On the Regularity of Matrix Refinable Functions

Abstract

It is shown that the transition operator $\T$ associated with the matrix refinement mask $\p(\go )=2^{-d}\Sigma_{\alpha \in [0,N]^d} \p_{\alpha}\hbox{exp}(-i\alpha \go )$ is equivalent to the matrix $(2^{-d}\a _{2i-j})_{i, j}$ with $\a _j=\Sigma_{\gk \in [0, N]^d}\p _{\gk -j}\otimes \p_{\gk}$ and $\p _{\gk -j}\otimes \p_{\gk}$ denoting the Kronecker product of matrices $\p _{\gk -j}$, $\p_{\gk}$. Some spectral properties of T are studied and a complete characterization of the matrix refinable functions in the Sobolev space $W^n(\RR^d)$ 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.