East Asian mathematical journal

The Youngnam Mathematical Society (영남수학회)
권호 표지
DOI QR코드

DOI QR Code

A NEW CLASS OF INTERPOLATORY HERMITE SUBDIVISION SCHEMES REPRODUCING POLYNOMIALS

  • Jeong, Byeongseon (Major in Mathematics, College of Natural Science, Keimyung University)
  • Received : 2022.05.02
  • Accepted : 2022.05.15
  • Published : 2022.05.18
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we present a new class of interpolatory Hermite subdivision schemes of order 2 reproducing polynomials. Each member in this class, denoted by Hn for n ≥ 1, preserves polynomials of degree up to 4n + 1 admitting the approximation order of 4n + 2. Furthermore, it has free parameters which provide flexibility in designing curves/surfaces. H1, the simplest and the most attractive scheme in this class, achieves C4 smoothness with the parameters in certain ranges, and its performance is demonstrated with numerical examples.

Keywords

Acknowledgement

This research was supported by the National Research Foundation of Korea(NRF) funded by the Ministry of Education (NRF-2019R1I1A1A01060757).

References

  1. M. Charina, C. Conti, T. Sauer, Regularity of multivariate vector subdivision schemes, Numer. Algorithms 39 (2005), 97-113. https://doi.org/10.1007/s11075-004-3623-z
  2. C. Conti, J.-L. Merrien, L. Romani, Dual Hermite subdivision schemes of de Rham-type, BIT Numer. Math. 54 (2014), 955-977. https://doi.org/10.1007/s10543-014-0495-z
  3. C. Conti, L. Romani, M. Unser, Ellipse-preserving Hermite interpolation and subdivision, J. Math. Anal. Appl. 426 (2015), 211-227. https://doi.org/10.1016/j.jmaa.2015.01.017
  4. M. Cotronei, C. Moosmuller, T. Sauer, N. Sissouno, Level-dependent interpolatory Hermite subdivision schemes and wavelets, Constr. Approx. 50 (2019), 341-366. https://doi.org/10.1007/s00365-018-9444-4
  5. N. Dyn, D. Levin, Analysis of Hermite-type subdivision schemes, in: C.K. Chui, L.L. Schumaker (Eds.), Approximation Theory VIII, Wavelets and Multilevel Approximation, vol.2, College Station, TX, World Sci., River Edge, NJ, (1995), 117-124.
  6. N. Dyn, D. Levin Analysis of Hermite-interpolatory subdivision schemes, in: S. Dubuc, G. Deslauriers (Eds.), Spline Functions and the Theory of Wavelets, Am. Math. Soc., Providence, RI, (1999), 105-113.
  7. S. Dubuc, J.-L. Merrien, Hermite subdivision schemes and Taylor polynomials, Constr. Approx. 29 (2009), 219-245. https://doi.org/10.1007/s00365-008-9011-5
  8. N. Guglielmi, C. Manni, D. Vitale, Convergence analysis of C2 Hermite interpolatory subdivision schemes by explicit joint spectral radius formulas, Linear Algebra Appl. 434 (2011), 884-902. https://doi.org/10.1016/j.laa.2010.10.002
  9. B. Han, Approximation properties and construction of Hermite interpolants and biorthogonal multiwavelets, J. Approx. Theory 110 (2001), no. 1, 18-53. https://doi.org/10.1006/jath.2000.3545
  10. J.-L. Merrien, A family of Hermite interpolants by bisection algorithms, Numer. Algorithms 2 (1992), 187-200. https://doi.org/10.1007/BF02145385
  11. C. Manni, M.-L. Mazure, Shape constraints and optimal bases for C1 Hermite interpolatory subdivision schemes, SIAM J. Numer. Anal. 48 (2010), no. 4, 1254-1280. https://doi.org/10.1137/09075874X
  12. J.-L. Merrien, T. Sauer, A generalized Taylor factorization for Hermite subdivision schemes, J. Comput. Appl. Math. 236 (2011), no. 4, 565-574. https://doi.org/10.1016/j.cam.2011.06.011
  13. J.-L. Merrien, T. Sauer, From Hermite to stationary subdivision schemes in one or several variables, Adv. Comput. Math. 36 (2012), 547-579. https://doi.org/10.1007/s10444-011-9190-7
  14. M. K. Jena, A Hermite interpolatory subdivision scheme constructed from quadratic rational BernsteinBezier spline, Math. Comput. Simulat. 187 (2021), 433-448. https://doi.org/10.1016/j.matcom.2021.03.018
  15. B. Jeong and J. Yoon, Construction of Hermite subdivision schemes reproducing polynomials, J. Math. Anal. Appl. 451 (2017), 565-582. https://doi.org/10.1016/j.jmaa.2017.02.014
  16. L. Romani, A circle-preserving C2 Hermite interpolatory subdivision scheme with tension control, Comput. Aided Geom. D. 27 (2010), 36-47. https://doi.org/10.1016/j.cagd.2009.08.006