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 ${\mathcal C}^m$ - or ${\mathcal C}^{m,{\omega}}$ -function. His criteria involve uniformity of the ${\mathcal C}^m$ - or ${\mathcal C}^{m,{\omega}}$ -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 $k^\# = 2^{\dim {\mathcal P}}$ in both cases, where ${\mathcal P}$ denotes the space of polynomials of degree at most $m$ in $n$ variables. We also show that the geometric ${\mathcal C}^m$ “paratangent bundle” of $E$ (see [BMP2]) can be defined using limits of distributions supported on $2^{\dim {\mathcal P} - 1}$ points