Null Spaces of Differential Operators, Polar Forms, and Splines Original Research Article

In this article we consider a class of functions, called D-polynomials, which are contained in the null space of certain second order differential operators with constant coefficients. The class of splines generated by these D-polynomials strictly contains the polynomial, trigonometric, and hyperbolic splines. The main objective of this paper is to present a unified theory of this class of splines via the concept of a polar form. By systematically employing polar forms, we extend essentially all of the well-known results concerning polynomial splines. Among other topics, we introduce a Schoenberg operator and define control curves for these splines. We also examine the knot insertion and subdivision algorithms and prove that the subdivision schemes converge quadratically.