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

We investigate univariate C2 quintic L1 splines and C3 seventh-degree L1 splines and revisit C1 cubic L1 splines. We first investigate these L1 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 L1 splines with conventional “L2 splines” calculated by minimizing analogous integrals involving squares of such expressions. The L2 splines do not preserve the shape of irregular data well. The quintic and seventh-degree L1 splines also do not preserve shape well, although they do so better than quintic and seventh-degree L2 splines. Consistent with previously known results, the cubic L1 splines do preserve shape well. For both L1 and L2 splines, the lower the level of the derivative in the minimization principle, the better the shape preservation is. Function-value-based cubic L1 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 L1 splines is proposed. As a basis for this procedure, the function-value-based cubic L1 spline is calculated. The quintic L1 spline is calculated by fixing the first derivatives at the nodes to be those of the cubic L1 spline and calculating the second derivatives at the nodes by minimizing a second-derivative-based quintic L1 spline functional. The seventh-degree L1 spline is calculated by fixing the first and second derivatives at the nodes to be those of the quintic L1 spline and calculating the third derivatives at the nodes by minimizing a second-derivative-based seventh-degree L1 spline functional. Computational results indicate that C2 quintic L1 splines and C3 seventh-degree L1 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 L1 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.