$C^2$ Interpolation with Range Restriction

TL;DR

This paper develops a method to extend finite data functions to twice continuously differentiable functions with range restrictions, providing an efficient algorithm for practical computation.

Contribution

It introduces a nonlinear extension operator for $C^2$ functions that preserves range constraints and offers an $O(N \, \log N)$ algorithm for implementation.

Findings

Constructed a $C^2$ extension operator preserving range restrictions.

Provided an efficient $O(N \log N)$ algorithm for extension computation.

Achieved a solution for the case $m=2$ in the interpolation problem.

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) $.