Rational splines for Hermite interpolation with shape constraints

This paper is concerned in shape-preserving Hermite interpolation of a given function f at the endpoints of an interval using rational functions. After a brief presentation of the general Hermite problem, we investigate two cases. In the first one, f   and f′f′ are given and it is proved that for any monotonic set of data, it is always possible to construct a monotonic rational function of type [3/2][3/2] interpolating those data. Positive and convex interpolants can be computed by a similar method. In the second case, results are proved using rational function of type [5/4][5/4] for interpolating the data coming from f  , f′f′ and f″f″ with the goal of constructing positive, monotonic or convex interpolants. Error estimates are given and numerical examples illustrate the algorithms.

► Shape-preserving.
► Hermite interpolation.
► Rational functions.
► Algorithms.
► Error estimates.