Previous |  Up |  Next

Article

Keywords:
Daubechies wavelet; quadrature formula
Summary:
The scaling function corresponding to the Daubechies wavelet with two vanishing moments is used to derive new quadrature formulas. This scaling function has the smallest support among all orthonormal scaling functions with the properties $M_2 = M_1^2$ and $M_0 = 1$. So, in this sense, its choice is optimal. Numerical examples are given.
References:
[1] J. J.  Benedetto, M. W.  Frazier: Wavelets: Mathematics and Applications. Studies in Advanced Mathematics. CRC Press, Boca Raton, 1994. MR 1247511
[2] A.  Cohen: Wavelet Methods in Numerical Analysis. Handbook of Numerical Analysis, Vol.  VII, P. G. Ciarlet et al. (eds.), North-Holland/Elsevier, Amsterdam, 2000, pp. 417–711. MR 1804747 | Zbl 0976.65124
[3] I.  Daubechies: Orthonormal bases of compactly supported wavelets. Commun. Pure Appl. Math. 41 (1988), 909–996. DOI 10.1002/cpa.3160410705 | MR 0951745 | Zbl 0644.42026
[4] I.  Daubechies: Ten Lectures on Wavelets. SIAM Publ., Philadelphia, 1992. MR 1162107 | Zbl 0776.42018
[5] V.  Finěk: Daubechies wavelets and two-point boundary value problems. Appl. Math. 49 (2004), 465–481. MR 2086089
[6] V.  Finěk: Approximation properties of wavelets and relations among scaling moments. Numer. Funct. Anal. Optimization 25 (2004), 503–513. DOI 10.1081/NFA-200041709 | MR 2106272 | Zbl 1069.42022
[7] G.  Hanwei, Y.  Jiaxian, H.  Jianguo, and L.  Peiguo: The Numerical Integral Algorithm Based on Multiresolution Analysis. Preprint (Wavelet Digest), Department of Electronic Technique, NUDT, Changsha, 2001.
[8] A. K.  Louis, P.  Maaß, and A.  Rieder: Wavelets. Theorie und Anwendungen. Teubner, Stuttgart, 1994. MR 1371382
[9] Y. Meyer: Ondelettes et Opérateurs  I: Ondelettes. Hermann Press, Paris, 1990; English translation: Wavelets and Operators. Cambridge University Press, Cambridge, 1992. MR 1085487 | Zbl 0694.41037
[10] W.-Ch. Shann, J.-Ch.  Yan: Quadratures involving polynomials and Daubechies’ wavelets. Preprint, National Central University, Chung-Li, R.O.C., April 1994.
[11] W.  Sweldens, R.  Piessens: Quadrature formulae and asymptotic error expansions for wavelet approximations of smooth functions. SIAM J.  Numer. Anal. 31 (1994), 1240–1264. DOI 10.1137/0731065 | MR 1286226
[12] P. Wojtaszczyk: A Mathematical Introduction to Wavelets. Cambridge University Press, Cambridge, 1997. MR 1436437 | Zbl 0865.42026
Partner of
EuDML logo