Tight frames of multidimensional wavelets

In this paper we deal with multidimensional wavelets arising from a multiresolution analysis with an arbitrary dilation matrix A, namely we have scaling equations $$\varphi ^s (x) = \sum\limits_{k \in \mathbb{Z}^n } {h_k^s \sqrt {|\det A|} \varphi ^1 } (Ax - k) for s = 1, \ldots ,q,$$ where ?¹ is a scaling function for this multiresolution and ?², …, ?^q (q=|det A |) are wavelets. Orthogonality conditions for ?¹, …, ?^q naturally impose constraints on the scaling coefficients $\{ h_k^s \} _{k \in \mathbb{Z}^n }^{s = 1, \ldots ,q} $ , which are then called the wavelet matrix. We show how to reconstruct functions satisfying the scaling equations above and show that ?², …, ?^q always constitute a tight frame with constant 1. Furthermore, we generalize the sufficient and necessary conditions of orthogonality given by Lawton and Cohen to the case of several dimensions and arbitrary dilation matrix A.