Stability analysis of biorthogonal multiwavelets whose duals are not in \(L_2\) and its application to local semiorthogonal lifting
The authors investigate the Riesz stability of compactly supported (multi-)wavelet bases that are constructed by a lifting scheme on regularly refined meshes of arbitrary topological type. The authors are interested in the Riesz stability of the two-step lifted wavelet transform consisting of one prediction step and one update step. The prediction step is based on subdivision rules. The update step can be described as local semiorthogonalization. In order to compute the exact range of Sobolev exponents \(s\) for which the wavelets form a Riesz basis of the Sobolev space \(H^s({\mathbb R}^d)\), one needs to determine the smoothness of the dual wavelet system. It might occur that the duals do not exist in \(L_2({\mathbb R}^d)\). By Fourier techniques, the range of corresponding Sobolev exponents is estimated. Examples of uni- and bivariate wavelet systems 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 in \(L_2({\mathbb R}^d)\).
scientific article; zbMATH DE number 5300753
scientific article