Decomposable pairs and construction of multiwavelets

Abstract

We study the construction of refinable function vectors and multiwavelets from the perspective of decomposable pairs. There exists a pair of matrices known as the decomposable pair, associated with a refinement mask of the form H(ξ)=12∑k=0lHke−iξk, which gives the spectral information about H(ξ). We show that the existence of a solution to a matrix refinement equation is related to the dimension of 1-eigenspace of the square matrix in the decomposable pair. We also show that the existence of a refinement mask with sum rules of order 1 is related to the dimension of (−1)-eigenspace of the square matrix in the decomposable pair. A systematic procedure is developed to create refinement masks satisfying the sum rules of order 1 using decomposable pairs, such that the corresponding matrix refinement equation has a solution. Several examples are provided to illustrate the theoretical results obtained.