Interpolation and extrapolation of smooth functions by linear operators

Abstract

Let $C^{m,1}({\mathbb{R}}^n)$ be the space of functions on ${\mathbb{R}}^n$ whose $m^{\mathsf{th}}$ derivatives are Lipschitz 1. For $E\subset {\mathbb{R}}^n$, let $C^{m,1}(E)$ be the space of all restrictions to $E$ of functions in $C^{m,1}({\mathbb{R}}^n)$. We show that there exists a bounded linear operator $T:C^{m,1}(E)\to C^{m,1}({\mathbb{R}}^n)$ such that, for any $f\in C^{m,1}(E)$, we have $Tf=f$ on $E$.