Complex Chebyshev polynomials and generalizations with an application to the optimal choice of interpolating knots in complex planar splines

Abstract

This paper contains a brief account on complex planar splines which are complex valued functions defined piecewise on a grid. For noncontinuous (so called nonconforming) splines the problem of the placement of knots at which these splines are required to be continuous is investigated. It is shown that this problem reduces to finding complex Chebyshev polynomials under the additional requirement that the zeros of the polynomials are on the boundary of the corresponding domains. It is proved that the zeros of a generalized Chebyshev polynomial are in the convex hull of the domain on which the Chebyshev polynomials are defined. Some open problems are stated. A numerical and graphical display for the optimal location of three and six points on certain triangles is provided.