On a stable family of four-point nonlinear subdivision schemes eliminating the Gibbs phenomenon. (English) Zbl 1415.65038

Summary: The Chaikin’s scheme presents the interesting property of not producing Gibbs phenomenon. The problem arises when we need a scheme that provides a higher order of approximation. In this case linear schemes are not a good option, as they produce Gibbs phenomenon close to discontinuities. In this article a new four-point nonlinear family of subdivision schemes that eliminate the Gibbs phenomenon is presented. It is based on the linear family of four-point subdivision schemes depending on a tension parameter introduced in [N. Dyn et al., in: Mathematical methods for curves and surfaces: Tromsø 2004. Sixth international conference on mathematical methods for curves and surfaces. Brentwood, TN: Nashboro Press. 145–156 (2005; Zbl 1080.65526)]. A simple algebraic transformation leads to an easy way of introducing nonlinearity in the original family of schemes. The non-interpolatory characteristic of the nonlinear scheme can be modulated just varying the value of the tension parameter. Results about the stability, convergence and the elimination of the Gibbs phenomenon are presented. Some numerical comparisons of the results obtained in the generation of curves are also shown, leading to the conclusion that the high order nonlinear schemes are more suitable for this purpose.


65D17 Computer-aided design (modeling of curves and surfaces)
41A05 Interpolation in approximation theory
65D05 Numerical interpolation


Zbl 1080.65526
Full Text: DOI


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