Multivariate Matrix Refinable Functions with Arbitrary Matrix Dilation

Characterizations of the stability and orthonormality of a multivariate matrix refinable function Φ 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 Φ is equivalent to the order of the vanishing moment conditions of the matrix refinement mask {Pα}. The restricted transition operator associated with the matrix refinement mask {Pα} is represented by a finite matrix (AMi−j )i,j , with Aj = |det(M)|−1 ∑ κ Pκ−j ⊗Pκ and Pκ−j ⊗Pκ being the Kronecker product of matrices Pκ−j and Pκ. The spectral properties of the transition operator are studied. The Sobolev regularity estimate of a matrix refinable function Φ is given in terms of the spectral radius of the restricted transition operator to an invariant subspace. This estimate is analyzed in an example.