Stability and Independence of the Shifts of a Multivariate Refinable Function

Due to their so-called time-frequency localization properties , wavelets have become a powerful tool in signal analysis and image processing. Typical constructions of wavelets depend on the stability of the shifts of an underlying reenable function 2 L 2 l R d. In this paper, we derive necessary and suucient conditions for the stability o f the shifts of certain compactly supported reenable functions. These conditions are in terms of the zeros of the reenement mask. We also provide a similar characterization of the global linear independence of the shifts. x1 I n troduction In this paper we present a c haracterization of the stability and linear independence of the shifts of a compactly supported reenable function in terms of the reenement mask. Our results are applicable to a large class of mul-tivariate functions which includes but is not limited to tensor products and box splines. A function 2 L p l R d is said to havè p-stable shifts if there exist positive constants C and D such that Ckak`p k X 2Z Z d a , k L p Dkak`p for all a 2 ` p Z Z d ; C j j it is often said that provides a Riesz basis in L p in this case. A compactly supported function is said to have linearly independent shifts if the map