Non-stationary subdivision schemes originated from uniform trigonometric B-spline

TL;DR

This paper introduces a new class of non-stationary subdivision schemes based on uniform trigonometric B-splines, capable of reproducing conic sections and trigonometric functions, with smoothness analyzed through asymptotic equivalence.

Contribution

It presents a novel algorithm for m-point binary non-stationary subdivision schemes derived from trigonometric B-splines, unifying and extending existing schemes with reproduction capabilities.

Findings

Most well-known binary schemes are non-stationary counterparts of the proposed algorithm.

The schemes can reproduce or regenerate conic sections and trigonometric functions.

Examples demonstrate the scheme's usefulness with a tension parameter.

Abstract

The paper proposes, an algorithm to produce novel m-point (for any integer m>=2) binary non-stationary subdivision scheme. It has been developed using uniform trigonometric B-spline basis functions and smoothness is being analyzed using the theory of asymptotically equivalence. The results show that the most of well-known binary approximating schemes can be considered as the non-stationary counterpart of the proposed algorithm. Furthermore, the schemes developed by the proposed algorithm has the ability to reproduce or regenerate the conic sections, trigonometric polynomials and trigonometric splines as well. Some examples are considered, by choosing an appropriate tension parameter 0<alpha<pi/3, to show the usefulness.