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Solutions in Sobolev Spaces of Vector Refinement Equations with a
General Dilation Matrix

### Bin Han

## Abstract

In this paper, we present a necessary and sufficient condition for
the existence of solutions in a Sobolev space $W_p^k(\RR^s) (1\le
p \le \infty)$ to a vector refinement equation with a general
dilation matrix. The criterion is constructive and can be
implemented. Rate of convergence of vector cascade algorithms in a
Sobolev space $W_p^k(\RR^s)$ will be investigated. When the
dilation matrix is isotropic, a characterization will be given for
the $L_p (1\le p \le \infty)$ critical smoothness exponent of a
refinable function vector without the assumption of stability on
the refinable function vector. As a consequence, we show that if a
compactly supported function vector $\phi$ in $L_p(\RR^s)$
satisfies a refinement equation with a finitely supported mask,
then all the components of $\phi$ must belong to a Lipschitz space
$\hbox{Lip}(\nu, L_p(\RR^s))$ for some $\nu>0$. This paper
generalizes the results in [R.~Q.~Jia, K.~S.~Lau, and D.~X.~Zhou,
J. Fourier Anal. Appl., {\bf 7} (2001), pp. 143--167] in the
univariate setting to the multivariate setting.

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