Non-polynomial divided difference and blossoming

TL;DR

This paper introduces a generalized non-polynomial blossom for various spline spaces and establishes a relationship between this blossom and divided differences, extending classical polynomial theory.

Contribution

It defines a non-polynomial homogeneous blossom for diverse spline spaces and links it to divided differences, broadening the scope of approximation theory.

Findings

Established a non-polynomial blossom for trigonometric, hyperbolic, and M"untz spline spaces.

Derived a relation between non-polynomial divided differences and the blossom.

Extended classical polynomial identities to a wider class of spline functions.

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.