= Parseval identity for Fourier series
{c}
{title2=$T^{-1}\int_0^T|f|^2=\sum_{r\in\mathbb Z}|u_r|^2$}
For a <square-integrable function> of <period> $T$, use <Fourier coefficients> $u_r=T^{-1}\int_0^Tf(t)e^{-2\pi irt/T}\,dt$. <Fourier orthogonality> proves the displayed identity for a finite <trigonometric polynomial>. The <mean-square convergence> of the <Fourier series> then passes it to the limit. It includes the zero-frequency contribution $|u_0|^2$; excluding that contribution gives the variance rather than the full <mean-square signal power>.
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