AJCM  Vol.2 No.2 , June 2012
Nonstationary Wavelets Related to the Walsh Functions
Abstract: 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.
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.