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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 a generalized Pythagorean theorem for inner-product spaces.
Informally, the identity asserts that the sum of squares of the Fourier coefficients of a function is equal to the integral of the square of the function,
More formally, the result holds as stated provided f {\displaystyle f} is a square-integrable function or, more generally, in Lp space L 2 . {\displaystyle L^{2}.} 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 f ∈ L 2 , {\displaystyle f\in L^{2},}