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Characterizations of the stability and orthonormality of a multi- variate matrix renable function with arbitrary matrix dilation M are pro- vided in terms of the eigenvalue and 1-eigenvector properties of the restricted transition operator. Under mild conditions, it is shown that the approximation order of is equivalent to the order of the vanishing moment conditions of the matrix renement mask fPg. The restricted transition operator associ- ated with the matrix renement mask fPg is represented by a nite matrix (AMi j)i;j ,w ith A j = jdet(M)j 1 P P j P and P j P being the Kronecker product of matrices P j and P. The spectral properties of the transition operator are studied. The Sobolev regularity estimate of a matrix renable function is given in terms of the spectral radius of the restricted transition operator to an invariant subspace. This estimate is analyzed in an example.

Multivariate matrix refinable functions with arbitrary matrix dilation