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Nonlinear stationary subdivision schemes reproducing hyperbolic and trigonometric functions

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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 \(\mathcal {C}^{1}\) 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.

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Acknowledgments

We would like to thank Professor Ulrich Reif for his suggestion about the definition of the nonlinear scheme in (27).

Funding

This work received financial support from Project MTM2017-83942-P (MINECO, Spain) and from the FPU14/02216 grant (MECD, Spain).

Author information

Authors and Affiliations

  1. Departament de Matemàtiques, Universitat de València, Doctor Moliner Street 50, 46100, Burjassot, Valencia, Spain

    Rosa Donat & Sergio López-Ureña

Authors
  1. Rosa Donat
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  2. Sergio López-Ureña
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Corresponding author

Correspondence to Sergio López-Ureña.

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Communicated by: Tomas Sauer

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Appendix. Gradient computations

Appendix. Gradient computations

This appendix describes the computation of the gradients that appear in Section 7. We recall that we are assuming that the data is strictly positive and \(\epsilon \in [0, \sqrt {2}]\); hence, the subdivision rules of \(S_{\epsilon }^{(1)}\) are

$$ \begin{array}{@{}rcl@{}} {\Psi}_{j} (x,y,z) = y +(-1)^{j} 2{\Gamma}^{[1]}_{\epsilon}(x,y,z) (x-z), \qquad j=0,1, \end{array} $$

where \({\Gamma }^{[1]}_{\epsilon }\), which does not depend on 𝜖, is given in the first row of (33).

Let us denote . An easy computation leads to

Then,

Since , G1 and G2, in (55), and (56) are positive and smooth too. Hence their gradients are well defined. Some details are provided below, and the results are summarized in Table 4. In the computations below, we use that , the chain rule, and the values obtained for .

Table 4 The 1-norms of the gradients of the subdivision rules Ψ0, Ψ1 and the functions G1 and G2
Full size table

For G1 we have,

$$ G_{1}(x,y)= \frac{{\Psi}_{1}(x,1,y)}{{\Psi}_{0}(x,1,y)} \qquad \text{for } x,y>0, \qquad \nabla (x,1,y) = \left( \begin{array}{lll} 1& 0\\ 0& 0\\ 0& 1 \end{array}\right), $$
(55)

thus the chain rule leads to

To compute , where

$$ G_{2}(x,y,z) = \frac{{\Psi}_{0}(1,y,yz)}{{\Psi}_{1}(x,1,y)}, $$
(56)

we proceed analogously

To carry out the computations required in Lemma 3, we use the following notation: The double application of S𝜖 is determined by

$$ {{\Psi}^{2}_{j}}(d_{i-2},d_{i-1},\ldots, d_{i+2}) := (S_{\epsilon}^{[1]}S_{\epsilon}^{[1]} d)_{4i+j}, \quad j=0,1,2,3,4, $$
(57)

where:

$$ \begin{array}{@{}rcl@{}} (S_{\epsilon}^{[1]}S_{\epsilon}^{[1]} d)_{4i} &={\Psi}_{0}\Big({\Psi}_{1}(d_{i-2},d_{i-1},d_{i}),{\Psi}_{0}(d_{i-1},d_{i},d_{i+1}),{\Psi}_{1}(d_{i-1},d_{i},d_{i+1})\Big) ,\\ (S_{\epsilon}^{[1]}S_{\epsilon}^{[1]} d)_{4i+1} &={\Psi}_{1}\Big({\Psi}_{1}(d_{i-2},d_{i-1},d_{i}),{\Psi}_{0}(d_{i-1},d_{i},d_{i+1}),{\Psi}_{1}(d_{i-1},d_{i},d_{i+1})\Big) ,\\ (S_{\epsilon}^{[1]}S_{\epsilon}^{[1]} d)_{4i+2} &= {\Psi}_{0}\Big({\Psi}_{0}(d_{i-1},d_{i},d_{i+1}),{\Psi}_{1}(d_{i-1},d_{i},d_{i+1}),{\Psi}_{0}(d_{i},d_{i+1},d_{i+2})\Big),\\ (S_{\epsilon}^{[1]}S_{\epsilon}^{[1]} d)_{4i+3} &= {\Psi}_{1}\Big({\Psi}_{0}(d_{i-1},d_{i},d_{i+1}),{\Psi}_{1}(d_{i-1},d_{i},d_{i+1}),{\Psi}_{0}(d_{i},d_{i+1},d_{i+2})\Big). \end{array} $$

Applying the chain rule and the previous results, we get

Then, for j = 0, 1, 2, 3,

(58)

so that

The relevant results are summarized in Table 5.

Table 5 The 1-norm of the ratio functions \({G^{2}_{j}}\) for j = 0,1,2,3
Full size table

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Donat, R., López-Ureña, S. Nonlinear stationary subdivision schemes reproducing hyperbolic and trigonometric functions. Adv Comput Math 45, 3137–3172 (2019). https://doi.org/10.1007/s10444-019-09731-8

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