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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Pointwise convergence of bounded cascade sequences


Authors: Di-Rong Chen and Min Han
Journal: Proc. Amer. Math. Soc. 135 (2007), 181-189
MSC (2000): Primary 42C15
Published electronically: June 28, 2006
MathSciNet review: 2280186
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Abstract: The cascade algorithm plays an important role in computer graphics and wavelet analysis. For an initial function $ \phi_0$, a cascade sequence $ (\phi_n)_{n=0}^{\infty}$ is constructed by the iteration $ \phi_n=C_a\phi_{n-1}, n=1, 2, \dots, $ where $ C_a$ is defined by $ C_ag=\sum_{\alpha\in\mathbb{Z}}a(\alpha)g(2\cdot-\alpha), \, g\in L_p(\mathbb{R}).$ In this paper, under a condition that the sequence $ (\phi_n)_{n=0}^\infty$ is bounded in $ L_\infty(\mathbb{R})$, we prove that the following three statements are equivalent: (i) $ (\phi_n)_{n=0}^{\infty}$ converges $ {\rm a.e.} x\in \mathbb{R}$. (ii) For $ {\rm a.e.} x\in \mathbb{R}$, there exist a positive constant $ c$ and a constant $ \gamma\in (0,1)$ such that $ \vert\phi_{n+1}(x)-\phi_n(x)\vert\leq c\gamma^n\,\,\forall n=1,2, \dots.$ (iii) For some $ p\in [1, \infty), (\phi_n)_{n=0}^{\infty}$ converges in $ L_p(\mathbb{R})$. An example is presented to illustrate our result.


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Additional Information

Di-Rong Chen
Affiliation: Department of Mathematics, and LMIB, Beijing University of Aeronautics and Aestronautics, Beijing 100083, People’s Republic of China

Min Han
Affiliation: Department of Mathematics, and LMIB, Beijing University of Aeronautics and Aestronautics, Beijing 100083, People’s Republic of China

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08467-X
PII: S 0002-9939(06)08467-X
Keywords: Refinable function, cascade algorithm, subdivision scheme, pointwise convergence, refinable curve, joint spectral radius
Received by editor(s): February 10, 2005
Received by editor(s) in revised form: July 29, 2005
Published electronically: June 28, 2006
Additional Notes: This research was supported in part by NSF of China under grant 10571010
Communicated by: David R. Larson
Article copyright: © Copyright 2006 American Mathematical Society