= Sampling expansion by periodic Fourier projection
{title2=$f(t)=\sum_{n\in\mathbb Z}f(n)\frac{\sin(\pi(t-n))}{\pi(t-n)}$}
When the continuous <Fourier transform> $F$ of an <integrable function> vanishes outside $[-\pi,\pi]$, its periodic <Fourier coefficients> are the integer samples of the continuous representative of $f$, with the sign of the index reversed. Projecting $F$ in $L^2[-\pi,\pi]$ and then applying the <Fourier inversion theorem> gives the displayed reconstruction using the normalized <sinc function>, with each quotient defined as one when $t=n$. The <Cauchy-Schwarz inequality> controls the reconstruction error uniformly in $t$. <Bessel's inequality> bounds the vector of shifted <sinc functions>, giving absolute uniform tails when the sample sequence belongs to $\ell^2$.
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