Design and characterizations of a class of orthogonal multiple vector-valued wavelets with 4-scale

The notion of vector-valued multiresolution analysis of space L2(R,Cs×s)L2(R,Cs×s) is introduced and the definition of orthogonal multiple vector-valued wavelets with 4-scale is given. First we obtain a necessary and sufficient condition on the existence of orthogonal multiple vector-valued wavelets by means of paraunitary vector filter bank theory. Second we propose an algorithm for constructing a class of compactly supported orthogonal multiple vector-valued wavelets. Finally, the notion of orthogonal multiple vector-valued wavelet packets is introduced. Their characterizations are presented by virtue of matrix theory, time–frequency analysis method and operator theory. In particular, orthonormal bases of space L2(R,Cs×s)L2(R,Cs×s) are constructed from these wavelet packets. Relation to some physical theories such as E-infinity theory is also discussed.