Accuracy conditions of multivariate refinable vector in frequence domain

Assume that Ф(x) = (ф1(x),…,фr(x)) , x Rd is a vector-valued function satisfying the refinement equation Ф(x) = ΣkεΛ ckФ(Ax − k) for some finite set Λ of Zd and some r × r matrices ck. In [1], the requirements for Φ to have accuracy p are given in terms of some complicated matrix generated by mask. But how to characterize accuracy via symbol is an unsolved question in [1]. In this paper, we give accuracy conditions in frequency domain, i.e., conditions (1.19) and (1.20). By using accuracy conditions (1.19) and (1.20), we construct the superfunction which is very important to characterize approximation of shift-invariant space. Furthermore, we prove that (see Section 3) condition (1.19) is sufficient for refinable smooth functions to have accuracy p in the case of isotropic matrix dilation (i.e., (1.20) is redundant).