Shape-preserving properties of univariate cubic splines

The results in this paper quantify the ability of cubic L 1 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 L 1 splines is developed. This transformation allows one to establish in a transparent manner relationships between the shape-preserving properties of a cubic L 1 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 L 1 splines preserve linearity and convexity/concavity of data. When four consecutive data points lie on a straight line, the cubic L 1 spline is linear in the interval between the second and third data points. Cubic L 1 splines of convex/concave data preserve convexity/concavity if the first divided differences of the data do not increase/decrease too rapidly. When cubic L 1 splines do not preserve convexity/concavity, they still do not cross the piecewise linear interpolant and, therefore, they do not have extraneous oscillation.