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 $\mathrm{\parallel }{F}^{\mathrm{\prime }\mathrm{\prime }}{\mathrm{\parallel }}_{L^{\mathrm{\infty }}}$. 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 $$\inf \{\mathrm{\parallel }{F}^{\mathrm{\prime }\mathrm{\prime }}{\mathrm{\parallel }}_{L^{\mathrm{\infty }}}:F(x)=f(x)\text{ on }E\text{ and }{F}^{\mathrm{\prime }}\ge 0\text{ everywhere}\}\mathrm{.}$$ 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.