On the regularity of matrix refinable functions

Abstract

It is shown that the transition operator Tp associated with the matrix refinement mask P(ω) = 2-d∑α∈[0, N]dPαexp(-iαω) is equivalent to the matrix (2-dA2i-j)i,j with Aj = ∑κ∈[0, N]dPκ-j⊗Pκ and Pκ-j⊗Pκ denoting the Kronecker product of matrices Pκ-j, Pκ. Some spectral properties of Tp are studied and a complete characterization of the matrix refinable functions in the Sobolev space Wn(Rd) 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.