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

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.