  We study multivariate trigonometric polynomials, satisfying a set of
constraints close to the known Strung-Fix conditions. Based on the polyphase
representation of these polynomials relative to a general dilation matrix, we
develop a simple constructive method for a special type of decomposition of
such polynomials. These decompositions are of interest to the analysis of
convergence and smoothness of multivariate subdivision schemes associated with
general dilation matrices. We apply these decompositions, by verifying
sufficient conditions for the convergence and smoothness of multivariate scalar
subdivision schemes, proved here. For the convergence analysis our sufficient
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 stated in the literature, we give here detailed proofs of the results,
which we could not find elsewhere.
