Whitney extension for finite-order Euclidean jets

The cases $A=\mathrm{\varnothing }$ and $A={\mathbb{R}}^d$ are immediate: use zero in the first case, and in the second define $H(a)=P_a(a)$; the compatibility condition is precisely the Taylor criterion for its derivatives $D^{\alpha }H(a)=D^{\alpha }{P}_a(a)$. Assume $A$ is nonempty and proper. Subdivide the standard integer-translated dyadic grid into closed cubes. Retain every dyadic cube $Q\subset {\mathbb{R}}^d\setminus A$ satisfying $4\sqrt{d}\mathrm{\ell }(Q)\le \mathrm{dist}(Q,A)$ which is maximal under dyadic parent inclusion. Every $z\mathrm{\notin }A$ belongs to the closure of one of these cubes: arbitrarily small cubes about $z$ satisfy the inequality, while sufficiently large ancestors fail it. Their interiors are disjoint. A retained cube has $4\sqrt{d}\mathrm{\ell }(Q)\le \mathrm{dist}(Q,A)\le 10\sqrt{d}\mathrm{\ell }(Q)$: the upper bound follows because its parent fails the retention test and every point of the parent lies within $2\sqrt{d}\mathrm{\ell }(Q)$ of $Q$. Consequently cubes whose fixed small enlargements meet have comparable side lengths, and only a dimension-dependent bounded number of those enlargements meet any one point. All cubes are obtained from a countable, explicitly ordered grid.

Fix one nonnegative $C^{\mathrm{\infty }}$ bump on the unit cube, equal to one on that cube and supported in its concentric $9\mathrm{/}8$ enlargement. Rescale it to each retained cube to get ${\varphi }_Q$. Set ${\psi }_Q={\varphi }_Q\mathrm{/}{\sum }_R{\varphi }_R$ on ${\mathbb{R}}^d\setminus A$. The denominator is at least one because the cubes cover the complement, and the sum is locally finite by step 1.1. Thus ${\sum }_Q{\psi }_Q=1$ and $\mathrm{\mid }{D}^{\alpha }{\psi }_Q\mathrm{\mid }\le C_{\alpha }\mathrm{\ell }(Q{)}^{-\mathrm{\mid }\alpha \mathrm{\mid }}$; the constants are locally uniform because intersecting enlarged cubes have comparable sizes and bounded overlap. For each cube choose the nearest point $a_Q\in A$ to its centre, breaking ties by successive minimum coordinates on the compact nearest-point set. This specifies $a_Q$ without arbitrary selections. Define $H(z)={\sum }_Q{\psi }_Q(z)P_{a_Q}(z)$ off $A$, and $H(a)=P_a(a)$ on $A$.

Fix $a\in A$ and let $z\to a$ through the complement. Write $t=\mathrm{dist}(z,A)$ and choose its lexicographically first nearest point $a_z$. If ${\psi }_Q(z)\mathrm{\ne }0$, step 1.1 gives $\mathrm{\ell }(Q)\asymp t$ and $\mathrm{\mid }{a}_Q-a_z\mathrm{\mid }=O(t)$. Taylor expansion of the polynomial $P_{a_Q}-P_{a_z}$ about $a_z$, combined with the assumed compatibility of every derivative order, gives $D^{\beta }(P_{a_Q}-P_{a_z})(z)=o(t^{r-\mathrm{\mid }\beta \mathrm{\mid }})(\mathrm{\mid }\beta \mathrm{\mid }\le r),$ locally uniformly as $z\to a$. Differentiating the partition sum and using ${\sum }_Q{\psi }_Q=1$ and the derivative bound in step 2.1 yields $D^{\alpha }H(z)-D^{\alpha }{P}_{a_z}(z)={\sum }_Q{\sum }_{\beta \le \alpha }\left(\genfrac{}{}{0px}{}{\alpha }{\beta }\right)D^{\alpha -\beta }{\psi }_Q(z)D^{\beta }(P_{a_Q}-P_{a_z})(z)=o(t^{r-\mathrm{\mid }\alpha \mathrm{\mid }})$ for $\mathrm{\mid }\alpha \mathrm{\mid }\le r$. There are only boundedly many terms at $z$, and each has the displayed order.

Compatibility again gives $D^{\alpha }{P}_{a_z}(z)-D^{\alpha }{P}_a(z)=o(\mathrm{\mid }z-a{\mathrm{\mid }}^{r-\mathrm{\mid }\alpha \mathrm{\mid }})$: $\mathrm{\mid }{a}_z-a\mathrm{\mid }\le 2\mathrm{\mid }z-a\mathrm{\mid }$, and the polynomial Taylor sum converts every jet discrepancy into that bound. Since $t\le \mathrm{\mid }z-a\mathrm{\mid }$, step 3.1 implies $D^{\alpha }H(z)-D^{\alpha }{P}_a(z)=o(\mathrm{\mid }z-a{\mathrm{\mid }}^{r-\mathrm{\mid }\alpha \mathrm{\mid }})$ as $z\to a$ through the complement. The same estimate at points of $A$ is exactly the given compatibility. Starting with $\mathrm{\mid }\alpha \mathrm{\mid }=0$, these estimates prove by induction that each derivative of order at most $r$ extends continuously across $A$ with value $D^{\alpha }{P}_a(a)$: for $\mathrm{\mid }\alpha \mathrm{\mid }<r$ subtract the linear part of $P_a$ and divide by $\mathrm{\mid }z-a\mathrm{\mid }$ to verify differentiability, and for $\mathrm{\mid }\alpha \mathrm{\mid }=r$ use the zero-order continuity estimate. Thus $H\in C^r$ and has the required jets. Every selection in the construction was by an ordered grid or a compact-set coordinate minimum.