This paper is concerned with refinement equations of the type
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where f is the unknown function defined on the s-dimensional Euclidean spac
e R-s, a is a finitely supported sequence on Z(s), and M is an s x s dilati
on matrix with m := \det M\. The solution of a refinement equation can be o
btained by using the subdivision scheme associated with the mask. In this p
aper we give a characterization for the convergence of the subdivision sche
me when the mask is nonnegative. Our method is to relate the problem of con
vergence to m column-stochastic matrices induced by the mask. In this way,
the convergence of the subdivision scheme can be determined in a finite num
ber of steps by checking whether each finite product of those column-stocha
stic matrices has a positive row. As a consequence of our characterization,
we show that the convergence of the subdivision scheme with a nonnegative
mask depends only on the location of its positive coefficients. Several exa
mples are provided to demonstrate the power and applicability of our approa
ch.