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Projectable multivariate refinable functions and biorthogonal wavelets

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Bin Han

## Abstract

A biorthogonal wavelet is derived from a pair of biorthogonal
refinable functions using the standard technique in multiresolution
analysis. In this paper, we introduce the concept of projectable
refinable functions and demonstrate that many multivariate
refinable functions are projectable; that is, they
essentially carry the tensor product (separable) structure though
themselves may be non-tensor product (nonseparable) refinable
functions. For any pair of biorthogonal refinable functions ($\phi,
\phi^d$) in $L_2(\RR^s)$, when the refinable function $\phi$ is
projectable, we prove
that without loss of several desirable properties such as spatial
localization, smoothness and approximation order, from the pair of
biorthogonal refinable functions ($\phi, \phi^d$), we can easily
obtain another pair of biorthogonal refinable functions in
$L_2(\RR^s)$ which are tensor product separable refinable functions.
As an application, we show that there is no dual refinable function
$\phi^d$ to the refinable basis function in the Loop scheme such that
$\phi^d$ can be supported on $[-4,4]^2$.

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