Article

Construction of orthonormal multi-wavelets with additional vanishing moments

(Impact Factor: 1.49). 01/2006; 24(1-4):239-262. DOI: 10.1007/s10444-004-7610-7
Source: DBLP

ABSTRACT

An iterative scheme for constructing compactly supported orthonormal (o. n. ) multi-wavelets with vanishing moments of arbitrarily high order is established. Precisely, let f = [f1, ···,fr], be an r-dimensional o. n. scaling function vector with polynomial preservation of order (p. p. o. ) m, and = [1, ·· ·, r], an o. n. multi-wavelet corresponding to f, with two-scale symbols P and Q, respectively. Then a new (r + 1)-dimensional o. n. scaling function vector f, := [f,,fr+1], and some corresponding o. n. multi-wavelet, are constructed in such a way that f, has p. p. o. = n > m and their two-scale symbols P, and Q, are lower and upper triangular block matrices, respectively, without increasing the size of the supports. For instance, for r = 1, if we consider the m, order Daubechies o. n. scaling function f,m, then f, := [f,m,f2], is a scaling function vector with p. p. o. > m. As another example, for r = 2, if we use the symmetric o. n. scaling function vector f in our earlier work [3], then we obtain a new pair of scaling function vector f, = [f,, f3], and multi-wavelet, that not only increase the order of vanishing moments but also preserve symmetry.

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Available from: Jian-ao Lian, Oct 27, 2015
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• "The properties of multiscaling functions and multiwavelets are discussed in many papers . Ashino, Nagase and Vaillancourt [14], Cohen, Daubechies and Plonka [15] Plonka and Strela [16], Strela [17], Shen [18], Keinert [19], Chui and Lian [20], Lian [21] and many others, have obtained important results on the existence, regularity, orthogonality, approximation order and symmetry of multiwavelets. "
Article: Construction of orthogonal multiscaling functions and multiwavelets with higher approximation order based on the matrix extension algorithm
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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.
01/2006; 46(2006).
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Article: An algorithm for constructing biorthogonal multiwavelets with higher approximation orders
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ABSTRACT: Given a pair of biorthogonal multiscaling functions, we present an algorithm for raising their approximation orders to any desired level. Precisely, let φ(χ) and φ̃(χ) be a pair of biorthogonal multiscaling functions of multiplicity r, with approximation orders m and m̃, respectively. Then for some integer s, we can construct a pair of new biorthogonal multiscaling functions φnew(χ) = [φT(χ), φr+1(χ), φr+2(χ),..., φr+s(χ)]T and φ̃new (χ) = [φ̃(χ)T, φ̃r+1 (χ), φ̃r+2(χ),..., φr+s (χ)]]T with approximation orders n (n > m) and ñ (ñ > m̃), respectively. In addition, corresponding to φnew (χ) and φ̃new (χ), a biorthogonal multiwavelet pair ψnew (χ) and ψ̃new (χ) is constructed explicitly. Finally, an example is given.
The ANZIAM Journal 04/2006; 47(4). DOI:10.1017/S1446181100010105 · 1.03 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 06/2012; 18(3). DOI:10.1007/s00041-011-9208-z · 1.12 Impact Factor