Higher-order tangents and Fefferman’s paper on Whitney’s extension problem

Abstract

Whitney [Wh2] proved that a function defined on a closed subset of $\mathbb{R}$ is the restriction of a $\mathcal{C}^m$ function if the limiting values of all $m^{\rm th}$ divided differences form a continuous function. We show that Fefferman’s solution of Whitney’s problem for $\mathbb{R}^n$ [F, Th. 1] is equivalent to a variant of our conjecture in [BMP2] giving a criterion for $\mathcal{C}^m$ extension in terms of iterated limits of finite differences.