Characterization of smoothness of multivariate refinable functions and convergence of cascade algorithms of nonhomogeneous refinement equations
A multivariate nonhomogeneous refinement equation takes the form \[ \varphi(x)=\sum_{\alpha \in \mathbb{Z}^s} a(\alpha)\varphi(Mx-\alpha)+g(x), \qquad x\in \mathbb{R}^s, \] where \(\varphi=(\varphi_1, \dots, \varphi_r)^T\) is the unknown refinable function vector, \(M\) is an \(s\times s\) dilation matrix, \(g=(g_1, \dots, g_r)^T\) is a given compactly supported vector-valued function, and \(a\) is a finitely supported refinement mask with multiplicity \(r\). For multivariate homogeneous equations (that is, \(g=0\)), when the dilation matrix \(M\) is isotropic, a characterization of smoothness in the Sobolev space \(W_2^\nu(\mathbb{R}^s)\) of the multivariate refinable function vector \(\varphi\) is presented in Theorem~3.6, which generalizes several results in the literature. A necessary and sufficient condition is also established in Theorem~4.1 for the convergence of the cascade algorithm associated with a nonhomogeneous refinement equation in the Sobolev space \((W_2^k(\mathbb{R}^s))^r\). The characterization in Theorem~4.1 involves the function \(g\) and the initial function vector \(\varphi_0\) in the cascade algorithm as well as the matrix mask \(a\) and the isotropic dilation matrix \(M\).
scientific article; zbMATH DE number 2057573