# On the exact values of coefficients of coiflets

Dana Černá; Václav Finěk; Karel Najzar

Open Mathematics (2008)

- Volume: 6, Issue: 1, page 159-169
- ISSN: 2391-5455

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topDana Černá, Václav Finěk, and Karel Najzar. "On the exact values of coefficients of coiflets." Open Mathematics 6.1 (2008): 159-169. <http://eudml.org/doc/269141>.

@article{DanaČerná2008,

abstract = {In 1989, R. Coifman suggested the design of orthonormal wavelet systems with vanishing moments for both scaling and wavelet functions. They were first constructed by I. Daubechies [15, 16], and she named them coiflets. In this paper, we propose a system of necessary conditions which is redundant free and simpler than the known system due to the elimination of some quadratic conditions, thus the construction of coiflets is simplified and enables us to find the exact values of the scaling coefficients of coiflets up to length 8 and two further with length 12. Furthermore for scaling coefficients of coiflets up to length 14 we obtain two quadratic equations, which can be transformed into a polynomial of degree 4 for which there is an algebraic formula to solve them.},

author = {Dana Černá, Václav Finěk, Karel Najzar},

journal = {Open Mathematics},

keywords = {orthonormal wavelet; coiflet; exact value of filter coefficients; numerical examples},

language = {eng},

number = {1},

pages = {159-169},

title = {On the exact values of coefficients of coiflets},

url = {http://eudml.org/doc/269141},

volume = {6},

year = {2008},

}

TY - JOUR

AU - Dana Černá

AU - Václav Finěk

AU - Karel Najzar

TI - On the exact values of coefficients of coiflets

JO - Open Mathematics

PY - 2008

VL - 6

IS - 1

SP - 159

EP - 169

AB - In 1989, R. Coifman suggested the design of orthonormal wavelet systems with vanishing moments for both scaling and wavelet functions. They were first constructed by I. Daubechies [15, 16], and she named them coiflets. In this paper, we propose a system of necessary conditions which is redundant free and simpler than the known system due to the elimination of some quadratic conditions, thus the construction of coiflets is simplified and enables us to find the exact values of the scaling coefficients of coiflets up to length 8 and two further with length 12. Furthermore for scaling coefficients of coiflets up to length 14 we obtain two quadratic equations, which can be transformed into a polynomial of degree 4 for which there is an algebraic formula to solve them.

LA - eng

KW - orthonormal wavelet; coiflet; exact value of filter coefficients; numerical examples

