Multivariate matrix refinable functions with arbitrary matrix dilation

Characterizations of the stability and orthonormality of a multivariate matrix refinable function Phi with arbitrary matrix dilation M are provided 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 Phi is equivalent to the order of the vanishing moment conditions of the matrix refinement mask {P-alpha}. The restricted transition operator associated with the matrix refinement mask {P-alpha} is represented by a finite matrix (A (Mi-j))(i,j), with A( j) = \det(M)\(-1) Sigma(k) Pk- j x P-k and Pk-j x P-k being the Kronecker product of matrices Pk-j and P-k. The spectral properties of the transition operator are studied. The Sobolev regularity estimate of a matrix refinable function Phi is given in terms of the spectral radius of the restricted transition operator to an invariant subspace. This estimate is analyzed in an example.