Parseval's equality
An identity equating a function's mean-square average to the sum of the squares of its Fourier coefficients.
Parseval's equality states \dfrac1L\displaystyle\int_{-L}^Lf(x)^2\,dx=\dfrac{a_0^2}2+\sum_{n=1}^\infty(a_n^2+b_n^2) for a function f with full Fourier coefficients a_n,b_n -- no information about f's size is lost in passing to its Fourier coefficients.
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