Shape-preserving univariate cubic and higher-degree L1 splines with function-value-based and multistep minimization principles

We investigate univariate C-2 quintic L-1 splines and C-3 seventh-degree L-1 splines and revisit C-1 cubic L-1 splines. We first investigate these L-1 splines when they are calculated by minimizing integrals of absolute values of expressions involving various levels of derivatives from zeroth derivatives (function values) to fourth derivatives and compare these L-1 splines with conventional "L-2 splines" calculated by minimizing analogous integrals involving squares of such expressions. The L-2 splines do not preserve the shape of irregular data well. The quintic and seventh-degree L-1 splines also do not preserve shape well, although they do so better than quintic and seventh-degree L-2 splines. Consistent with previously known results, the cubic L-1 splines do preserve shape well. For both L-1 and L-2 splines, the lower the level of the derivative in the minimization principle, the better the shape preservation is, Function-value-based cubic L-1 splines preserve shape well for all situations tested except the one in which an "S-curve" occurs when a flatter representation might be expected. A multi-step procedure for calculating the coefficients of quintic and seventh-degree L-1 splines is proposed. As a basis for this procedure, the function-value-based cubic L-1 spline is calculated. The quintic L-1 spline is Calculated by fixing the first derivatives at the nodes to be those of the cubic L-1 spline and calculating the second derivatives at the nodes by minimizing a second-derivative-based quintic L-1 spline functional. The seventh-degree L-1 spline is calculated by fixing the first and second derivatives at the nodes to be those of the quintic L-1 spline and calculating the third derivatives at the nodes by minimizing a second-derivative-based seventh-degree L-1 spline functional. Computational results indicate that C-2 quintic L-1 splines and C-3 seventh-degree L-1 splines calculated in this manner preserve shape well for all situations tested except one in which an "S-curve" occurs when a flatter representation might be expected. To ensure that the parameters of Cubic and higher-degree L-1 splines depend continuously on the positions of the data. weights that depend on the local interval length need to be used in the integrals in the minimization principles of these splines. Published by Elsevier B.V.