The characterization of vector-valued multivariate wavelet packets associated with a dilation matrix

In this work, we introduce the notion of vector-valued multiresolution analysis and vectorvalued multivariate wavelet packets associated with an arbitrary integer-valued dilation matrix. A novel method for constructing higher-dimensional vector-valued wavelet packet is presented. Their characteristics are researched by means of operator theory, time-frequency analysis method and matrix theory. Three orthogonality formulas regarding the wavelet packets are provided. Orthogonality decomposition relation formulas of the space L2(Rs)r are obtained by constructing a series of subspaces of the vector-valued wavelet packets. Finally, several orthonormal wavelet packet bases of L2(Rs)r are constructed from these wavelet packets. Relation to some physical theories such as fractal theory and Einfinity theory is also discussed.