Geometric dual formulation for first-derivative-based univariate cubic L1 splines

With the objective of generating “shape-preserving” smooth interpolating curves that represent data with abrupt changes in magnitude and/or knot spacing, we study a class of first-derivative-based \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{C}^1$$\end{document}-smooth univariate cubic L1 splines. An L1 spline minimizes the L1 norm of the difference between the first-order derivative of the spline and the local divided difference of the data. Calculating the coefficients of an L1 spline is a nonsmooth non-linear convex program. Via Fenchel’s conjugate transformation, the geometric dual program is a smooth convex program with a linear objective function and convex cubic constraints. The dual-to-primal transformation is accomplished by solving a linear program.