## Journal of Mathematics of Kyoto University

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

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

#### 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