Article

# Construction of orthonormal multi-wavelets with additional vanishing moments.

Advances in Computational Mathematics (Impact Factor: 1.47). 01/2006; 24:239-262. DOI: 10.1007/s10444-004-7610-7

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). - [Show abstract] [Hide abstract]

**ABSTRACT:**Let P(z) and P ˜(z) be two r×r matrices of Laurent polynomials such that P(z)P ˜(z) * +P(-z)P ˜(-z) * =I r×r . That is, (P,P ˜) is a pair of dual matrix masks with multiplicity r. With the matrices A,A ˜,B,B ˜ of Laurent polynomials satisfying the conditions in Lemma 3.1, define P new (z)=P(z)OA(z)B(z),P ˜ new (z)=P ˜(z)OA ˜(z)B ˜(z)· Then the author shows in Theorem 3.2 that (P new ,P ˜ new ) is a pair of new dual matrix masks with multiplicity r+s. That is, P new (z)P ˜ new (z) * +P new (-z)P ˜ new (-z) * =I (r+s)×(r+s) · Moreover, with a careful choice of A,A ˜,B,B ˜, the author shows in Theorem 5.3 that the approximation order of the new pair (P new ,P ˜ new ) can be higher than the original given pair (P,P ˜). An example is given in Section 6 to illustrate the proposed construction. Without increasing the multiplicity of a given pair (P,P ˜) of dual matrix masks, one can improve their approximation order to an arbitrarily high order by using the CBC (coset by coset) algorithm in [B. Han, J. Approxmation Theory 110, No. 1, 18–53 (2001; Zbl 0986.42020)].The ANZIAM Journal 01/2006; · 1.04 Impact Factor -
##### Article: Hermite-Like 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 d-refinable function vector ϕ=(ϕ 1,…,ϕ r )T is (d,r)-interpolating if $$\phi_{\ell}\biggl(\frac{m}{r}+k\biggr)=\delta_{k}\delta_{\ell-1-m},\quad \forall k\in\mathbb{Z},\ m=0,1,\ldots,r-1,\ \ell=1,\ldots,r.$$ 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 d-refinable function vector ϕ ♮=(ϕ T ,ϕ r+1,…,ϕ 2r )T belongs to (C 1(ℝ))2r and has the Hermite-like 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 Hermite-like interpolating refinable function vectors with symmetry are constructed from ϕ such that they have higher approximation order than it. Several examples of Hermite-like 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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