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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 .$

References [Enhancements On Off] (What's this?)

  • 1. C. de Boor, R. DeVore and A. Ron, On the construction of multivariate (pre)wave-lets, Constructive Approximation, 9(1993), 123-166. MR 94k:41048
  • 2. C. W. Curtis and J. Reiner, Representation Theory of Finite Groups and Associative Algebras, John Wiley and Sons, New York, 1962. MR 26:2519
  • 3. R. Q. Jia and C. A. Micchelli, Using the refinement equation for the construction of pre-wavelets, V: extensibility of trigonometric polynomials, Computing 48(1992), 62-71. MR 94a:42049
  • 4. R. Q. Jia and Z. W. Shen, Multiresolution and wavelets, Proc. Edinburgh Math. Soc. 37(1994), 271-300. MR 95h:42035
  • 5. Y. Meyer, Ondelettes et Opérateurs I: Ondelettes, Hermann Editeurs, 1990. MR 93i:42002
  • 6. S. D. Riemenschneider and Z. W. Shen, Wavelets and prewavelets in low dimensions, J. Approx. Theory, 71(1992), 18-38. MR 94d:42046
  • 7. S. L. Xiao, Construction of real-valued multivariate wavelets, Applied Math.-JCU, 10B(1995), 229-236. MR 96c:42076
  • 8. D. X. Zhou, Construction of real-valued wavelets by symmetry, J. Approximation Theory, 81(1995), 323-331. MR 96m:42047
  • 9. W. P. Ziemer, Weakly Differentiable Functions, Springer-Verlag, New York, Berlin, Heidelberg, 1989. MR 91e:46046

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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

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

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