Research proof index

For an $L^2$ function on a circle of period $T$, the squared $L^2$ norm of its $N$th partial Fourier sum equals the sum of the squared magnitudes of its Fourier coefficients from $-N$ through $N$:

$\lVert S_Nf\rVert_2^2=\sum_{n=-N}^{N}|\widehat f(n)|^2.$