Extension of functions with bounded finite differences

In 1934, Whitney posed the problem of how to recognize whether a function f defined on a closed subset X of R is the restriction of a function of class C. Whitney himself solved the one-dimensional case (i.e., for n = 1) in terms of finite differences [W1, W2, W3], giving the classical Whitney’s extension theorem. A geometrical solution for the case C(R) was given by G. Glaeser [G], who introduced a geometric object called the “iterated paratangent space”. Glaeser’s paper influenced all the later work on Whitney’s problem. A variant of Whitney’s problem replaces C(R) by C(R), the space of C functions whose m derivative have a given modulus of continuity ω. The problem for C(R) is well-understood due to the work of Brudnyi and Shvartsman [B], [BS1, BS2, BS3, BS4], [S1, S2, S3] and Fefferman [F1]. The correct notion of an iterated paratangent bundle, relevant for C(R), was introduced by Bierstone-Milman-Pawlucki [BMP1] who proved an extension theorem for subanalytic sets. In [BMP1], Bierstone-Milman-Pawlucki introduced a necessary geometric