Extension of 𝐶^{∞} functions defined in a half space

Abstract

The following question arises in connection with manifolds with boundary: given a function f defined and Co in a half space, and all of whose derivatives have continuous limits at the boundary, can f be extended to a Coo function in the whole space? More specifically, let xERn, tER, S+ = Rn X {t > 0 }, and D+ = {f: f in CW?(S+), f and all its derivatives have continuous limits as t-0 .+ }. D+ has the topology of uniform convergence of each derivative on compact subsets of the closure of S+ in Rn+l, and C??(Rn+l) has a corresponding topology.