The investigation into vector-valued poly-scale composite wavelet packets

In this paper, the notion of vector-valued multiresolution analysis is introduced. An approach for constructing biorthogonal vector-valued wavelet packs in higher dimensions is proposed and their properties are investigated by means of time frequency analysis method, matrix theory and operator theory. Three biorthogonality formulas concerning these wavelet packs are obtained. Finally, new Riesz bases of three dimensional vector-valued function space are obtained by designing a series of subspaces of biorthogonal matrix wavelet packs. The sufficient condition for the existence of multiple pseudoframes for subspaces of L²(R) is derived based on such a generalized multiresolution structure. The pyramid decomposition scheme is also obtained.