Article
Construction of orthonormal multiwavelets with additional vanishing moments.
Advances in Computational Mathematics (Impact Factor: 1.47). 01/2006; 24:239262. DOI: 10.1007/s1044400476107
Source: DBLP
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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.Journal of Mathematics of Kyoto University  J MATH KYOTO UNIV. 01/2006; 46(2006).  The ANZIAM Journal 01/2006; · 1.04 Impact Factor

Article: HermiteLike Interpolating Refinable Function Vector and Its Application in Signal Recovering
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ABSTRACT: Interpolating refinable function vectors with compact support are of interest in applications such as sampling theory, numerical algorithm, and signal processing. Han et al. (J. Comput. Appl. Math. 227:254–270, 2009), constructed a class of compactly supported refinable function vectors with (d,r)interpolating property. A continuous drefinable function vector ϕ=(ϕ 1,…,ϕ r )T is (d,r)interpolating if $$\phi_{\ell}\biggl(\frac{m}{r}+k\biggr)=\delta_{k}\delta_{\ell1m},\quad \forall k\in\mathbb{Z},\ m=0,1,\ldots,r1,\ \ell=1,\ldots,r.$$ In this paper, based on the (d,r)interpolating refinable function vector ϕ∈(C 1(ℝ))r , we shall construct r functions ϕ r+1,…,ϕ 2r such that the new drefinable function vector ϕ ♮=(ϕ T ,ϕ r+1,…,ϕ 2r )T belongs to (C 1(ℝ))2r and has the Hermitelike interpolating property: Then any function f∈C 1(ℝ) can be interpolated and approximated by That is, $\widetilde{f}^{(\kappa)}(k+\frac{m}{r})=f^{(\kappa)}(k+\frac{m}{r})$ , ∀κ∈{0,1}, ∀k∈ℤ, and m=0,1,…,r−1. When ϕ has symmetry, it is proved that so does ϕ ♮ by appropriately selecting some parameters. Moreover, we address the approximation order of ϕ ♮. A class of Hermitelike interpolating refinable function vectors with symmetry are constructed from ϕ such that they have higher approximation order than it. Several examples of Hermitelike interpolating refinable function vectors are given to illustrate our results. The truncated error estimate of the interpolating series above is given in Sect. 3. A numerical example of recovering signal is given in Sect. 5 to check the efficiency of the interpolating formula above.Journal of Fourier Analysis and Applications 18(3). · 1.08 Impact Factor
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