Classification of Refinable Splines |
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| Authors: | Xin-Rong Dai De-Jun Feng Yang Wang |
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| Institution: | (1) Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou, 310014, P. R. China;(2) School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA;(3) Department of Mathematical Sciences, Tsinghua University, Beijing, 100084 and Department of Mathematics, Chinese University of Hong Kong, Shatin, N.T., Hong Kong, China |
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| 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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| Keywords: | Spline Refinable spline Quasi-trigonometric polynomial Weierstrass factorization theorem |
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