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Izv. RAN. Ser. Mat., 2005, Volume 69, Issue 3, Pages 193–220 (Mi izv644)  

This article is cited in 50 scientific papers (total in 50 papers)

Orthogonal wavelets with compact support on locally compact Abelian groups

Yu. A. Farkov

Moscow State Geological Prospecting Academy

Abstract: We extend and improve the results of W. Lang (1998) on the wavelet analysis on the Cantor dyadic group $\mathcal C$. Our construction is realized on a locally compact abelian group $G$ which is defined for an integer $p\geqslant2$ and coincides with $\mathcal C$ when $p=2$. For any integers $p,n\geqslant 2$ we determine a function $\varphi$ in $L^2(G)$ which
  • is the sum of a lacunary series by generalized Walsh functions,
  • has orthonormal “integer” shifts in $L^2(G)$,
  • satisfies “the scaling equation” with $p^n$ numerical coefficients,
  • has compact support whose Haar measure is proportional to $p^n$,
  • generates a multiresolution analysis in $L^2(G)$.

Orthogonal wavelets $\psi$ with compact supports on $G$ are defined by such functions $\varphi$. The family of these functions $\varphi$ is in many respects analogous to the well-known family of Daubechies' scaling functions. We give a method for estimating the moduli of continuity of the functions $\varphi$, which leads to sharp estimates for small $p$ and $n$. We also show that the notion of adapted multiresolution analysis recently suggested by Sendov is applicable in this situation.

DOI: https://doi.org/10.4213/im644

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English version:
Izvestiya: Mathematics, 2005, 69:3, 623–650

Bibliographic databases:

UDC: 517.58
MSC: 42A38, 42A55, 42C15, 42C40, 43A70
Received: 05.07.2004

Citation: Yu. A. Farkov, “Orthogonal wavelets with compact support on locally compact Abelian groups”, Izv. RAN. Ser. Mat., 69:3 (2005), 193–220; Izv. Math., 69:3 (2005), 623–650

