Cm-norms on finite sets and Cm extension criteria

Abstract

C. Fefferman [F1], [F2] has recently given criteria for a function defined on a compact set $E\subset {\mathbb{R}}^n$ to extend to a $C^m$- or $C^{m,\omega }$-function. His criteria involve uniformity of the $C^m$- or $C^{m,\omega }$-norms for extension from finite subsets $S\subset E$ of cardinality at most a large natural number $k^{\#}$ depending only on $m$ and $n$. We prove that one can take $k^{\#}=2^{\dim P}$ in both cases, where $P$ denotes the space of polynomials of degree at most $m$ in $n$ variables. We also show that the geometric $C^m$ “paratangent bundle” of $E$ (see [BMP2]) can be defined using limits of distributions supported on $2^{\dim P-1}$ points