UR - http://eudml.org/doc/269141

ER -

## References

top- [1] Adams W.W., Loustaunau P., An Introduction to Gröbner Bases, American Mathematical Society, 1994 Zbl0803.13015
- [2] Antonini M., Barlaud M., Mathieu P., Daubechies I., Image coding using wavelet transforms, IEEE Trans. Image Process., 1992, 1, 205–220 http://dx.doi.org/10.1109/83.136597
- [3] Beylkin G., On the representation of operators in bases of compactly supported wavelets, SIAM J. Numer. Anal., 1992, 29, 1716–1740 http://dx.doi.org/10.1137/0729097 Zbl0766.65007
- [4] Beylkin G., Coifman R.R., Rokhlin V., Fast wavelet transforms and numerical algorithms I, Comm. Pure Appl. Math., 1991, 44, 141–183 http://dx.doi.org/10.1002/cpa.3160440202 Zbl0722.65022
- [5] Bittner K., Urban K., Adaptive wavelet methods using semiorthogonal spline wavelets: Sparse evaluation of nonlinear functions, preprint Zbl1135.42335
- [6] Buchberger B., An algorithm for finding a basis for the residue class ring of a zero-dimensional polynomial ideal, PhD thesis, University of Inssbruck, Austria, 1965 (in German)
- [7] Burrus C.S., Odegard J.E., Coiflet Systems and Zero Moments, IEEE Trans. Signal Process., 1998, 46, 761–766 http://dx.doi.org/10.1109/78.661342
- [8] Burrus C.S., Gopinath R.A., On the moments of the scaling function ψ 0, Proceedings of the ISCAS-92, 1992, 963–966
- [9] Černá D., Finěk V., On the computation of scaling coefficients of Daubechies wavelets, Cent. Eur. J. Math., 2004, 2, 399–419 http://dx.doi.org/10.2478/BF02475237 Zbl1076.65126
- [10] Chyzak F., Paule P., Scherzer O., Schoisswohl A., Zimmermann B., The construction of orthonormal wavelets using symbolic methods and a matrix analytical approach for wavelets on the interval, Experiment. Math., 2001, 10, 67–86 Zbl0974.42027
- [11] Cohen A., Ondelettes analyses multirésolutions et filtres miroir en quadrature, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1990, 7, 439–459 Zbl0736.42021
- [12] Cohen A., Daubechies I., Feauveau J.C., Biorthogonal bases of compactly supported wavelets, Comm. Pure Appl. Math., 1992, 45, 485–500 http://dx.doi.org/10.1002/cpa.3160450502 Zbl0776.42020
- [13] Dahmen W., Kunoth A., Urban K., Biorthogonal spline wavelets on the interval - stability and moment conditions, Appl. Comput. Harmon. Anal., 1999, 6, 132–196 http://dx.doi.org/10.1006/acha.1998.0247 Zbl0922.42021
- [14] Daubechies I., Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math., 1988, 41, 909–996 http://dx.doi.org/10.1002/cpa.3160410705 Zbl0644.42026
- [15] Daubechies I., Orthonormal bases of compactly supported wavelets II Variations on a theme, SIAM J. Math. Anal., 1993, 24, 499–519 http://dx.doi.org/10.1137/0524031 Zbl0792.42018
- [16] Daubechies I., Ten lectures on wavelets, SIAM, Philadelphia, 1992
- [17] Eirola T., Sobolev characterization of compactly supported wavelets, SIAM J. Math. Anal., 1992, 23, 1015–1030 http://dx.doi.org/10.1137/0523058 Zbl0761.42014
- [18] Finěk V., Approximation properties of wavelets and relations among scaling moments, Numer. Funct. Anal. Optim., 2004, 25, 503–513 http://dx.doi.org/10.1081/NFA-200041709 Zbl1069.42022
- [19] Finěk V., Approximation properties of wavelets and relations among scaling moments II, Cent. Eur. J. Math., 2004, 2, 605–613 http://dx.doi.org/10.2478/BF02475967 Zbl1081.42027
- [20] Lawton W.M., Necessary and sufficient conditions for constructing orthonormal wavelet bases, J. Math. Phys., 1991, 32, 57–61 http://dx.doi.org/10.1063/1.529093 Zbl0757.46012
- [21] Lebrun J., Selesnick I., Grobner bases and wavelet design, J. Symb. Comp., 2004, 37, 227–259 http://dx.doi.org/10.1016/j.jsc.2002.06.002 Zbl1047.94505
- [22] Monzón L., Beylkin G., Hereman W., Compactly supported wavelets based on almost interpolating and nearly linear phase filters (coiflets), Appl. Comput. Harmon. Anal., 1999, 7, 184–210 http://dx.doi.org/10.1006/acha.1999.0266 Zbl0944.42023
- [23] Regensburger G., Scherzer O., Symbolic computation for moments and filter coefficients of scaling functions, Ann. Comb., 2005, 9, 223–243 http://dx.doi.org/10.1007/s00026-005-0253-7 Zbl1077.42030
- [24] Regensburger G., Parametrizing compactly supported orthonormal wavelets by discrete moments, Applicable Algebra in Engineering, Communication and Computing, 2007, 18, 583–601 http://dx.doi.org/10.1007/s00200-007-0054-9 Zbl1142.42325
- [25] Resnikoff H.L., Wells R.O., Wavelet analysis. The scalable structure of information, Springer-Verlag, New York, 1998 Zbl0922.42020
- [26] Shann W.C., Yen C.C., On the exact values of orthonormal scaling coefficients of length 8 and 10, Appl. Comput. Harmon. Anal., 1999, 6, 109–112 http://dx.doi.org/10.1006/acha.1997.0240
- [27] Tian J., The mathematical theory and applications of biorthogonal Coifman wavelet systems, Ph.D. thesis, Rice University, Houston, TX, 1996
- [28] Tian J., Wells R.O. Jr., Vanishing moments and biorthogonal Coifman wavelet systems, Proceedings of 4th International Conference on Mathematics in Signal Processing, University of Warwick, England, 1997
- [29] Tian J., Wells R.O. Jr., Vanishing moments and wavelet approximation, Technical Report, CML TR95-01, Rice University, January 1995
- [30] Unser M., Approximation power of biorthogonal wavelet expansions, IEEE Transactions on Signal Processing, 1996, 44, 519–527 http://dx.doi.org/10.1109/78.489025
- [31] Villemoes L.F., Energy moments in time and frequency for two-scale difference equation solutions and wavelets, SIAM J. Math. Anal., 1992, 23, 1519–1543 http://dx.doi.org/10.1137/0523085 Zbl0759.39005

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