Bessel multiwavelet sequences and dual multiframelets in Sobolev spaces

The dual 2Id-framelets in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ (H^{s}(\mathbb{R}^{d}), H^{-s}(\mathbb{R}^{d})) $\end{document}, s > 0, were introduced by Han and Shen (Constr Approx 29(3):369–406, 2009). In this paper, we systematically study the Bessel property of multiwavelet sequences in Sobolev spaces. The conditions for Bessel multiwavelet sequence in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ H^{-s}(\mathbb{R}^{d}) $\end{document} take great difference from those for Bessel wavelet sequence in this space. Precisely, the Bessel property of multiwavelet sequence in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ H^{-s}(\mathbb{R}^{d}) $\end{document} is not only related to multiwavelets themselves but also to the corresponding refinable function vector. We construct a class of Bessel M-refinable function vectors with M being an isotropic dilation matrix, which have high Sobolev smoothness, and of which the mask symbols have high sum rules. Based on the constructed Bessel refinable function vector, an explicit algorithm is given for dual M-multiframelets in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ (H^{s}(\mathbb{R}^{d}),H^{-s}(\mathbb{R}^{d})) $\end{document} with the multiframelets in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ H^{-s}(\mathbb{R}^{d}) $\end{document} 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 constructing procedures of dual multiframelets.