On hyperbolic splines

Abstract

In recent years there has been an intense interest in spline functions (see [4] and references therein). Among the various classes of splines, the polynomial spline functions have received by far the greatest attention, primarily because they are the most useful in numerical computations. This, in turn, is due largely to the fact that the polynomial splines admit a basis of so-called B-splines which can be computed efficiently and accurately via certain recursion relations. Recently (see [3]) it was discovered that certain classes of trigonometric splines also admit of B-spline bases which satisfy similar recursion relations. The purpose of this paper is to give a detailed discussion of a third class of splines, the hyperbolic splines, which also have a basis of B-splines which can be computed recursively. In addition to their value in certain applications and as an illustration of the space of L-splines (cf. [4]), the hyperbolic splines are of special interest in view of the fact (see [5]) that the only classes of splines which have B-spline bases computable by recursions are the polynomial, trigonometric, and hyperbolic splines. Our treatment of hyperbolic spiines depends heavily on obtaining explicit formulae for a related Green’s function, for determinants formed from the hyperbolic functions, and for certain associated hyperbolic divided differences. These results are developed in Sections 24. The hyperbolic Bsplines are introduced in Section 5, and the key recursion relation is established in Section 6. In the remaining sections of the paper we discuss the shape of the B-splines, a Peano kernel representation for divided differences, integrals of the B-splines, a Marsden-type identity, a partition of unity result, and, finally, give a basis and dual basis for 9. We turn now to the definition of hyperbolic splines. First we need some notation. Throughout the paper we shall use the abbreviations