Design and properties of vector-valued wavelets associated with an orthogonal vector-valued scaling function

The notion of vector-valued wavelets associated with an orthogonal vector-valued scaling function with 3-scale is introduced. The existence of orthogonal vector-valued wavelets is discussed and a necessary and sufficient condition is presented by means of the theory of vector-valued multiresolution analysis and paraunitary vector filter bank theory. An algorithm for constructing a class of orthogonal vector-valued wavelets with compact support is proposed. The concept of vector-valued wavelet packets is introduced and their properties are characterized by virtue of operator theory, time-frequency analysis method. Moreover, it is shown how to construct various orthonormal bases of space L2(R,Cμ)(2≤μ element of Z) from these wavelet packets. Relation to some physical theories such as fractal theory and E-infinity theory is also discussed.