Fr.: théorème de Parseval
A theorem relating the → Fourier coefficients to the function that they describe. It states that: (1/L) ∫ |f(x)|2dx (integrated from x0 to x0 + L) = (a0/2)2 + (1/2) Σ (ar2 + br2) (summed from r = 1 to ∞). In other words, the sum of the moduli squared of the complex Fourier coefficients is equal to the average value of |f(x)|2 over one period.
Named after Marc-Antoine Parseval (1755-1836), French mathematician; → theorem.