$C^2$ 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\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) $.