Non-polynomial divided difference and blossoming
This work provides a theoretical extension for approximation theory and geometric design, but it is incremental as it generalizes existing polynomial concepts to specific non-polynomial domains.
The paper tackled the problem of extending the blossom and divided difference operators to non-polynomial spline spaces, including trigonometric, hyperbolic, and Müntz splines, and established a relationship between non-polynomial divided differences and the blossom analogous to the polynomial case.
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 Müntz 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.