Smooth extension theorems for one variable maps

Abstract

We characterize the real valued functions f defined on perfect subsets P of R which admit n-times differentiable extensions F:R→R. In this characterization no continuity of F(n) is imposed. In particular, it generalizes Jarník's Extension Theorem, according to which f admits differentiable extension F:R→R if, and only if, f is differentiable. The new characterization is also closely related to the Whitney's Extension Theorem, which characterizes partial maps f admitting n-times differentiable extensions F:R→R with continuous nth derivative F(n). We also provide an elegant description of a linear extension operator Tn:C(P)→C(R) such that Tn(f)∈Dn(R) for every Dn(R)-extendable f∈C(P) and Tn(f)∈Cn(R) whenever f∈C(P) is Cn(R)-extendable.