$C^2$ Interpolation with Range Restriction

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 $\lambda\leq F \leq \Lambda$ and $||F||_{C^m(\mathbb{R}^n)}$ 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) .