Construction of a class of multivariate compactly supported wavelet bases for L2(ℝd)

In this paper, for a given d×d expansive matrix M with |detM| = 2, we investigate the compactly supported M-wavelets for L2(ℝd). Starting with a pair of compactly supported refinable functions ϕ and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\tilde \varphi $\end{document} satisfying a mild condition, we obtain an explicit construction of a compactly supported wavelet ψ such that {2j/2ψ(Mj · −k): j ∈ ℤ, k ∈ ℤd} forms a Riesz basis for L2(ℝd). The (anti-)symmetry of such ψ is studied, and some examples are also provided.