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

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- by Chen Di-Rong
- Proc. Amer. Math. Soc.
**125**(1997), 2883-2889 - DOI: https://doi.org/10.1090/S0002-9939-97-03876-8
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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

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## Bibliographic 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
- 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
- © Copyright 1997 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**125**(1997), 2883-2889 - MSC (1991): Primary 41A63, 42C05, 46C99
- DOI: https://doi.org/10.1090/S0002-9939-97-03876-8
- MathSciNet review: 1396974