Nonlinear stationary subdivision schemes reproducing hyperbolic and trigonometric functions

• Published in: Advances in computational mathematics 2019-12, Vol.45 (5-6), p.3137-3172
• Main Authors: Donat, Rosa, López-Ureña, Sergio
• Format: Article
• Language: English
• Subjects: Computational mathematics; Computational Mathematics and Numerical Analysis; Computational Science and Engineering; Hyperbolic functions; Mathematical and Computational Biology; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Polynomials; Preprocessing; Stability analysis; Trigonometric functions; Visualization
• ISSN: 1019-7168; 1572-9044
• DOI: 10.1007/s10444-019-09731-8

In this paper we introduce a new family of interpolatory subdivision schemes with the capability of reproducing trigonometric and hyperbolic functions, as well as polynomials up to second degree. It is well known that linear, non-stationary, subdivision schemes do achieve this goal, but their application requires the practical determination of the parameters defining the level-dependent rules, by preprocessing the available data. Since different conic sections require different refinement rules to guarantee exact reproduction, it is not possible to reproduce a shape composed, piecewisely, by several trigonometric functions. On the other hand, our construction is based on the design of a family of stationary nonlinear rules. We show that exact reproduction of different conic shapes may be achieved using the same nonlinear scheme, without any previous preprocessing of the data. Convergence, stability, approximation, and shape preservation properties of the new schemes are analyzed. In addition, the conditions to obtain C 1 limit functions are also studied, which are related with the monotonicity of the data. Some numerical experiments are also carried out to check the theoretical results, and a preferred nonlinear scheme in the family is identified.