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.