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The Initial Functions in a Cascade Algorithm

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Bin Han

## Abstract

In this paper we shall study the initial functions in a cascade
algorithm in a Sobolev space. By investigating the mutual relations
among the initial functions in a cascade algorithm, we are able to
study in a relatively unified approach several questions
related to cascade algorithms and subdivision schemes in a Sobolev
space $W_p^k(\RR^s)$ ($1\le p \le \infty, k\in \NN\cup \{0\}$)
such as convergence, error estimate and convergence rate of
cascade algorithms and subdivision schemes in a Sobolev space
with a general isotropic dilation matrix.
The approach in this paper can also be used to study vector
cascade algorithms and vector subdivision schemes with a dilation
matrix in a multidimensional space.

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