NULL SPACES OF DIFFERENTIAL-OPERATORS, POLAR FORMS, AND SPLINES

In this article we consider a class of functions, called D-polynomials , which are contained in the null space of certain second order differ ential operators with constant coefficients. The class of splines gene rated by these D-polynomials strictly contains the polynomial, trigono metric, 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 essen tially all of the well-known results concerning polynomial splines. Am ong other topics, we introduce a Schoenberg operator and define contro l curves for these splines. We also examine the knot insertion and sub division algorithms and prove that the subdivision schemes converge qu adratically. (C) 1996 Academic Press. Inc.