Convergence of cascade algorithms associated with nonhomogeneous refinement equations

This paper is devoted to a study of multivariate nonhomogeneous refinement equations of the form where $\phi = (\phi_1,\ldots,\phi_r)^T$ is the unknown, $g = (g_1,\ldots,g_r)^T$ is a given vector of functions on $\mathbb{R}^s$, $M$ is an $s \times s$ dilation matrix, and $a$ is a finitely supported refinement mask such that each $a(\alpha)$ is an $r \times r$ (complex) matrix. Let $\phi_0$ be an initial vector in $(L_2(\mathbb{R}^s))^r$. The corresponding cascade algorithm is given by In this paper we give a complete characterization for the $L_2$-convergence of the cascade algorithm in terms of the refinement mask $a$, the nonhomogeneous term $g$, and the initial vector of functions $\phi_0$.