On convergence and quasiconformality of complex planar spline interpolants

Abstract

There appear to be two main approaches for developing complex splines. One of these, which has been in use for quite some time, consists in defining splines on the boundary of a given region which are then extended into the interior by Cauchy's integral formula (see e.g. [1]). The other approach, which is of a more recent origin, is motivated in spirit by the theory of finite elements (see e.g. [10], p. 320) and is contained in [8] and [9]. Observing that the foregoing extension into the interior is not easy to execute numerically, certain continuous piecewise non-holomorphic functions, called complex planar splines have been studied in [8] and [9]. The choice of non-holomorphic functions is justified, since if we take the pieces to be holomorphic functions like polynomials, then by the well known identity theorem ([5], p. 132, theorem 60) the continuity of such a piecewise function implies that all the pieces represent just one holomorphic function. Thus, we shall consider polynomials in z and its conjugate z¯ of the formwhich are generally non-holomorphic functions. The numberwill be called the degree of q. For simplicity we also write q(z) for q(z, z¯).