 Projection residual for discrete total variation

For $C=D^{\ast }\{p:|p_i{|}_2\le \lambda \}$, discrete total variation denoising is the proximal operator of the support function of $C$. The Moreau decomposition gives the displayed projection residual. It is generally not itself a metric projection onto a closed convex set: such a projection is idempotent, but TV denoising shrinks a sufficiently large two-level jump again when reapplied.