Non-polynomial divided difference and blossoming

Abstract

Two notable examples of dual functionals in approximation theory and computer-aided geometric design are the blossom and the divided difference operator. Both of these dual functionals satisfy a similar set of formulas and identities. Moreover, the divided differences of polynomials can be expressed in terms of the blossom. In this paper, an extended non-polynomial homogeneous blossom for a wide collection of spline spaces, including trigonometric splines, hyperbolic splines, and special M\"{u}ntz spaces of splines, is defined. It is shown that there is a relation between the non-polynomial divided difference and the blossom, which is analogous to the polynomial case.