Share This Article:

Nonstationary Wavelets Related to the Walsh Functions

Abstract Full-Text HTML Download Download as PDF (Size:189KB) PP. 82-87
DOI: 10.4236/ajcm.2012.22011    4,579 Downloads   7,462 Views   Citations


Using the Walsh-Fourier transform, we give a construction of compactly supported nonstationary dyadic wavelets on the positive half-line. The masks of these wavelets are the Walsh polynomials defined by finite sets of parameters. Application to compression of fractal functions are also discussed.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

Y. Farkov and E. Rodionov, "Nonstationary Wavelets Related to the Walsh Functions," American Journal of Computational Mathematics, Vol. 2 No. 2, 2012, pp. 82-87. doi: 10.4236/ajcm.2012.22011.


[1] W. C. Lang, “Orthogonal Wavelets on the Cantor Dyadic Group,” SIAM Journal on Mathematical Analysis, Vol. 27, No. 1, 1996, pp. 305-312. doi:10.1137/S0036141093248049
[2] W. C. Lang, “Wavelet Analysis on the Cantor Dyadic Group,” Houston Journal of Mathematics, Vol. 24, No. 3, 1998, pp. 533-544.
[3] W. C. Lang, “Fractal Multiwavelets Related to the Cantor Dyadic Group,” International Journal of Mathematics and Mathematical Sciences, Vol. 21, No. 2, 1998, pp. 307-317. doi:10.1155/S0161171298000428
[4] Y. A. Farkov, “Orthogonal Wavelets with Compact Support on Locally Compact Abelian Groups,” Izvestiya: Mathematics, Vol. 69, No. 3, 2005, pp. 623-650. doi:10.1070/IM2005v069n03ABEH000540
[5] Y. A. Farkov, “Wavelets and Frames in Walsh Analysis,” In: M. del Valle, Ed., Wavelets: Classification, Theory and Applications, Chapter 11. Nova Science Publishers, New York, 2012, pp. 267-304.
[6] Y. A. Farkov and E. A. Rodionov, “Estimates of the Smoothness of Dyadic Orthogonal Wavelets of Daubechies Type,” Mathematical Notes, Vol. 82, No. 6, 2007, pp. 407-421. doi:10.1134/S0001434609090144
[7] Y. A. Farkov, A. Yu. Maksimov and S. A. Stroganov, “On Biorthogonal Wavelets Related to the Walsh Functions,” International Journal of Wavelets, Multiresolution and Information Processing, Vol. 9, No. 3, 2011, pp. 485- 499. doi:10.1142/S0219691311004195
[8] Y. A. Farkov and E. A. Rodionov, “Algorithms for Wave- let Construction on Vilenkin Groups,” P-Adic Numbers, Ultrametric Analysis, and Applications, Vol. 3, No. 3, 2011, pp. 181-195. doi:10.1134/S2070046611030022
[9] Y. A. Farkov, “Periodic Wavelets on the p-Adic Vilenkin Group,” P-Adic Numbers, Ultrametric Analysis, and Applications, Vol. 3, No. 4, 2011, pp. 281-287. doi:10.1134/S2070046611040030
[10] Y. A. Farkov and M. E. Borisov, “Periodic Dyadic Wave- lets and Coding of Fractal Functions,” Russian Mathematics (Izvestiya VUZ. Matematika), No. 9, 2012, pp. 54- 65.
[11] Ya. Novikov, “On the Construction of Nonstationary Orthonormal Infinitely Differentiable Compactly Supported Wavelets,” Proceedings of the 12th International Association for Pattern Recognition, Jerusalem, 9-13 Oc- tober 1994, pp. 214-215. doi:10.1109/ICPR.1994.577164
[12] B. Sendov, “Adapted Multiresolution Analysis,” Proceedings of Alexits Memorial Conference Functions, Series, Operators, Budapest, 9-14 August 1999, pp. 23-38.
[13] B. Sendov, “Adaptive Multire-solution Analysis on the Dyadic Topological Group,” Journal of Approximation Theory, Vol. 96, No. 2, 1998, pp. 21-45. doi:10.1006/jath.1998.3234
[14] Ya. Novikov, V. Yu. Protasov and M. A. Skopina, “Wave- let Theory,” American Mathematical Society, Providence, 2011.
[15] F. Schipp, W. R. Wade and P. Simon, “Walsh Series: An Introduction to Dyadic Harmonic Analysis,” Adam Hilger, Bristol, 1990.
[16] B. I. Golubov, A. V. Efimov and V. A. Skvortsov, “Walsh Series and Transforms,” Kluwer, Dordrecht, 1991.
[17] S. Welstead, “Fractal and Wavelet Image Compression Techniques,” SPIE Optical Engineering Press, Belling- ham, 2002.

comments powered by Disqus

Copyright © 2020 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.