Comonotone approximation and interpolation by entire functions

A theorem of Hoischen states that given a positive continuous function epsilon : R -> R, an integer n >= 0, and a closed discrete set E subset of R, any Cn function f : R -> R can be approximated by an entire function g so that for k = 0, . . . , n, and x is an element of R, vertical bar D-k g(x)-D-k f(x)vertical bar, and if x is an element of E then D-k g(x) = D-k f (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. (C) 2019 Elsevier Inc. All rights reserved.