Accuracy conditions of multivariate refinable vector in frequence domain

Assume that Phi (x) = (Phi (1)(x),...,phi (r)(x))(inverted perpendicular), x is an element of R-d is a vector-valued function satisfying the refinemen t equation Phi (x) = Sigma (k is an element of Lambda) c(k)Phi (Ax - k) for some finite set Lambda of Z(d) and some r x r matrices ck. In [Ij, the req uirements for Phi to have accuracy p are given in terms of some complicated matrix generated by mask. But how to characterize accuracy via symbol is a n unsolved question in [I]. In this paper, we give accuracy conditions in f requency domain, i.e., conditions (1.19) and (1.20). By using accuracy cond itions (1.19) and (1.20), we construct the superfunction which is very impo rtant 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 dilatio n (i.e., (1.20) is redundant). (C) 2001 Elsevier Science Ltd. All rights re served.