$C^2$ Interpolation with Range Restriction

Abstract

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

62 pages. arXiv admin note: text overlap with arXiv:2102.05777