an equation of the form
where a0 an, and bn are the Fourier coefficients for f(x). It was established in 1805 by the French mathematician M. Parseval by assuming that trigonometric series could be termwise integrated. In 1896, A. M. Liapunov showed that the equality is valid if the function is bounded on the interval (—π, π) and if the integral ∫π-πf(x) dx exists. It was later proved that the Parseval equality holds for any function whose square is integrable. V. A. Steklov demonstrated that the Parseval equality is valid for series in other orthogonal systems of functions.