Scaling sets and orthonormal wavelets with dilations induced by expanding matrices

The paper studies orthonormal wavelets in L2(Rn) with dilations induced by expanding integer matrices of arbitrary determinant. We provide a method for construction of all scaling sets and, hence, of all orthonormal MSF wavelets with the additional property that the core space of the underlying multiresolution structure is singly generated. Several examples on the real line and in R2 are included. We also prove that all MSF orthonormal wavelets whose dimension function is essentially bounded by 1 are obtained by our construction method. Finally, we derive a description of all wavelets (not necessarily MSF ones) that arise from a single scaling function in terms of the underlying multiresolution structure