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Classification of Refinable Splines

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Constructive Approximation Aims and scope

Abstract

A refinable spline is a compactly supported refinable function that is piecewise polynomial. Refinable splines, such as the well-known B-splines, play a key role in computer aided geometric design. So far all studies on refinable splines have focused on positive integer dilations and integer translations, and under this setting a rather complete classification was obtained in [12]. However, refinable splines do not have to have integer dilations and integer translations. The classification of refinable splines with noninteger dilations and arbitrary translations is studied in this paper. We classify completely all refinable splines with integer translations and arbitrary dilations. Our study involves techniques from number theory and complex analysis.

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Authors and Affiliations

  1. Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou, 310014, P. R. China

    Xin-Rong Dai

  2. School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA

    De-Jun Feng

  3. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084 and Department of Mathematics, Chinese University of Hong Kong, Shatin, N.T., Hong Kong, China

    Yang Wang

Authors
  1. Xin-Rong Dai
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  2. De-Jun Feng
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  3. Yang Wang
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Corresponding authors

Correspondence to Xin-Rong Dai, De-Jun Feng or Yang Wang.

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Dai, XR., Feng, DJ. & Wang, Y. Classification of Refinable Splines. Constr Approx 24, 187–200 (2006). https://doi.org/10.1007/s00365-005-0622-9

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  • DOI: https://doi.org/10.1007/s00365-005-0622-9

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