Construction of orthonormal multi-wavelets with additional vanishing moments

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


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. "
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    01/2006; 46(2006).
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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
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