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Vector Cascade Algorithms and Refinable Function Vectors in Sobolev Spaces
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Bin Han

## Abstract

In this paper we shall study vector cascade algorithms and refinable
function vectors with a general isotropic dilation matrix in Sobolev
spaces. By investigating several properties of the initial function
vectors in a vector cascade algorithm, we are able to take a
relatively unified approach to study several questions such as
convergence, rate of convergence
and error estimate for a perturbed mask
of a vector cascade algorithm in a Sobolev space $W_p^k(\RR^s)
(1\le p\le \infty, k\in \NN\cup \{0\})$. We shall characterize the
convergence of a vector cascade algorithm in a Sobolev space in
various ways. As a consequence, a
simple characterization for refinable Hermite interpolants and a sharp
error estimate for a perturbed mask of a vector cascade algorithm in a
Sobolev space will be presented. The approach in this paper enables us
to answer some unsolved questions in the literature on vector cascade
algorithms and to comprehensively generalize and improve results on
scalar cascade algorithms and scalar refinable functions to the vector
case.

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