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.

Authors

Edward Bierstone

Department of Mathematics
University of Toronto
Toronto, Ontario
Canada M5S 2E4

Pierre D. Milman

Department of Mathematics
University of Toronto
Toronto, Ontario
Canada M5S 2E4

Wiesław Pawłucki

Institute of Mathematics
Jagiellonian University
30-348 Kraków
Poland