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Construction of orthonormal multi-wavelets with additional vanishing moments

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Abstract

An iterative scheme for constructing compactly supported orthonormal (o.n.) multi-wavelets with vanishing moments of arbitrarily high order is established. Precisely, let φ=[φ1,. . .,φr]⊤ be an r-dimensional o.n. scaling function vector with polynomial preservation of order (p.p.o.) m, and ψ=[ψ1,. . .,ψr]⊤ an o.n. multi-wavelet corresponding to φ, with two-scale symbols P and Q, respectively. Then a new (r+1)-dimensional o.n. scaling function vector φ ♯:=[φ ⊤,φr+1]⊤ and some corresponding o.n. multi-wavelet ψ ♯ are constructed in such a way that φ ♯ 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 mth order Daubechies o.n. scaling function φ Dm , then φ ♯:=[φ Dm ,φ2]⊤ 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 φ in our earlier work, then we obtain a new pair of scaling function vector φ ♯=[φ ⊤,φ3]⊤ and multi-wavelet ψ ♯ that not only increase the order of vanishing moments but also preserve symmetry.

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Authors and Affiliations

  1. Department of Mathematics & Computer Science, University of Missouri–St. Louis, St. Louis, MO 63121, USA, and Department of Statistics, Stanford University, Stanford, CA, 94305, U.S.A.

    Charles K. Chui

  2. Department of Mathematics, Prairie View A&M University, Prairie View, TX, 77446, U.S.A.

    Jian-ao Lian

Authors
  1. Charles K. Chui
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  2. Jian-ao Lian
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Additional information

Communicated by Yuesheng Xu

Dedicated to Charles A. Micchelli in Honor of His Sixtieth Birthday

Mathematics subject classifications (2000)

42C15, 42C40.

Charles K. Chui: Supported in part by NSF grants CCR-9988289 and CCR-0098331 and Army Research Office under grant DAAD 19-00-1-0512.

Jian-ao Lian: Supported in part by Army Research Office under grant DAAD 19-01-1-0739.

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Chui, C.K., Lian, Ja. Construction of orthonormal multi-wavelets with additional vanishing moments. Adv Comput Math 24, 239–262 (2006). https://doi.org/10.1007/s10444-004-7610-7

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  • DOI: https://doi.org/10.1007/s10444-004-7610-7

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