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ISSN 1088-6826(e) ISSN 0002-9939(p)

Convergence of cascade sequence on the Heisenberg group

Authors: Heping Liu and Yu Liu
Journal: Proc. Amer. Math. Soc. 134 (2006), 1413-1423
MSC (2000): Primary 40A30, 42C15, 39B99, 65F15
Posted: October 7, 2005
MathSciNet review: 2199188
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Abstract | References | Similar Articles | Additional Information

Abstract: The investigation of convergence of cascade sequence plays an important role in wavelet analysis on the Euclidean space and also in wavelet analysis on the Heisenberg group. This paper characterizes the $L^p(\mathbf H^d)$ $(1\leq p\leq\infty)$ -convergence of cascade sequence on the Heisenberg group in terms of the -norm joint spectral radius of a collection of matrices associated with the refinement sequence and gives a sufficient condition.

References

1. L. Baggett, A. Carey, W. Moran, and P. Ohring, General existence theorems for orthonormal wavelets, an abstract approach, publications of the Research Institute of Mathematical science, Kyoto University, 31 (1995) no. 1, 95-111. MR 1317525 (96c:42060)
2. B. Han and R.Q. Jia, Multivariate refinement equations and convergence of subdivision schemes, SIAM. J. Math. Anal. 29 (1998), no.5, 1177-1199. MR 1618691 (99f:41018)
3. R. Q. Jia, Subdivision Scheme in spaces, Adv. Comput. Math. 3 (1995), 309-341. MR 1339166 (96d:65028)
4. W. Lawton, S. L. Lee, and Z. W. Shen, Convergence of multidimensional cascade algorithm, Numer. Math. 78 (1997), no. 3, 427-438. MR 1603354 (98k:41027)
5. W. Lawton, Infinite convolution products and refinable distributions on Lie groups, Trans. Amer. Math. Soc. 352 (2000), 2913-2936. MR 1638258 (2000j:43002)
6. H. P. Liu and L. Z. Peng, Admissible wavelets associated with the Heisenberg Group, Pacific J. Math. 180 (1997), 101-123. MR 1474896 (99d:42065)
7. L. Z. Peng, Wavelets on the Heisenberg Group, Geometry and Nonlinear Partial Differential Equations, AMS/IP Studies in Advanced Mathematics, 29 (2002), 123-131. MR 1926440 (2003j:43016)
8. G. C. Rota and W. G. Strang, A note on the joint spectral radius, Indag. Math. 22 (1960), 379-381. MR 147922 (26:5434)
9. R. S. Strichartz, Self-similarity on nilpotent Lie group, Contemp. Math. 140 (1992), 123-157. MR 1197594 (94e:43011)
10. Q. D. Yang, Multiresolution analysis on non-abelian locally compact groups, Ph.D. Thesis, Dept. of Math., Univ. of Saskatchewan, 1999.

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Additional Information

Heping Liu
Affiliation: LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, People's Republic of China
Email: hpliu@math.pku.edu.cn

Yu Liu
Affiliation: LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, People's Republic of China
Email: liuyu@math.pku.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9939-05-08108-6
PII: S 0002-9939(05)08108-6
Keywords: Cascade sequence, refinement equations, joint spectral radius, wavelets, refinable functions.
Received by editor(s): July 13, 2004
Received by editor(s) in revised form: December 13, 2004
Posted: October 7, 2005
Additional Notes: This research was supported by the National Natural Science Foundation of China (No. 10371004) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20030001107)
Communicated by: David R. Larson
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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