Armlets and balanced multiwavelets: flipping filter construction

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)^kp_1-k. However, due to the noncommutativity of matrix multiplication, there is little such development in the multiwavelet literature to express the two-scale matrix sequence {Q_k} of an orthogonal multiwavelet in terms of the two-scale matrix sequence {P_k} of its corresponding scaling function vector. This paper, in part, is devoted to this study for the setting of orthogonal multiwavelets of dimension r=2. In particular, the two lowpass filters are flipping filters, whereas the two highpass filters are linear phase. These results will be applied to constructing both a family of the most recently introduced notion of armlet of order n and a family of n-balanced orthogonal multiwavelets.