Subdivision Schemes with General Dilation Matrices

Subdivision schemes are useful for some applications, in particular, for geometric design. They are also closely related to refinable functions, namely, one of the limit functions of a subdivision scheme (which is the image of the delta-sequence) is the refinable function with the same mask. We give a sufficient conditions for the convergence of multivariate scalar subdivision schemes. Our conditions apply to arbitrary dilation matrices, while the previously known necessary and sufficient conditions are relevant only in case of dilation matrices with a self similar tiling. For the analysis of smoothness, we state and prove two theorems on multivariate matrix subdivision schemes, which lead to sufficient conditions for &C ¹ ; limits of scalar multivariate subdivision schemes associated with isotropic dilation matrices. Although similar results are studied in the literature, we give detailed proofs of the results, which we could not find elsewhere.