Nonstationary Multiwavelets and Multiwavelet Packets in Sobolev Space Hs(Rd)\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}

Nonstationary multiresolution analysis has different scaling and wavelet functions at different scales. In Bastin and Laubin (Duke Math J 87:481–508, 1997), nonstationary wavelets in Sobolev spaces were introduced. Here we construct a nonstationary multiresolution analysis of multiwavelets for the higher-dimensional Sobolev spaces. We give the construction of nonstationary multiwavelets and derived their orthogonal properties in Sobolev space Hs(Rd)\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}. We perform some splitting trick over nonstationary multiwavelets and multiwavelet spaces to construct nonstationary multiwavelet packet and multiwavelet packet spaces in Sobolev space Hs(Rd)\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}. A nonstationary multiwavelet and multiwavelet packet expansion of Bessel potentials is given as an application of an orthonormal multiwavelet and multiwavelet packet basis in Sobolev space.