The research of a class of biorthogonal compactly supported vector-valued wavelets

In this paper, we introduce the biorthogonal vector-valued wavelets. We prove that, like in the scalar wavelet case, the existence of a pair of biorthogonal compactly supported vector-valued scaling functions guarantees the existence of a pair of biorthogonal compactly supported vector-valued wavelet functions. An algorithm for constructing a pair of biorthogonal compactly supported vector-valued wavelet functions is presented by means of vector-valued multiresolution analysis and matrix theory. The notion of biorthogonal vector-valued wavelet packets is introduced, and their properties are investigated by virtue of time-frequency analysis and algebra theory. Three biorthogonality formulas concerning the wavelet packets are established. Relation to some physical theories such as E-infinity Cantorian space-time theory is also discussed.