Complex cubic splines

Abstract

Introduction. The past two decades have witnessed an increasing intensity of investigation, [1]-[26], into the properties of spline functions. These functions, which in their simplest form yield the analytic counterpart of the draftsman's tool for drawing a smooth curve through a number of prescribed points, play an important and fundamental role in many parts of numerical analysis. Many of the properties which these functions possess, such as the minimum curvature property [3], [8], [13], [14], are associated with obvious attributes of the elastic beam. Some relate to the best approximation of linear operators [15], [20], and have rather profound meanings in approximation theory. Others, such as orthogonality [24], rate of convergence [12], [17], [23], [25], and completeness [25] of certain bases of splines, are at first rather surprising. A few (cf. [22, p. 241]) are still quite puzzling. The application of spline theory to the approximation of a function analytic interior to a rectifiable Jordan curve and continuous in the corresponding closed region has been considered to a limited extent in an earlier paper, [24], by the authors. There splines were treated which are piecewise polynomial in the arc length s on the curve. The present development is concerned with cubic splines in the complex variable z and provides some insight into the structure of the spline approximation generally. In particular, it serves to establish a connection with the classical theory of approximation to an analytic function. As part of this development, proofs are given for the convergence of the complex cubic spline and its derivatives for the situations in which the approximated function is of class C' (a = 0, 1, 2, 3, or 4) on the boundary. These may be modified in an obvious manner for the standard real cubic spline and for the convergence properties of second and third derivatives. There result noteworthy simplifications over proofs already existing in the literature [12], [17], for a= 2, 3, 4. The convergence properties of cubic splines for cases in which the approximated function is assumed merely to be continuous or to have continuous first derivative constitute significant new developments in spline theory. In addition, a curious spline property is here presented relating to the approximation of the fourth derivative.