Multivariate matrix refinable functions with arbitrary matrix dilation

Characterizations of the stability and orthonormality of a multivariate matrix refinable function P h i Phi with arbitrary matrix dilation M M are provided in terms of the eigenvalue and 1 1 -eigenvector properties of the restricted transition operator. Under mild conditions, it is shown that the approximation order of Φ Φ is equivalent to the order of the vanishing moment conditions of the matrix refinement mask { P α } \{Pα\} . The restricted transition operator associated with the matrix refinement mask { P α } \{Pα\} is represented by a finite matrix ( A M i − j ) i , j ({ A} Mi-j)i, j , with A j = | det ⁡ ( M ) | − 1 ∑ κ