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

 

On the existence and constructions
of orthonormal wavelets on $L_2(\mathbb {R}^s)$


Author: Chen Di-Rong
Journal: Proc. Amer. Math. Soc. 125 (1997), 2883-2889
MSC (1991): Primary 41A63, 42C05, 46C99
MathSciNet review: 1396974
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Abstract: For a multiresolution analysis of $ L _2 (\mathbb {R}^ s)$ associated with the scaling matrix $T$ having determinant $n$ we prove the existence of a wavelet basis with certain desirable properties if $ 2n-1 >s $ and its real-valued counterpart if the scaling function is real-valued and $ n - 1 > s$. That those results cannot be extended to $ 2n - 1 \leq s $ and $ n -1 \leq s $ respectively in general is demonstrated by Adams's theorem about vector fields on spheres. Moreover we present some new explicit constructions of wavelets, among which is a variation of Riemenschneider-Shen's method for $ s\leq 3 .$


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Additional Information

Chen Di-Rong
Affiliation: Department of Applied Mathematics, Beijing University of Aeronautics and Astronautics, Beijing 100083, People’s Republic of China
Email: chengry@maindns.buaa.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9939-97-03876-8
PII: S 0002-9939(97)03876-8
Received by editor(s): January 31, 1994
Received by editor(s) in revised form: April 9, 1996
Additional Notes: Research supported in part by Natural Science Foundation of China.
Communicated by: J. Marshall Ash
Article copyright: © Copyright 1997 American Mathematical Society