Wavelet basis packets and wavelet frame packets
The article generalizes the results of \textit{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 \textit{A. Cohen} and \textit{I. Daubechies} [SIAM J. Math. Anal. 24, No. 5, 1340-1354 (1993; Zbl 0792.42020)] is generalized to \({\mathbb{R}}^d\).
scientific article; zbMATH DE number 1038863