= Adjoint of a discrete forward gradient
{title2=$D^*=-\operatorname{div}$}
On an $N$-pixel line with last forward difference zero, $D^*p$ has first entry $-p_1$, interior entries $p_{i-1}-p_i$, and last entry $p_{N-1}$, with $D^*=0$ for $N=1$. On a square grid add these expressions along rows and columns. Unused last-edge components contribute nothing. The identity $\langle Du,p\rangle=\langle u,D^*p\rangle$ fixes every boundary sign. On the unscaled square grid, $\|D\|^2=8\cos^2(\pi/(2N))\le8$ for $N>1$.
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