ANZIAM  J.  47 (2006), 513-526
An algorithm for constructing biorthogonal multiwavelets with higher approximation orders

Yang Shouzhi
  Department of Mathematics
  Shantou University
  Shantou 515063
  P. R. China

Given a pair of biorthogonal multiscaling functions, we present an algorithm for raising their approximation orders to any desired level. Precisely, let $\Phi(x)$ and ${\tilde{\Phi}}(x)$ be a pair of biorthogonal multiscaling functions of multiplicity r, with approximation orders m and $\tilde{m}$, respectively. Then for some integer s, we can construct a pair of new biorthogonal multiscaling functions $\Phi^{\text{new}}(x)=[\Phi^T(x), \phi_{r+1}(x),\phi_{r+2}(x),\dots,\phi_{r+s}(x)]^T$ and $\tilde{\Phi}^{\text{new}}(x) =[{\tilde{\Phi}}(x)^T,\tilde{\phi}_{r+1}(x),\tilde{\phi}_{r+2}(x),\dots, \tilde{\phi}_{r+s}(x)]^T$ with approximation orders n ($n>m$) and $\tilde{n}$ ($\tilde{n}>\tilde{m}$), respectively. In addition, corresponding to $\Phi^{\text{new}}(x)$ and $\tilde{\Phi}^{\text{new}}(x)$, a biorthogonal multiwavelet pair $\Psi^{\text{new}}(x)$ and $\tilde{\Psi}^{\text{new}}(x)$ is constructed explicitly. Finally, an example is given.
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