Nonlinear stationary subdivision schemes reproducing hyperbolic and trigonometric functions
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Nonlinear stationary subdivision schemes reproducing hyperbolic and trigonometric functions ´ ˜ 1 Rosa Donat1 · Sergio Lopez-Ure na Received: 15 November 2018 / Accepted: 23 September 2019 / © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract 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. Keywords Nonlinear subdivision schemes · Exponential polynomials · Reproduction · Monotonicity preservation · Smoothness Mathematics Subject Classification (2010) 41A25 · 41A30 · 65D15 · 65D17 Communicated by: Tomas Sauer Sergio L´opez-Ure˜na
[email protected] Rosa Donat [email protected] 1
Departament de Matem`atiques, Universitat de Val`encia, Doctor Moliner Street 50, 46100, Burjassot, Valencia, Spain
˜ R. Donat, S. L´opez-Urena
1 Introduction Subdivision refinement is a powerful technique for the design and representation of curves and surfaces. Subdivision algorithms are simple to implement and extremely well suited for computer applications, which make them very attractive to users interested in geometric modeling and Computer Aided Geometric Design (CAGD). As in [14–16], in this paper we consider that a subdivision scheme S = (S k )∞ k=0 is a sequence of operators S k : l∞ (Z) −→ l∞ (Z)1 such that for each f 0 ∈ l∞ (Z), a sequence (f k )∞ k=0 ⊂ l∞ (Z) is recursively defined as follows: f k+1 := S k f k ∈ l∞ (Z),
k ≥ 0.
We shall restrict our attention to binary subdivision schemes such that there exists q ≥ 0 such that for any f ∈ l∞ (Z) (S k f )2i+j = jk (fi−q , . . . , fi+q ),
j = 0, 1,
i ∈ Z,
(1)
for some jk : R2q+1 −→ R. Notice that such schemes are uniform and local. If the functions jk are linear, the level-dependent rules can be expressed as follows jk (fi−q , . . . , fi+q ) =
q
ajk−2l fi+l
i ∈ Z,
(2)
l=−q
where the sequence (aik )i∈Z (that has finite support) is the ma
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