Construction of orthonormal multi-wavelets with additional vanishing moments

Department of Statistics, Stanford University, Stanford, California, United States
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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