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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)

##### MSC:
 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
##### Keywords:
wavelets; Riesz bases; frames; wavelet packets; unitary matrices
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##### References:
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