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Non-polynomial divided difference and blossoming

Fatma Zürnacı-Yetiş

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 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.

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 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.
Paper Structure (5 sections, 10 theorems, 50 equations)

This paper contains 5 sections, 10 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Non-polynomial divided differences
  3. Blossoming
  4. Non-polynomial divided difference and blossoming
  5. Conclusions

Key Result

Theorem 2.3

Let $x, x_{0} \in [a,b]$ and $f, \gamma_{1}, \gamma_{2} \in C^{n}[a,b]$ and let $f^{n+1}$, $\gamma_{1}^{n+1}$ and $\gamma_{2}^{n+1}$ exist in the open interval $(a,b)$ . If $1\in \text{span}\{\gamma_{1}, \gamma_{2}\}$, then where

Theorems & Definitions (25)

  • Definition 2.1: cf. fatma
  • Definition 2.2: cf. fatma
  • Theorem 2.3: Generalized Taylor Theorem
  • Definition 3.1
  • Definition 3.2
  • Theorem 4.1
  • proof
  • Definition 4.2
  • Example 1
  • Theorem 4.3
  • ...and 15 more