On the regularity of matrix refinable functions

It is shown that the transition operator T-P associated with the matrix refinement mask P(omega) = 2(-d)Sigma(alpha is an element of[0,N]d)P(alpha)exp(-i alpha omega) is equivalent to the matrix (2(-d) A(2i-j))i,j with A(j) = Sigma(kappa is an element of[0,N]d)P(kappa-j) x P-kappa denoting the Kronecker products of matrices P (kappa-j), P-kappa. Some spectral properties of T-P are studied and a complete characterization of the matrix refinable functions in the Sobolev space W-n (R-d) for nonnegative integers n is provided. The Sobolev regularity estimate of the matrix refinable function is given in terms of the spectral radius of a restricted transition operator. These estimates are analyzed in some examples.