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Construction of orthonormal multi-wavelets with additional vanishing moments. (English) Zbl 1115.65133
In this interesting paper, an iterative scheme for constructing compactly supported orthonormal multiwavelets with vanishing moments of arbitrarily high order is established. Precisely, let ϕ=(ϕ 1 ,...,ϕ r ) T be an r-dimensional orthonormal scaling function vector with polynomial preservation of order m, and ψ=(ψ 1 ,...,ψ r ) T an orthonormal multiwavelet corresponding to ϕ, with two-scale symbols P and Q, respectively. Then a new (r+1)-dimensional orthonormal scaling function vector ϕ * =(ϕ T ,ϕ r+1 ) T and some corresponding orthonormal multiwavelet ψ * are constructed in such a way that ϕ * has polynomial preservation of order 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. This method is illustrated by some examples.
MSC:
65T60Wavelets (numerical methods)
42C40Wavelets and other special systems
42C15General harmonic expansions, frames
References:
[1]B.K. Alpert, Wavelets and other bases for fast numerical linear algebra, in: Wavelets: A Tutorial in Theory and Applications, ed. C.K. Chui (Academic Press, New York, 1992) pp. 181–216.
[2]B.K. Alpert, A class of bases in L2 for the sparse representation of integral operators, SIAM J. Math. Anal. 24 (1993) 246–262. · Zbl 0764.42017 · doi:10.1137/0524016
[3]C.K. Chui and J.-A. Lian, A study of orthonormal multi-wavelets, J. Appl. Numer. Math. 20 (1996) 272–298. · Zbl 0877.65098 · doi:10.1016/0168-9274(95)00111-5
[4]W. Dahmen and C.A. Micchelli, Biorthogonal wavelet expansions, Constr. Approx. 13 (1997) 293–328. · Zbl 0882.65148 · doi:10.1007/s003659900045
[5]J.S. Geronimo, D.P. Hardin and P.R. Massopust, Fractal functions and wavelet expansions based on several scaling functions, J. Approx. Theory 78 (1994) 373–401. · Zbl 0806.41016 · doi:10.1006/jath.1994.1085
[6]T. Hogan, A note on matrix refinement equations, SIAM J. Math. Anal. 29 (1998) 849–854. · Zbl 0914.65141 · doi:10.1137/S003614109630135X
[7]Q. Jiang, Matlab routines for Sobolev and Hölder smoothness computations of refinable functions, http://www.cs.umsl.edu/jiang/Jsoftware.htm.
[8]J.-A. Lian, On the order of polynomial reproduction for multi-scaling functions, Appl. Comput. Harmonic Anal. 3 (1996) 358–365. · Zbl 0858.42026 · doi:10.1006/acha.1996.0027
[9]J.-A. Lian, Polynomial identities of Bezout type, in: Trends in Approximation Theory, eds. K. Kopotun, T. Lyche and M. Neamtu (Vanderbilt University Press, Nashville, TN, 2001) pp. 243–252.
[10]S.K. Lodha, Bernstein–Bézier multi-wavelets, in: Approximation Theory VIII, Vol. 2: Wavelets, eds. C.K. Chui and L.L. Schumaker (World Scientific, Singapore, 1995) pp. 259–266.
[11]G. Plonka, Approximation order provided by refinable function vectors, Constr. Approx. 13 (1997) 221–244