Construction of nonseparable orthonormal compactly supported wavelet bases for L2(ℝd)

Suppose M and N are two r × r and s × s dilation matrices, respectively. Let ΓM and ΓN represent the complete sets of representatives of distinct cosets of the quotient groups M−T ℤr/ℤr and N−Tℤs/ℤs, respectively. Two methods for constructing nonseparable Ω-filter banks from M-filter banks and N-filter banks are presented, where Ω is a (r + s) × (r + s) dilation matrix such that one of its complete sets of representatives of distinct cosets of the quotient groups Ω−T ℤr+s/ℤr+s are ΓΩ = {[γhT, ζqT]T: γh ∈ ΓM, ζq ∈ ΓN}. Specially, Ω can be \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left[ {\begin{array}{*{20}c} {M\Theta } \\ {0N} \\ \end{array} } \right] $$\end{document}, where Θ is a r × s integer matrix with M−1Θ being also an integer matrix. Moreover, if the constructed filter bank satisfies Lawton’s condition, which can be easy to verify, then it generates an orthonormal nonseparable Ω-wavelet basis for L2(ℝr+s). Properties, including Lawton’s condition, vanishing moments and regularity of the new Ω-filter banks or new Ω-wavelet basis are discussed then. Finally, a class of nonseparable Ω-wavelet basis for L2(ℝr+1) are constructed and three other examples are given to illustrate the results. In particular, when M = N = 2, all results obtained in this paper appeared in [1].