Restricted range polynomial interpolation (Q1813828)

The author discusses the question of interpolation of real-valued continuous functions \(f\), \(0\leq f(x)\leq 1\) for all \(0\leq x\leq 1\), by means of polynomials whose ranges are similarly restricted. The main result of the paper is that there exists such an n-th polynomial \(p_ n\) which interpolates \(f\) at \(n+1\) distinct points in all cases except when \(f(0)=f(1)=0\) or 1 and \(n\) is odd; or when \(f(0)=0\) and \(f(1)=1\) or \(f(1)=0\) and \(f(0)=1\) and \(n\) is even. Quite the opposite occurs at the extreme case when the \(n+1\) interpolation points coalesce, namely, we take the Taylor polynomial of \(f\in C^{\infty}[0,1]\). The author constructs a function \(f\in C^{\infty}[0,1]\), \(0\leq f\leq 1\), \(f^{(n+1)}(x)<0\) for each \(n\geq 3\) and for all \(0\leq x\leq 1\), and \(f(0)=0\), \(f(1)=f(x_ 0)=1\) for some \(0<x_ 0<1\) for which no Taylor polynomial of \(f\) at any \(0\leq c\leq 1\) is so restricted in its range when \(n\geq 3\). The paper is related to a work by \textit{J. Briggs} and \textit{L. A. Rubel} [J. Approximation Theory 30, 160-168 (1980; Zbl 0492.41006)].