Journal of Mathematics of Kyoto University

Construction of orthogonal multiscaling functions and multiwavelets with higher approximation order based on the matrix extension algorithm

Shouzhi Yang and Zengjian Lou

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Abstract

An algorithm is presented for constructing orthogonal multiscaling functions and multiwavelets with higher approximation order in terms of any given orthogonal multiscaling functions. That is, let $\Phi (x) = [\phi _{1}(x), \phi _{2}(x),\ldots , \phi _{r}(x)]^{T} \in (L^{2}(R))^{r}$ be an orthogonal multiscaling function with multiplicity $r$ and approximation order $m$. We can construct a new orthogonal multiscaling function $\Phi ^{new}(x) = [\Phi ^{T} (x), \phi _{r+1}(x), \phi _{r+2}(x),\ldots ,\phi _{r+s}(x)]^{T}$ with approximation order $n(n > m)$. Namely, we raise approximation order of a given multiscaling function by increasing its multiplicity. Corresponding to the new orthogonal multiscaling function $\Phi ^{new}(x)$, orthogonal multiwavelet $\Psi ^{new}(x)$ is constructed. In particular, the spacial case that $r = s$ is discussed. Finally, we give an example illustrating how to use our method to construct an orthogonal multiscaling function with higher approximation order and its corresponding multiwavelet.

Article information

Source
J. Math. Kyoto Univ., Volume 46, Number 2 (2006), 275-290.

Dates
First available in Project Euclid: 14 August 2009

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1250281777

Digital Object Identifier
doi:10.1215/kjm/1250281777

Mathematical Reviews number (MathSciNet)
MR2284344

Zentralblatt MATH identifier
1118.42013

Subjects
Primary: 42C40: Wavelets and other special systems
Secondary: 65T60: Wavelets

Citation

Yang, Shouzhi; Lou, Zengjian. Construction of orthogonal multiscaling functions and multiwavelets with higher approximation order based on the matrix extension algorithm. J. Math. Kyoto Univ. 46 (2006), no. 2, 275--290. doi:10.1215/kjm/1250281777. https://projecteuclid.org/euclid.kjm/1250281777


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