$C^2$ Interpolation with Range Restriction

Given $-\mathrm{\infty }<\lambda <\mathrm{\Lambda }<\mathrm{\infty }$, $E\subset {\mathbb{R}}^n$ finite, and $f:E\to [\lambda ,\mathrm{\Lambda }]$, how can we extend $f$ to a $C^m({\mathbb{R}}^n)$ function $F$ such that $\lambda \le F\le \mathrm{\Lambda }$ and $\mathrm{\mid }\mathrm{\mid }F\mathrm{\mid }{\mathrm{\mid }}_{C^m({\mathbb{R}}^n)}$ 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 $[\lambda ,\mathrm{\Lambda }]$, and we provide an efficient algorithm to compute such an extension using $O(N\log N)$ operations, where N = #(E).