Optimal $C^{1,1}$ and Quasi-Optimal $C^2$ Monotone Interpolation with Curvature Control

TL;DR

This paper develops optimal monotone Hermite interpolation methods with curvature control, providing explicit constructions and formulas for $C^{1,1}$ splines and techniques to mollify solutions to $C^2$.

Contribution

It introduces explicit optimal $C^{1,1}$ spline constructions, formulas for trace seminorms, and methods to mollify solutions while maintaining monotonicity.

Findings

Explicit quadratic spline construction for optimal curvature.

Formula for computing trace seminorm with monotonicity constraints.

Method to mollify $C^{1,1}$ solutions to $C^2$ preserving monotonicity.

Abstract

We study monotone Hermite interpolation on an interval, where both function values and first derivatives are prescribed at the nodes. Among all $C^{1,1}$ interpolants, we seek one with optimal curvature, measured by $\|F''\|_{L^\infty}$. In this paper, we analyze the limitations of some classical techniques, and provide an explicit optimal construction in $C^{1,1}$ given by quadratic splines by studying the optimal velocity profile. Moreover, given $E = \{x_1,\cdots,x_N\}$ and $f: E\to \mathbb{R}$ (without derivatives), we also provide a formula to compute the corresponding trace seminorm \[ \inf\Bigl\{ \|F''\|_{L^\infty} : F(x)=f(x) \text{ on $E$ and } F'\ge 0 \text{ everywhere} \Bigr\}. \] In addition, we also describe how to mollify $C^{1,1}$ solutions to $C^2$ while preserving monotonicity and sacrificing a controlled amount of optimality.