Geodesic
Orthogonal Wavelets on Direct Products of Cyclic Groups
Matematičeskie zametki, Tome 82 (2007) no. 6, pp. 934-952.

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We describe a method for constructing compactly supported orthogonal wavelets on a locally compact Abelian group $G$ which is the weak direct product of a countable set of cyclic groups of $p$th order. For all integers $p,n\ge 2$, we establish necessary and sufficient conditions under which the solutions of the corresponding scaling equations with $p^n$ numerical coefficients generate multiresolution analyses in $L^2(G)$. It is noted that the coefficients of these scaling equations can be calculated from the given values of $p^n$ parameters using the discrete Vilenkin–Chrestenson transform. Besides, we obtain conditions under which a compactly supported solution of the scaling equation in $L^2(G)$ is stable and has a linearly independent system of “integer” shifts. We present several examples illustrating these results.
Keywords: orthogonal wavelets, multiresolution analysis, scaling equation, locally compact Abelian group, cyclic group, Walsh function, Haar measure, Borel set, blocked set of a mask. 
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Yu. A. Farkov. Orthogonal Wavelets on Direct Products of Cyclic Groups. Matematičeskie zametki, Tome 82 (2007) no. 6, pp. 934-952. http://geodesic.mathdoc.fr/item/MZM_2007_82_6_a13/

[1] M. Holshneider, Wavelets: an Analysis Tool, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1995 | MR | Zbl

[2] J. J. Benedetto, R. L. Benedetto, “A wavelet theory for local fields and related groups”, J. Geom. Anal., 14:3 (2004), 423–456 | MR | Zbl

[3] W. C. Lang, “Orthogonal wavelets on the Cantor dyadic group”, SIAM J. Math. Anal., 27:1 (1996), 305–312 | DOI | MR | Zbl

[4] W. C. Lang, “Wavelet analysis on the Cantor dyadic group”, Houston J. Math., 24:3 (1998), 533–544 | MR | Zbl

[5] W. C. Lang, “Fractal multiwavelets related to the Cantor dyadic group”, Internat. J. Math. Math. Sci., 21:2 (1998), 307–314 | DOI | MR | Zbl

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

[7] E. Khyuitt, K. Ross, Abstraktnyi garmonicheskii analiz, t. 1: Struktura topologicheskikh grupp. Teoriya integrirovaniya. Predstavleniya grupp, Nauka, M., 1975 | MR | Zbl

[8] B. I. Golubov, A. V. Efimov, V. A. Skvortsov, Ryady i preobrazovaniya Uolsha: Teoriya i primeneniya, Nauka, M., 1987 | MR | Zbl

[9] Yu. A. Farkov, “Ortogonalnye vspleski na lokalno kompaktnykh abelevykh gruppakh”, Funkts. analiz i ego pril., 31:4 (1997), 86–88 | MR | Zbl

[10] I. Dobeshi, Desyat lektsii po veivletam, NITs “Regulyarnaya i khaoticheskaya dinamika”, Izhevsk, 2001 | MR | Zbl

[11] V. Yu. Protasov, Yu. A. Farkov, “Diadicheskie veivlety i masshtabiruyuschie funktsii na polupryamoi”, Matem. sb., 197:10 (2006), 129–160 | MR

[12] F. Schipp, W. R. Wade, P. Simon, Walsh series: An Introduction to Dyadic Harmonic Analysis, Adam Hilger, Ltd., Bristol, 1990 | MR | Zbl

[13] V. N. Malozemov, S. M. Masharskii, “Obobschennye veivletnye bazisy, svyazannye s diskretnym preobrazovaniem Vilenkina–Krestensona”, Algebra i analiz, 13:1 (2001), 111–157 | MR | Zbl

[14] Yu. A. Farkov, “Orthogonal $p$-wavelets on $\mathbb R_+$”, Wavelets and Splines, International conference on wavelets and splines (St. Petersburg, Russia, July 3–8, 2003), St. Petersburg Univ. Press, St. Petersburg, 2005, 4–26 | MR | Zbl

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

[16] S. A. Bychkov, Yu. A. Farkov, “O teoreme Koena dlya veivlet-razlozhenii na gruppakh”, VI Mezhdunarodnaya konferentsiya “Novye idei v naukakh o Zemle” (Moskva, aprel 2003 g.), Izbrannye doklady, MGGRU, Moskva, 2003, 226–238