Analytic Extension of Smooth Functions

Abstract

Let F be a closed proper subset of ℝn and let ℰ* be a class of ultradifferentiable functions. We give a new proof for the following result of Schmets and Valdivia on analytic modification of smooth functions: for every function ƒ ∈ ℰ* (ℝn) there is \({\widetilde f} \in {\cal E}_{*}(\rm R ^{n})\)which is real analytic on ℝnF and such that ∂a ƒ ¦F = ∂a ƒ ¦ F for any a ∈ ℕ0n. For bounded ultradifferentiable functions ƒ we can obtain \({\widetilde f}\)by means of a continuous linear operator.