$C^2$ interpolation with range restriction

Abstract

Given $-\infty <\lambda <\Lambda <\infty$ , $E\subset {\mathbb{R}}^n$ finite, and $f:E\to [\lambda ,\Lambda ]$ , how can we extend $f$ to a $C^m({\mathbb{R}}^n)$ function $F$ such that $\lambda \le F\le \Lambda$ and $\Vert F{\Vert }_{C^m({\mathbb{R}}^n)}$ is within a constant multiple of the least possible, with the constant depending only on $m$ and $n$? In this paper, we provide the solution to the problem for the case $m=2$. Specifically, we construct a (parameter-dependent, nonlinear) $C^2({\mathbb{R}}^n)$ extension operator that preserves the range $[\lambda ,\Lambda ]$, and we provide an efficient algorithm to compute such an extension using $O(N\log N)$ operations, where $N=\#(E)$ .