/π) ∫_{-π}^{π} f(x) sin(nx) dx, Parseval's identity states that (1/π) ∫_{-π}^{π} f(x)^2 dx = (a0^2)/2 + ∑_{n=1}^{∞} (a_n^2 + b_n^2). Equivalently, the mean-square of f over a period equals the sum of the squared magnitudes of its Fourier coefficients.

For the Fourier transform on the real line, if F(ω) is the transform of f(t) given by F(ω) = ∫_{-∞}^{∞} f(t) e^{-i ω t} dt, then Parseval's theorem states ∫_{-∞}^{∞} |f(t)|^2 dt = (1/2π) ∫_{-∞}^{∞} |F(ω)|^2 dω, under common normalization. This expresses energy conservation between time and frequency representations.

In the language of Hilbert spaces, Parseval's identity generalizes to: if {e_n} is an orthonormal basis for a Hilbert space H and f ∈ H, then ∥f∥^2 = ∑_{n} |⟨f, e_n⟩|^2.

Historically, Parseval's theorem is attributed to Marc-Antoine Parseval des Toulars, whose early-19th-century work established the link between a function and its Fourier coefficients. It remains a standard tool in applied mathematics and engineering.