Local Taylor approximation on a Sobolev grid

For each $\mathrm{\mid }\alpha \mathrm{\mid }<n$, $D^{\alpha }f\in W^{1,\mathrm{\infty }}(Q)$ by [F1], since its first weak derivatives are among the derivatives of $f$ of order at most $n$. By [F1], choose their Lipschitz representatives; their Lipschitz constants are at most $\sqrt{d}$, because each coordinate weak derivative has essential supremum at most $1$. Since $Q$ has finite measure and every weak derivative of $f$ through order $n$ is essentially bounded, the same weak-derivative identities give $f\in W^{n,1}(Q)$, hence $f\in W^{n-1,1}(Q)$. By [A1] and [F2], Countable Choice holds, so the interior approximation theorem applies with $k=n-1$ and $p=1$, while [F3] applies with $k=n$ and $p=1$. For every compact segment in $Q$, convolution of each chosen Lipschitz representative differs from it uniformly by at most $\sqrt{d}$ times the mollifier radius, so the mollified derivatives through order $n-1$ converge uniformly there.

For an interior grid center $a_m$ and interior $x$, choose the mollifier radius smaller than the distance from their segment to $\mathrm{\partial }Q$. By [F3] and step 1.1, $D^{\alpha }{f}_{\delta }={\rho }_{\delta }\ast D^{\alpha }f$ on that segment; nonnegativity and unit mass of ${\rho }_{\delta }$ imply $\mathrm{\mid }{D}^{\alpha }{f}_{\delta }\mathrm{\mid }\le \mathrm{\parallel }{D}^{\alpha }f{\mathrm{\parallel }}_{\mathrm{\infty }}\le 1$ for $\mathrm{\mid }\alpha \mathrm{\mid }=n$. The one-variable Taylor formula applied to $t\mapsto f_{\delta }(a_m+t(x-a_m))$ therefore gives remainder at most $\frac{1}{n!}{\sup }_t\mathrm{\mid }\frac{d^n}{d{t}^n}{f}_{\delta }(a_m+t(x-a_m))\mathrm{\mid }\le C(d,n)\mathrm{\parallel }x-a_m{\mathrm{\parallel }}_{\mathrm{\infty }}^n$, since ${\sum }_i\mathrm{\mid }{x}_i-a_{m,i}\mathrm{\mid }\le d\mathrm{\parallel }x-a_m{\mathrm{\parallel }}_{\mathrm{\infty }}$. Passing to the uniform limits from step 1.1 proves the estimate for interior points. Approximate any boundary grid center and any boundary input by interior points; the continuous extensions specified in the statement and the finite Taylor polynomial make the same estimate hold on the closed cube.

The one-dimensional hats sum to $1$ on each grid interval, so their tensor products satisfy ${\sum }_m{\varphi }_m=1$ on the cube; at most $2^d$ are nonzero at any point, and each nonzero term has $\mathrm{\parallel }x-a_m{\mathrm{\parallel }}_{\mathrm{\infty }}\le 1\mathrm{/}N$. Therefore $\mathrm{\mid }f(x)-g_N(x)\mathrm{\mid }\le {\sum }_{m:{\varphi }_m(x)>0}{\varphi }_m(x)C(d,n)N^{-n}=C(d,n)N^{-n}$. Each extended Lipschitz representative and each hat is continuous on the closed cube, so the finite sum $g_N$ is continuous there.