On the Approximation by Local Complex-Valued Splines

Sometimes it is desirable to visualize complex-valued functions in polar co-ordinates that always have difficulties. But actually it is sufficient to visualize separately the real and imaginary parts of a complex-valued function. For a fast visualization process, local spline interpolation of functions from two variables is the most convenient in applications and gives approximations with the required order of accuracy. This paper deals with local complex-valued splines, constructed using tensor product spline interpolation in a disc with a centre of zero and a radius of 1. For constructing the tensor product we use local basis splines of a radial variable and local complex-valued basis splines of an angular variable. For the construction of the grid we consider a number of circles in the disc of radius 1 with a center of zero, and get a number of points on the boundary of this disc, arranged from the centre to the edge. The points at which those lines cross each circle form the grid nodes. The approximation is constructed separately in each elementary segment, formed by two arcs and two line segments. For the approximation of a complex-valued function we use the values of this function in several nodes near this elementary segment and the tensor product of basis splines. The order of the approximation depends on the splines' properties which we use in the tensor product. In this paper we use local exponential and polynomial splines of second and third order approximation.