Abstract
We consider a dilation operatorT admitting a scaling function with compact support as fixed point. It is shown that the adjoint operatorT*admits a sequence of polynomial eigenfunctions and that a smooth functionf admits an expansion in these eigenfunctions, which reveals the asymptotic behavior ofT* forn→∞.
Due to this asymptotic expansion, an extrapolation technique can be applied for the accurate numerical computation of the integrals appearing in the wavelet decomposition of a smooth function. This extrapolation technique fits well in a multiresolution scheme.
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References
W. Dahmen and C.A. Michelli, Using the refinement equation for evaluating integrals of wavelets, to appear in SIAM J. Numer. Anal.
I. Daubechies, Orthonormal bases of compactly supported wavelets, Commun. Pure Appl. Math. 41 (1988) 909–996.
P.J. Davis and P. Rabinowitz,Methods of Numerical Integration (Academic Press, London, 1984).
W. Romberg, Vereinfachte numerische Integration, Norske Vid. Selsk. Forh. (Trondheim) 28 (1955) 30–36.
G. Strang, Wavelets and dilation equations: A brief introduction, SIAM Rev. 31 (1989) 614–627.
W. Sweldens and R. Piessens, Introduction to wavelets and efficient quadrature formulae for the calculation of the wavelet decomposition, Dept. of Computer Science, Katholieke Universiteit Leuven, Rep. TW 159 (1991).
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Communicated by C. Brezinski
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Verlinden, P., Haegemans, A. An asymptotic expansion in wavelet analysis and its application to accurate numerical wavelet decomposition. Numer Algor 2, 287–298 (1992). https://doi.org/10.1007/BF02139469
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DOI: https://doi.org/10.1007/BF02139469