On extending Ck functions from an open set to ℝ with applications

Abstract

For k ∈ ℕ ∪ {∞} and U open in ℝ, let Ck (U) be the ring of real valued functions on U with the first k derivatives continuous. It is shown that for f ∈ Ck(U) there is g ∈ C∞(ℝ) with U ⊆ coz g and h ∈ Ck(ℝ) with fg∣U = h∣U. The function f and its k derivatives are not assumed to be bounded on U. The function g is constructed using splines based on the Mollifier function. Some consequences about the ring Ck(ℝ) are deduced from this, in particular that Qcl(Ck(ℝ)) = Q(Ck(ℝ)).