Constructing Orthogonal Refinable Function Vectorswith Prescribed Approximation

We review and announce results from 6] and 7] and demonstrate their utility in constructing orthogonal reenable function vectors with prescribed approximation order and smoothness. In the case of local dimension 3 and approximation order 2, our approach naturally produces the family of orthogonal multiwavelets originally found in 2] as well as Daubechies' well known scaling function 2. In Section 5 we use our techniques to construct an orthogonal reenable function vector with two entries, one supported on 0;2] and one supported on 0;3], that is continuously diierential and provides approximation order 3.