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 $\{\mathbf{P}_{\alpha}\}$. The restricted transition operator associated with the matrix refinement mask $\{\mathbf{P}_{\alpha}\}$ is represented by a finite matrix $({\mathcal A} _{Mi-j})_{i, j}$, with ${\mathcal A} _j=|\hbox{det$(M)$}|^{-1}\sum _{\kappa }\mathbf{P}_{\kappa -j}\otimes \mathbf{P}_{\kappa }$ and $\mathbf{P}_{\kappa -j}\otimes \mathbf{P}_{\kappa }$ being the Kronecker product of matrices $\mathbf{P}_{\kappa -j}$ and $\mathbf{P}_{\kappa }$. 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.