Geodesic
Periodic dyadic wavelets and coding of fractal functions
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 9 (2012), pp. 54-65.

Voir la notice de l'article provenant de la source Math-Net.Ru

Recently, using the Walsh–Dirichlet type kernel, the first author has defined periodic dyadic wavelets on the positive semiaxis which are similar to the Chui–Mhaskar trigonometric wavelets. In this paper we generalize this construction and give examples of applications of periodic dyadic wavelets for coding the Riemann, Weierstrass, Schwarz, van der Waerden, Hankel, and Takagi fractal functions.
Keywords: periodic dyadic wavelets, Walsh functions, Walsh–Dirichlet kernel, discrete Walsh transform, signal processing, fractal functions. 
@article{IVM_2012_9_a5,
     author = {Yu. A. Farkov and M. E. Borisov},
     title = {Periodic dyadic wavelets and coding of fractal functions},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {54--65},
     publisher = {mathdoc},
     number = {9},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2012_9_a5/}
}
TY  - JOUR
AU  - Yu. A. Farkov
AU  - M. E. Borisov
TI  - Periodic dyadic wavelets and coding of fractal functions
JO  - Izvestiâ vysših učebnyh zavedenij. Matematika
PY  - 2012
SP  - 54
EP  - 65
IS  - 9
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IVM_2012_9_a5/
LA  - ru
ID  - IVM_2012_9_a5
ER  - 
%0 Journal Article
%A Yu. A. Farkov
%A M. E. Borisov
%T Periodic dyadic wavelets and coding of fractal functions
%J Izvestiâ vysših učebnyh zavedenij. Matematika
%D 2012
%P 54-65
%N 9
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IVM_2012_9_a5/
%G ru
%F IVM_2012_9_a5
Yu. A. Farkov; M. E. Borisov. Periodic dyadic wavelets and coding of fractal functions. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 9 (2012), pp. 54-65. http://geodesic.mathdoc.fr/item/IVM_2012_9_a5/

[1] Chui C. K., Mhaskar H. N., “On trigonometric wavelets”, Constr. Approx., 9:2–3 (1993), 167–190 | DOI | MR | Zbl

[2] Skopina M. A., “Ortogonalnye polinomialnye bazisy Shaudera v $C[-1,1]$ s optimalnym rostom stepenei”, Matem. sb., 192:3 (2001), 115–136 | MR | Zbl

[3] Fischer B., Prestin J., “Wavelets based on orthogonal polynomials”, Math. Comp., 66:220 (1997), 1593–1618 | DOI | MR | Zbl

[4] Petukhov A. P., “Periodicheskie diskretnye vspleski”, Algebra i analiz, 8:3 (1996), 151–183 | MR | Zbl

[5] Petukhov A. P., “Periodicheskie vspleski”, Matem. sb., 188:10 (1997), 69–94 | MR | Zbl

[6] Skopina M., “Multiresolution analysis of periodic functions”, East J. Approx., 3:2 (1997), 203–224 | MR | Zbl

[7] Maksimenko I. E., Skopina M. A., “Mnogomernye periodicheskie vspleski”, Algebra i analiz, 15:2 (2003), 1–39 | MR | Zbl

[8] Novikov I. Ya., Protasov V. Yu., Skopina M. A., Teoriya vspleskov, Fizmatlit, M., 2006 | MR

[9] Farkov Yu. A., “Wavelets and frames based on Walsh–Dirichlet type kernels”, Commun. Math. Appl., 1:1 (2010), 27–46 | MR | Zbl

[10] Farkov Yu. A., “Ortogonalnye veivlety s kompaktnymi nositelyami na lokalno kompaktnykh abelevykh gruppakh”, Izv. RAN. Ser. matem., 69:3 (2005), 193–220 | MR | Zbl

[11] Farkov Yu. A., “Diskretnye veivlety i preobrazovanie Vilenkina–Krestensona”, Matem. zametki, 89:6 (2011), 914–928 | MR | Zbl

[12] Farkov Yu. A., Rodionov E. A., “Algorithms for wavelet construction on Vilenkin groups”, $p$-Adic Numb. Ultr. Anal. Appl., 3:3 (2011), 181–195 | DOI | MR

[13] Farkov Yu. A., “Wavelets and frames in Walsh analysis”, Wavelets: Classification, Theory an d Applications, Chapter 11, eds. Manel del Valle et al., Nova Science Publishers, 2012, 267–304

[14] Farkov Yu. A., Stroganov S. A., “O diskretnykh diadicheskikh veivletakh dlya obrabotki izobrazhenii”, Izv. vuzov. Matem., 2011, no. 7, 57–66 | MR

[15] Farkov Yu. A., Maksimov A. Yu., Stroganov S. A., “On biorthogonal wavelets related to the Walsh functions”, Int. J. Wavelets Multiresolut. Inf. Process., 9:3 (2011), 485–499 | DOI | MR | Zbl

[16] Golubov B. I., Efimov A. V., Skvortsov V. A., Ryady i preobrazovaniya Uolsha: teoriya i primeneniya, Izd-vo LKI, M., 2008

[17] Golubov B. I., Elementy dvoichnogo analiza, Izd-vo LKI, M., 2007

[18] Schipp F., Wade W. R., Simon P., Walsh series: An introduction to dyadic harmonic analysis, Adam Hilger, N.Y., 1990 | MR | Zbl

[19] Farkov Yu. A., “Periodic wavelets on the $p$-adic Vilenkin group”, $p$-Adic Numb. Ultr. Anal. Appl., 3:4 (2011), 281–287 | DOI | MR

[20] Uelstid S., Fraktaly i veivlety dlya szhatiya izobrazhenii v deistvii, Fizmatlit, M., 2007

[21] Holshneider M., Wavelets: an analysis tool, Clarendon Press, Oxford, 1995 | MR

[22] Jaffard S., Meyer Y., Wavelet methods for pointwise reqularity and local oscillations of functions, Mem. Amer. Math. Soc., 587, 1996 | MR | Zbl

[23] Medvedev F. A., Ocherki istorii teorii funktsii deistvitelnogo peremennogo, KomKniga, M., 2006

[24] A. A. Potapov (red.), Noveishie metody obrabotki izobrazhenii, Fizmatlit, M., 2008