$C^2$ 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 $ lambdaleq 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(Nlog N) $ operations, where $ N = #(E) $.