Stability analysis of biorthogonal multiwavelets whose duals are not in L2 and its application to local semiorthogonal lifting

Wavelets have been used in a broad range of applications such as image processing, computer graphics and numerical analysis. The lifting scheme provides an easy way to construct wavelet bases on meshes of arbitrary topological type. In this paper we shall investigate the Riesz stability of compactly supported (multi-) wavelet bases that are constructed with the lifting scheme on regularly refined meshes of arbitrary topological type. More particularly we are interested in the Riesz stability of a standard two-step lifted wavelet transform consisting of one prediction step and one update step. The design of the update step is based on stability considerations and can be described as local semiorthogonalization, which is the approach of Lounsbery et al. in their groundbreaking paper [26]. Riesz stability is important for several wavelet based numerical algorithms such as compression or Galerkin discretization of variational elliptic problems. In order to compute the exact range of Sobolev exponents for which the wavelets form a Riesz basis one needs to determine the smoothness of the dual system. It might occur that the duals, that are only defined through a refinement relation, do not exist in L2. By using Fourier techniques we can estimate the range of Sobolev exponents for which the wavelet basis forms a Riesz basis without explicitly using the dual functions. Several examples in one and two dimensions are presented. These examples show that the lifted wavelets are a Riesz basis for a larger range of Sobolev exponents than the corresponding non-updated hierarchical bases but, in general, they do not form a Riesz basis of L2.