= Finite-dimensional Hilbert sampling reconstruction
{title2=$\widetilde f=(P_S|_T)^{-1}P_Sf$}
For equal finite-dimensional <closed subspaces of a Hilbert space> $T,S$, positivity of $\cos\theta_{T,S}$ makes $P_S|_T$ invertible. The displayed reconstruction is the unique element of $T$ whose <orthogonal projection> onto $S$ matches that of $f$. It is the <oblique projection> onto $T$ along $S^\perp$. Its <operator norm> is the value of the <secant function> at the <directed subspace angle>, and its error is at most that <secant function> value times the best <orthogonal projection> error. The lower error bound follows from the <Pythagorean identity>. Zero-dimensional cases are handled directly.
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