Exponential polynomial reproducing property of non-stationary symmetric subdivision schemes and normalized exponential B-splines

Byeongseon Jeong, Hong Oh Kim, Yeon Ju Lee, Jungho Yoon

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15 Scopus citations


An important capability for a subdivision scheme is the reproducing property of circular shapes or parts of conics that are important analytical shapes in geometrical modeling. In this regards, this study first provides necessary and sufficient conditions for a non-stationary subdivision to have the reproducing property of exponential polynomials. Then, the approximation order of such non-stationary schemes is discussed to quantify their approximation power. Based on these results, we see that the exponential B-spline generates exponential polynomials in the associated spaces, but it may not reproduce any exponential polynomials. Thus, we present normalized exponential B-splines that reproduce certain sets of exponential polynomials. One interesting feature is that the set of exponential polynomials to be reproduced is varied depending on the normalization factor. This provides us with the necessary accuracy and flexibility in designing target curves and surfaces. Some numerical results are presented to support the advantages of the normalized scheme by comparing them to the results without normalization.

Original languageEnglish
Pages (from-to)647-666
Number of pages20
JournalAdvances in Computational Mathematics
Issue number3
StatePublished - Apr 2013

Bibliographical note

Funding Information:
Acknowledgements The authors are grateful to the anonymous referee for the valuable suggestions on this paper. This work was supported by Basic Science Research Program 2010-0011689 (Y. Lee), Mid-career Researcher Program 2009-0084583 (H. Kim), Basic Science Research Program 2010-0016257 (J. Yoon) and Priority Research Centers Program 2010-0028298 (J. Yoon) through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology.


  • Approximation order
  • Exponential B-spline
  • Exponential polynomial reproduction
  • Non-stationary subdivision scheme
  • Normalization


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