Bessel multiwavelet sequences and dual multiframelets in Sobolev spaces. (English) Zbl 1263.42023
Summary: The dual \(2I_d\)-framelets in \( (H^s(\mathbb R^d), H^{-s}(\mathbb R^d)), s>0\), were introduced by B. Han and Z. Shen [Constr. Approx. 29, No. 3, 369–406 (2009; Zbl 1161.42018)]. In this paper, we systematically study the Bessel property of multiwavelet sequences in Sobolev spaces. The conditions for Bessel multiwavelet sequences in \( H^{-s}(\mathbb R^d) \) take great difference from those for Bessel wavelet sequences in this space. Precisely, the Bessel property of a multiwavelet sequence in \( H^{-s}(\mathbb R^d) \) is not only related to the multiwavelets themselves but also to the corresponding refinable function vector. We construct a class of Bessel \(M\)-refinable function vectors with \(M\) being an isotropic dilation matrix, which have high Sobolev smoothness, and the mask symbols of which have high sum rules. Based on the constructed Bessel refinable function vector, an explicit algorithm is given for dual \(M\)-multiframelets in \( (H^s(\mathbb R^d),H^{-s}(\mathbb R^d)) \) with multiframelets in \( H^{-s}(\mathbb R^d) \) having high vanishing moments. On the other hand, based on the dual multiframelets, an algorithm for dual \(M\)-multiframelets with symmetry is given. In Section 6, we give an example to illustrate the construction procedures of dual multiframelets.
MSC:
| 42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |
| 42C15 | General harmonic expansions, frames |
| 94A12 | Signal theory (characterization, reconstruction, filtering, etc.) |
Citations:
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