Rational splines for Hermite interpolation with shape constraints
The general Hermite problem constitutes the background of the study, the authors being interested in shape-preserving Hermite interpolation of a given function and its derivatives at the endpoints of an interval using rational splines. Two cases are studied and for the first case it is proved that, having \(f\), \(f^{\prime}\) and a monotonic set of data, it is possible to construct a monotonic rational function of type 3/2 interpolating the data. These results are extended for positive or convex interpolants. The novelty regarding the approach of this case comes from the fact that the shape properties of a control polygon associated with the function are used. In the second case which had not been considered before in its full general approach, taking into account that \(f\), \(f^{\prime}\) and \(f^{\prime\prime}\) are given, for any positive (or monotonic or convex) set of data, algorithms are designed to construct a positive (respectively, monotonic or convex) rational function of type 5/4 interpolating the data. An algorithm for the value of a free parameter \(\sigma\) related to the interpolant is also given together with error estimates. Some numerical examples are also conducted in order to illustrate the algorithm for each problem considered in the paper.
scientific article; zbMATH DE number 6161982