Periodic multilinear interpolation is convex and L-infinity contractive

Each factor ${\theta }_j$ and $1-{\theta }_j$ is nonnegative, so every $w_{\mathit{ϵ}}(x)\ge 0$. Summing over all binary vectors factorizes as ${\sum }_{\mathit{ϵ}}{w}_{\mathit{ϵ}}(x)={\prod }_{j=1}^d((1-{\theta }_j)+{\theta }_j)=1$. Thus ${\mathcal{I}}_{h}V(x)$ is a convex combination of its corner data, even when wrapping makes two corner indices coincide.

If $V_{\mathit{i}}\ge 0$ at every node, each summand in the interpolant is nonnegative, so its value is nonnegative; the same convex-combination identity bounds it by the minimum and maximum corner data.

At each $x$, [F1] and the triangle inequality give $\mathrm{\mid }{\mathcal{I}}_{h}V(x)-{\mathcal{I}}_{h}W(x)\mathrm{\mid }\le {\sum }_{\mathit{ϵ}}{w}_{\mathit{ϵ}}(x)\mathrm{\mid }{V}_{\mathit{i}(\mathit{ϵ})}-W_{\mathit{i}(\mathit{ϵ})}\mathrm{\mid }\le \mathrm{\parallel }V-W{\mathrm{\parallel }}_{{\mathrm{\ell }}^{\mathrm{\infty }}}{\sum }_{\mathit{ϵ}}{w}_{\mathit{ϵ}}(x)=\mathrm{\parallel }V-W{\mathrm{\parallel }}_{{\mathrm{\ell }}^{\mathrm{\infty }}}$. Taking the supremum over $x$ proves the stated contraction; no estimate in this argument uses the displacement size.