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In this article, the notion of orthogonal nonseparable four-dimensional wavelet packs, which is the generalization of orthogonal univariate wavelet packs, is introduced. A new approach for constructing them is presented by iteration method and wavelets as well wavelet frames. The biorthogonality properties of four-dimensi- -onal wavelet packets are discussed. Three biorthogonality formulas concerning these wavelet packs are estabished. A necessary and sufficient condition for the existence of the pyramid decomposition scheme of space is presented.

Wavelet analysis is nowadays a widely used tool in applied mathematics. The advantages of wavelet wraps and their promising features in various application have attracted a lot of interest and effort in recent years. In this paper, we develop the notion of a sort of multivariate vector wavelet wraps. A new approcah for designing the multidimensional vector wavelet wraps is formul- ated. Their characters are investigated by virtue of iterative method, time-frequency analysis metho- d and matrix theory. There biorthogonality formulas regarding the wavelet wraps are provided. Biorthogonality decomposition relation formulas of the space L2(Rn)r are obtained by constructing a series of subspaces of the vector-valued wavelet wraps. Moreover, several Riesz bases of space L2(Rn)r are constructed from the wavelet wraps.

Wavelet analysis is a popular subject in science research. The notion of univariate orthog- -onal wavelet packs is generalized. First, the notion of biorthogonal nonseparable trivariate wavelet packs is proposed and a procedure for constructing them is presented. Next, the biorthogonality pro- -perty of trivariate wavelet packs is investigated, and three biorthogonality formulas of wavelet packs are established. Moreover, it is shown how to gain new Riesz bases of space from these wavelet packs.

In this paper, we introduce a sort of vector four-dimensional wavelet packets according to a dilation matrix, which are generalizations of univariate wavelet packets. The definition of biortho- gonal vector four-dimensional wavelet packets is provided and their biorthogonality quality is researched by means of time-frequency analysis method, vector subdivision scheme and functional analysis method. Three biorthogonality formulas regarding the wavelet packets are established. Finally, it is shown how to draw new Riesz bases of space from these wavelet packets. The sufficient condition for the existence of four-dimensional wavelet packets is established based on the multiresolution analysis method.