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 phi(s)(x) = Sigma(k is an element of Zn)h(k)(s) root\det A\phi(1)(Ax-k) for s = 1,...,q, where phi(1) is a scaling function for this multiresolution and phi(2),...,phi(q) (q = \det A\)are wavelets. Orthogonality conditions for phi(1),...,phi(q) naturally impose constraints on the scaling coefficients {h(k)(s)}(k is an element of Zn)(s=1,...,q), which are then called the wavelet matric. We show how to reconstruct functions satisfying the scaling equations above and show that phi(2),...,phi(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.