Convergence of cascade algorithms associated with nonhomogeneous refinement equations

Jia, Rong-Qing; Jiang, Qingtang; Shen, Zuowei

doi:10.1090/S0002-9939-00-05567-2

Convergence of cascade algorithms associated with nonhomogeneous refinement equations
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by Rong-Qing Jia, Qingtang Jiang and Zuowei Shen PDF

Proc. Amer. Math. Soc. 129 (2001), 415-427 Request permission

Abstract:

This paper is devoted to a study of multivariate nonhomogeneous refinement equations of the form \begin{equation*} \phi (x) = g(x) + \sum _{\alpha \in \mathbb {Z}^s} a(\alpha ) \phi (Mx-\alpha ), \qquad x \in \mathbb {R}^s, \end{equation*} 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 \begin{equation*} \phi _k := g + \sum _{\alpha \in \mathbb {Z}^s} a(\alpha ) \phi _{k-1}({M \cdot } - \alpha ), \qquad k=1,2,\ldots . \end{equation*} 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 $\phi _0$.

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