Nonlinear stationary subdivision schemes that reproduce trigonometric functions

TL;DR

This paper introduces a family of nonlinear stationary subdivision schemes capable of reproducing various conic shapes, including polynomials up to second order, with consistent rules unlike linear schemes.

Contribution

The paper presents a novel nonlinear stationary subdivision scheme that can reproduce multiple conic shapes using a single set of rules, unlike traditional linear schemes.

Findings

The schemes are convergent and stable.

They preserve shape and approximate well.

Conditions for $ ext{C}^1$ smoothness are established.

Abstract

In this paper we define a family of nonlinear, stationary, interpolatory subdivision schemes with the capability of reproducing conic shapes including polynomials upto second order. Linear, non-stationary, subdivision schemes do also achieve this goal, but different conic sections require different refinement rules to guarantee exact reproduction. On the other hand, with our construction, exact reproduction of different conic shapes can be achieved using exactly the same nonlinear scheme. Convergence, stability, approximation and shape preservation properties of the new schemes are analyzed. In addition, the conditions to obtain $\mathcal{C}^1$ limit functions are also studied.