On the computation of scaling coefficients of Daubechies' wavelets

Dana Černá; Václav Finěk

Open Mathematics (2004)

  • Volume: 2, Issue: 3, page 399-419
  • ISSN: 2391-5455

Abstract

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In the present paper, Daubechies' wavelets and the computation of their scaling coefficients are briefly reviewed. Then a new method of computation is proposed. This method is based on the work [7] concerning a new orthonormality condition and relations among scaling moments, respectively. For filter lengths up to 16, the arising system can be explicitly solved with algebraic methods like Gröbner bases. Its simple structure allows one to find quickly all possible solutions.

How to cite

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Dana Černá, and Václav Finěk. "On the computation of scaling coefficients of Daubechies' wavelets." Open Mathematics 2.3 (2004): 399-419. <http://eudml.org/doc/268909>.

@article{DanaČerná2004,
abstract = {In the present paper, Daubechies' wavelets and the computation of their scaling coefficients are briefly reviewed. Then a new method of computation is proposed. This method is based on the work [7] concerning a new orthonormality condition and relations among scaling moments, respectively. For filter lengths up to 16, the arising system can be explicitly solved with algebraic methods like Gröbner bases. Its simple structure allows one to find quickly all possible solutions.},
author = {Dana Černá, Václav Finěk},
journal = {Open Mathematics},
keywords = {65T60},
language = {eng},
number = {3},
pages = {399-419},
title = {On the computation of scaling coefficients of Daubechies' wavelets},
url = {http://eudml.org/doc/268909},
volume = {2},
year = {2004},
}

TY - JOUR
AU - Dana Černá
AU - Václav Finěk
TI - On the computation of scaling coefficients of Daubechies' wavelets
JO - Open Mathematics
PY - 2004
VL - 2
IS - 3
SP - 399
EP - 419
AB - In the present paper, Daubechies' wavelets and the computation of their scaling coefficients are briefly reviewed. Then a new method of computation is proposed. This method is based on the work [7] concerning a new orthonormality condition and relations among scaling moments, respectively. For filter lengths up to 16, the arising system can be explicitly solved with algebraic methods like Gröbner bases. Its simple structure allows one to find quickly all possible solutions.
LA - eng
KW - 65T60
UR - http://eudml.org/doc/268909
ER -

References

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  1. [1] B. Han: “Analysis and construction of optimal multivariate biortogonal wavelets with compact support”,SIAM J. Math. Anal., Vol. 31, (2000), pp. 274–304. http://dx.doi.org/10.1137/S0036141098336418 Zbl0943.65163
  2. [2] B. Han: “Approximation properties and construction of Hermite interpolants and biorthogonal multiwavelets”,J. Approx. Theory,Vol. 110, (2001),pp. 18–53. http://dx.doi.org/10.1006/jath.2000.3545 Zbl0986.42020
  3. [3] B. Han: “Symmetric multivariate orthogonal refinable functions”,Appl. Comput. Harmon. Anal., to appear. Zbl1058.42028
  4. [4] A. Cohen and R.D. Ryan:Wavelets and Multiscale Signal Processing, Chapman & Hall, London, 1995. Zbl0848.42021
  5. [5] A. Cohen: “Wavelet methods in numerical analysis”, In: P.G. Ciarlet et al. (Eds.):Handbook of numerical analysis, Vol. VII, Amsterdam: North-Holland/Elsevier, 2000, pp. 417–711. 
  6. [6] I. Daubechies:Ten Lectures on Wavelets. Society for industrial and applied mathematics, Philadelphia, 1992. 
  7. [7] V. Finěk:Approximation properties of wavelets and relations among scaling moments II, Preprint, TU Dresden, 2003. 
  8. [8] W. Lawton: “Necessary and sufficient conditions for constructing orthonormal wavelet bases”,J. Math. Phys., Vol. 32, (1991), pp. 57–61. http://dx.doi.org/10.1063/1.529093 Zbl0757.46012
  9. [9] A.K. Louis, P. Maass and A. Rieder:Wavelets: Theorie und Anwendungen, B.G. Teubner, Stuttgart, 1994. 
  10. [10] G. Polya and G. Szegö:Aufgaben und Lehrsätze aus der Analysis, Vol. II, Springer-Verlag, Berlin, 1971. Zbl0219.00003
  11. [11] W.C. Shann and C.C. Yen: “On the exampleact values of the orthogonal scaling coefficients of lenghts 8 and 10”,Appl. Comput. Harmon. Anal., Vol. 6, (1999), pp 109–112. http://dx.doi.org/10.1006/acha.1997.0240 
  12. [12] G. Strang and T. Nguyen:Wavelets and Filter Banks, Wellesley-Cambridge Press, 1996. 
  13. [13] P. Wojtaszczyk:A Mathematical introduction to wavelets, Cambridge University Press, 1997. 

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