Tight frames of multidimensional wavelets

Abstract

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 ϕ1 is a scaling function for this multiresolution and ϕ2, …, ϕq (q=|det A |) are wavelets. Orthogonality conditions for ϕ1, …, ϕq 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 ϕ2, …, ϕq 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.