A Generalized Sharp Whitney Theorem for Jets

Abstract

Suppose that, for each point $x$ in a given subset $E\subset {\mathbb{R}}^n$, we are given an $m$-jet $f(x)$ and a convex, symmetric set $\sigma (x)$ of $m$-jets at $x$. We ask whether there exist a function $F\in C^{m,\omega }({\mathbb{R}}^n)$ and a finite constant $M$, such that the $m$-jet of $F$ at $x$ belongs to $f(x)+M\sigma (x)$ for all $x\in E$. We give a necessary and sufficient condition for the existence of such $F,M$, provided each $\sigma (x)$ satisfies a condition that we call "Whitney $\omega$-convexity''.