In this article the authors construct considarably simpler nonseparable examples of Haar-type wavelets in higher dimensions which are convenient for applications. If \(c \in \text{GL}_{d}({\mathbb R})\), the dilation \(D_{c}\) is the operator that maps a function \(f\) on \({\mathbb R}^d\) to the function \((D_{c}f)(x)=|\det c|^{-1/2}f(c^{-1}x)\). Let \(\Gamma\) be a lattice in \({\mathbb R}^d\), that is, \(\Gamma\) is an image of the integer lattice \({\mathbb Z}^d\) under some matrix in \(\text{GL}_{d}({\mathbb R})\). The translation by \(\gamma \in \Gamma\) is the operator \(T_{\gamma}\) that maps a function \(f \in {\mathbb R}^d\) to the function \((T_{\gamma}f)(x) =f(x-\gamma)\). Let \(B\) be a finite group in \(\text{GL}_{d}({\mathbb R})\) with \(|\det b|=1\) for all \(b \in B\) and \(B(\Gamma)= \Gamma\). A multiresolution analysis associated with a set of dilations \(A=\{ a^j \}_{j \in {\mathbb Z}}\) and the group \(B\) is an increasing collection \(\{ V_{j} \}_{j \in {\mathbb Z}}\) of closed subspaces of \(L^2({\mathbb R}^d)\) that satisfies the conditions:
(M1) \(V_{j}=D_{a^{-j}}V_{0}\);
(M2) \(\overline{\bigcup_{j \in {\mathbb Z}}V_{j}} =L^2({\mathbb R}^d) \);
(M3) \(\bigcap_{j \in {\mathbb Z}}V_{j} = \{ 0 \}\);
(M4) There exists \(\varphi \in V_{0}\) such that \(\{ D_{b}T_{\gamma}\varphi \}, \;b \in B\) and \(\gamma \in \Gamma\), is an orthonormal basis for \(V_{0}\).
The purpose of this article is to present the construction of several examples of compactly supported multiwavelets \(\Psi =\{ \psi^1, \psi^2, \dots, \psi^L \}\) such that the affine system \(\{ D_{a^j}D_{b}T_{\gamma}\psi^l: j \in {\mathbb Z}\), \(b \in B\), \(\gamma \in \Gamma\), \(l=1,2, \dots, L \}\) is an orthnormal basis for \(L^2({\mathbb R}^d)\). Examples are constructed for several finite groups \(B\) and the goal includes the requirement that each \(\psi^l\) is a finite linear combination of characteristic functions of very simple sets (for example, triangles). Consequently, these are good candidates for applications.