Wachspress type rational complex planar splines of degree (3,1) (Q1895910)

The authors investigate rational complex planar splines (in the sense of Wachspress) of the following type: A given region in \(\mathbb{C}\) is subdivided into quadrilaterals of a specific form, and on each such quadrilateral a rational function of degree (3,1) is defined; the (continuous) composition of these rational functions then gives the desired spline \(s\). It is proved that there exists a unique spline \(s\) of this type, which interpolates data given at the four vertices of each quadrilateral and at the midpoints of two of its edges. Furthermore, it is shown that this interpolating spline is quasiregular, if the data comes from a sufficiently smooth function, and if the meshsize is small enough. The results are illustrated by some numerical examples and a number of figures.