Shape-preserving properties of univariate cubic L1 splines

The results in this paper quantify the ability of cubic L1 splines to preserve the shape of nonparametric data. The data under consideration include multiscale data, that is, data with abrupt changes in spacing and magnitude. A simplified dual-to-primal transformation for a geometric programming model for cubic L1 splines is developed. This transformation allows one to establish in a transparent manner relationships between the shape-preserving properties of a cubic L1 spline and the solution of the dual geometric-programming problem. Properties that have often been associated with shape preservation in the past include preservation of linearity and convexity/concavity. Under various circumstances, cubic L1 splines preserve linearity and convexity/concavity of data. When four consecutive data points lie on a straight line, the cubic L1 spline is linear in the interval between the second and third data points. Cubic L1 splines of convex/concave data preserve convexity/concavity if the first divided differences of the data do not increase/decrease too rapidly. When cubic L1 splines do not preserve convexity/concavity, they still do not cross the piecewise linear interpolant and, therefore, they do not have extraneous oscillation.