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Wavelet basis packets and wavelet frame packets. (English) Zbl 0882.42022
The article generalizes the results of Z. Shen [SIAM J. Math. Anal. 26, No. 4, 1061-1074 (1995; Zbl 0826.42025)] on wavelet packets in \({\mathbb{R}}^d\) and succeeds to set up a more natural framework. In particular, the following result is proved:
Let \(\varphi \in L^2({\mathbb{R}}^d)\) be an orthogonal scaling function with two-scale symbol \(m_0(\xi)\) and suppose that there exists a \(2\pi {\mathbb{Z}}^d\)-periodic measurable matrix completion \(M(\xi) = (m_{\mu}(\xi+\nu \pi))_{\mu, \nu \in E_d}\) being unitary for \(\xi\) a.e.. Further, let \((n, j) \in {\mathbb{Z}}_+ \times {\mathbb{Z}}_+\) correspond to the dyadic interval \(I_{j,n}=\{ l \in {\mathbb{Z}}_+: 2^{jd}n \leq l < 2^{jd}(n+1)\}\). Then \(\{ 2^{jd/2} w_n(2^jx-k) \}\), \(k \in {\mathbb{Z}}^d\), is an orthonormal basis of \(L^2({\mathbb{R}}^d)\) if and only if \(\{ I_{j,n} \}_{(n,j)}\) is a disjoint covering of \({\mathbb{Z}}_+\).
Finally, an “unstability” result of nonorthogonal wavelet packets in A. Cohen and I. Daubechies [SIAM J. Math. Anal. 24, No. 5, 1340-1354 (1993; Zbl 0792.42020)] is generalized to \({\mathbb{R}}^d\).
Reviewer: G.Plonka (Rostock)

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
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[1] deBoor, C., DeVore, R., and Ron, A. (1993). On the construction of multivariate (pre) wavelets.Constr. Approx. 9, 123–166. · Zbl 0773.41013 · doi:10.1007/BF01198001
[2] Chen, D. (1994). On splitting trick and wavelet frame packets, preprint.
[3] Chui, C. R., and Li, C. (1993). Non-orthogonal wavelet packets,SIAM J. Math. Anal. 24, 712–738. · Zbl 0770.41022 · doi:10.1137/0524044
[4] Cohen, A., and Danbechies, I. (1993). On the instability of arbitrary biorthogonal wavelet packets.SIAM J. Math. Anal. 24, 1340–1350. · Zbl 0792.42020 · doi:10.1137/0524077
[5] Coifman, R., and Meyer, Y. Orthogonal wave packet bases, preprint. · Zbl 0864.42014
[6] Coifman, R., Meyer, Y., and Wickerhauser, M. V. (1992). Wavelet analysis and signal processing.Wavelets and Their Applications (M. B. Ruskai et al., eds.). Jones and Bartlett, Boston, MA, 153–178. · Zbl 0792.94004
[7] –, Size properties of wavelet packets.Wavelets and Their Applications (M. B. Ruskai et al., eds.). Jones and Bartlett, Boston, MA, 453–470.
[8] Daubechies, I. (1992). Ten lectures on wavelets.CBMS Lecture Notes 61. Society for Industrial and Applied Mathematics, Philadelphia, PA.
[9] Lawton, W. (1990). Tight frames of compactly supported wavelets.J. Math. 31, 1898–1910. · Zbl 0708.46020
[10] Long, R., and Chen, D. (1995). Biorthogonal wavelet bases on \(\mathbb{R}\) d .Appl. Comp. Harmonic Anal. 2, 230–242. · Zbl 0846.42018 · doi:10.1006/acha.1995.1016
[11] Shen, Z. (1995). Non-tensor product wavelet packets inL 2(\(\mathbb{R}\) s ).SIAM J. Math. Anal. 26, 1061–1074. · Zbl 0826.42025 · doi:10.1137/S0036141093243642
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