## 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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