Existence and characterization of orthogonal multiple vector-valued wavelets with three-scale

In this article, we introduce orthogonal multiple vector-valued wavelets with three-scale, which are wavelets for vector fields, based on the notion of full rank subdivision operators. It is demonstrated that, like in the scalar and multiwavelet case, the existence of an orthogonal multiple vector-valued scaling function guarantees the existence of orthogonal multiple vector-valued wavelet functions. In this context, however, scaling functions as well as wavelet functions turn out to be multiple vector-valued functions. A method for constructing a class of orthogonal multiple vector-valued compactly supported wavelets is presented by means of matrix theory. The properties of the multiple vector-valued wavelet packets are characterized by virtue of operator theory and time–frequency analysis method. Three orthogonality formulas concerning these wavelet packets are obtained. Relation to some physical theories such as E-infinity Cantorian spacetime theory and fractal theory is also discussed.