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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On compactly supported spline wavelets and a duality principle


Authors: Charles K. Chui and Jian-zhong Wang
Journal: Trans. Amer. Math. Soc. 330 (1992), 903-915
MSC: Primary 41A15; Secondary 41A05, 41A30, 42C05
MathSciNet review: 1076613
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Abstract: Let $ \cdots \subset{V_{ - 1}} \subset{V_0} \subset{V_1} \subset \cdots $ be a multiresolution analysis of $ {L^2}$ generated by the $ m$th order $ B$-spline $ {N_m}(x)$. In this paper, we exhibit a compactly supported basic wavelet $ {\psi _m}(x)$ that generates the corresponding orthogonal complementary wavelet subspaces $ \cdots,{W_{ - 1}},{W_0},{W_1}, \ldots $. Consequently, the two finite sequences that describe the two-scale relations of $ {N_m}(x)$ and $ {\psi _m}(x)$ in terms of $ {N_m}(2x - j),j \in \mathbb{Z}$, yield an efficient reconstruction algorithm. To give an efficient wavelet decomposition algorithm based on these two finite sequences, we derive a duality principle, which also happens to yield the dual bases $ \{ {\tilde N_m}(x - j)\} $ and $ \{ {\tilde \psi _m}(x - j)\} $, relative to $ \{ {N_m}(x - j)\} $ and $ \{ {\psi _m}(x - j)\} $, respectively.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1992-1076613-3
PII: S 0002-9947(1992)1076613-3
Keywords: Compactly supported wavelets, $ B$-splines, decomposition, reconstruction, duality principle, dual bases
Article copyright: © Copyright 1992 American Mathematical Society