Null Spaces of Differential Operators, Polar Forms, and Splines

✍ Scribed by Dan Gonsor; Marian Neamtu

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 963 KB
• Volume: 86
• Category: Article
• ISSN: 0021-9045
• DOI: 10.1006/jath.1996.0056

✦ Synopsis

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. 1996 Academic Press, Inc. 1964 [16] and later investigated by, among others, Lyche and Winther [14], have turned out to have a similar structure. In addition to other desirable properties, certain recurrence relations have been discovered for article no.