Armlets and balanced multi-wavelets

In the scalar-valued setting, it is well-known that the two-scale sequences {q(k)} of Daubechies orthogonal wavelets can be given explicitly by the two-scale sequences {p(k)} of their corresponding orthogonal scaling functions, such as q(k) = (-1)(k)p(1-k). However, due to the non-commutativity of matrix multiplication, there is little such development in the multi-wavelet literature to express the two-scale matrix sequence {Q(k)} of an orthogonal multi-wavelet in terms of the two-scale matrix sequence {P-k} of its corresponding multi-scaling function vector. This paper, in part, is devoted to this study for the setting of orthogonal multi-wavelets of dimension r = 2. We will apply our results to constructing a family of the most recently introduced notion of armlet of order n and a family of the n-balanced orthogonal multi-wavelets.