In mathematical analysis, Parseval's identity is a fundamental result on the summability of the Fourier series of a function. Geometrically, it is the Pythagorean theorem for inner-product spaces.
Informally, the identity asserts that the sum of the squares of the Fourier coefficients of a function is equal to the integral of the square of the function,
where the Fourier coefficients cn of ƒ are given by
More formally, the result holds as stated provided ƒ is square-integrable or, more generally, in L2. 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 ƒ ∈ L2(R),
Read more about Parseval's Identity: Generalization of The Pythagorean Theorem
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