Comonotone approximation and interpolation by entire functions

Abstract

A theorem of Hoischen states that given a positive continuous function ε:R→R, an integer n≥0, and a closed discrete set E⊆R, any Cn function f:R→R can be approximated by an entire function g so that for k=0,…,n, and x∈R, |Dkg(x)−Dkf(x)|<ε(x), and if x∈E then Dkg(x)=Dkf(x). The approximating function g is entire and hence piecewise monotone. We determine conditions under which when f is piecewise monotone we can choose g to be comonotone with f (increasing and decreasing on the same intervals), and under which the derivatives of g can be taken to be comonotone with the corresponding derivatives of f if the latter are piecewise monotone.