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. V. Yu. Protasov, Yu. A. Farkov, “Dyadic wavelets and refinable functions on a half-line”, Sb. Math., 197:10 (2006), 1529–1558  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    2. V. Yu. Protasov, “Approximation by dyadic wavelets”, Sb. Math., 198:11 (2007), 1665–1681  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    3. Yu. A. Farkov, “Biorthogonal dyadic wavelets on $\mathbb R_+$”, Russian Math. Surveys, 62:6 (2007), 1197–1198  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    4. Yu. A. Farkov, “Orthogonal Wavelets on Direct Products of Cyclic Groups”, Math. Notes, 82:6 (2007), 843–859  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    5. Yu. A. Farkov, “Biorthogonal Wavelets on Vilenkin Groups”, Proc. Steklov Inst. Math., 265 (2009), 101–114  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    6. E. A. Rodionov, Yu. A. Farkov, “Estimates of the Smoothness of Dyadic Orthogonal Wavelets of Daubechies Type”, Math. Notes, 86:3 (2009), 407–421  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    7. Farkov Yu.A., “On wavelets related to the Walsh series”, J. Approx. Theory, 161:1 (2009), 259–279  crossref  mathscinet  zmath  isi  elib  scopus
    8. Shah F.A., “Construction of wavelet packets on $p$-adic field”, Int. J. Wavelets Multiresolut. Inf. Process., 7:5 (2009), 553–565  crossref  mathscinet  zmath  isi  elib  scopus
    9. S. F. Lukomskii, “O ryadakh Khaara na kompaktnoi nul-mernoi gruppe”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 9:1 (2009), 14–19  mathnet  elib
    10. Manchanda P., Meenakshi, “New classes of Wavelets”, Modelling of Engineering and Technological Problems, AIP Conference Proceedings, 1146, 2009, 253–271  crossref  isi  scopus
    11. S. F. Lukomskii, “Multiresolution analysis on zero-dimensional Abelian groups and wavelets bases”, Sb. Math., 201:5 (2010), 669–691  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    12. Shah F.A., Wahid A., “Wavelet packets on locally compact Abelian groups”, An. Şt. Univ. Ovidius Constanţa. Ser. Mat., 18:2 (2010), 223–239  mathscinet  isi
    13. Sergei F. Lukomskii, “Haar system on a product of zero-dimensional compact groups”, Centr. Eur. J. Math., 9:3 (2011), 627–639  crossref  mathscinet  zmath  isi  scopus
    14. Yu. A. Farkov, S. A. Stroganov, “The use of discrete dyadic wavelets in image processing”, Russian Math. (Iz. VUZ), 55:7 (2011), 47–55  mathnet  crossref  mathscinet  elib
    15. F. A. Shah, Lokenath Debnath, “Dyadic wavelet frames on a half-line using the Walsh–Fourier transform”, Integral Transforms and Special Functions, 22:7 (2011), 477  crossref  mathscinet  zmath  isi  scopus
    16. S. F. Lukomskii, “Neortogonalnyi kratnomasshtabnyi analiz na nul-mernykh lokalno kompaktnykh gruppakh”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 11:3(1) (2011), 25–32  mathnet  elib
    17. Yuri A. Farkov, Evgeny A. Rodionov, “Algorithms for wavelet construction on Vilenkin groups”, P-Adic Num Ultrametr Anal Appl, 3:3 (2011), 181  crossref  mathscinet  zmath
    18. F. A. Shah, Lokenath Debnath, “p-Wavelet frame packets on a half-line using the Walsh–Fourier transform”, Integral Transforms and Special Functions, 2011, 1  crossref  mathscinet  isi  scopus
    19. 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  crossref  mathscinet  zmath  isi  elib  scopus
    20. Lukomskii S.F., “Multiresolution analysis on product of zero-dimensional Abelian groups”, J Math Anal Appl, 385:2 (2012), 1162–1178  crossref  mathscinet  zmath  isi  elib  scopus
    21. Yu. A. Farkov, M. E. Borisov, “Periodic dyadic wavelets and coding of fractal functions”, Russian Math. (Iz. VUZ), 56:9 (2012), 46–56  mathnet  crossref  mathscinet
    22. Yuri A. Farkov, “Examples of frames on the Cantor dyadic group”, J Math Sci, 187:1 (2012), 22  crossref  mathscinet  zmath  scopus
    23. Yu.A.. Farkov, E.A.. Rodionov, “Nonstationary Wavelets Related to the Walsh Functions”, AJCM, 02:02 (2012), 82  crossref
    24. S. S. Platonov, “On spectral synthesis on zero-dimensional Abelian groups”, Sb. Math., 204:9 (2013), 1332–1346  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    25. F.A.HMAD SHAH, “TIGHT WAVELET FRAMES GENERATED BY THE WALSH POLYNOMIALS”, Int. J. Wavelets Multiresolut Inf. Process, 2013, 1350042  crossref  mathscinet  isi  scopus
    26. S. V. Kozyrev, A. Yu. Khrennikov, V. M. Shelkovich, “$p$-Adic wavelets and their applications”, Proc. Steklov Inst. Math., 285 (2014), 157–196  mathnet  crossref  crossref  isi  elib  elib
    27. P. Manchanda, Vikram Sharma, “Construction of vector valued wavelet packets on ℝ+ using Walsh-Fourier transform”, Indian J Pure Appl Math, 45:4 (2014), 539  crossref  mathscinet  zmath  scopus
    28. S. F. Lukomskii, “Riesz multiresolution analysis on Vilenkin groups”, Dokl. Math, 90:1 (2014), 412  crossref  mathscinet  zmath  scopus
    29. F.A.. Shah, “Nonuniform Multiresolution Analysis on Local Fields of Positive Characteristic”, Complex Anal. Oper. Theory, 2014  crossref  scopus
    30. Yu. A. Farkov, “Wavelet Expansions on the Cantor Group”, Math. Notes, 96:6 (2014), 996–1007  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    31. F. A. Shah, “A characterization of tight wavelet frames on local fields of positive characteristic”, J. Contemp. Mathemat. Anal, 49:6 (2014), 251  crossref  mathscinet  zmath  scopus
    32. Yu. A. Farkov, E. A. Rodionov, “On biorthogonal discrete wavelet bases”, Int. J. Wavelets Multiresolut Inf. Process, 2014, 1550002  crossref  mathscinet  scopus
    33. Lukomskii S.F., “Step Refinable Functions and Orthogonal Mra on Vilenkin Groups”, J. Fourier Anal. Appl., 20:1 (2014), 42–65  crossref  mathscinet  zmath  isi  scopus
    34. S. F. Lukomskii, “Riesz multiresolution analysis on zero-dimensional groups”, Izv. Math., 79:1 (2015), 145–176  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    35. F.A.. Shah, “Frame Multiresolution Analysis on Local Fields of Positive Characteristic”, Journal of Operators, 2015 (2015), 1  crossref  mathscinet
    36. Shukla N.K., Vyas A., “Multiresolution Analysis Through Low-Pass Filter on Local Fields of Positive Characteristic”, Complex Anal. Oper. Theory, 9:3 (2015), 631–652  crossref  mathscinet  zmath  isi  scopus
    37. F.A.hmad Shah, M. Younus Bhat, “Vector-valued nonuniform multiresolution analysis on local fields”, Int. J. Wavelets Multiresolut Inf. Process, 2015, 1550029  crossref  mathscinet  scopus
    38. S. F. Lukomskii, G. S. Berdnikov, Yu. S. Kruss, “On the Orthogonality of a System of Shifts of the Scaling Function on Vilenkin Groups”, Math. Notes, 98:2 (2015), 339–342  mathnet  crossref  crossref  mathscinet  isi  elib
    39. Yu. S. Kruss, “O tochnosti otsenki chisla shagov algoritma postroeniya masshtabiruyuschei funktsii na lokalnykh polyakh”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 15:3 (2015), 279–287  mathnet  crossref  elib
    40. Farkov Yu., Lebedeva E., Skopina M., “Wavelet Frames on Vilenkin Groups and Their Approximation Properties”, Int. J. Wavelets Multiresolut. Inf. Process., 13:5 (2015), 1550036  crossref  mathscinet  zmath  isi  elib  scopus
    41. Lukomskii S.F., Berdnikov G.S., “N-Valid Trees in Wavelet Theory on Vilenkin Groups”, Int. J. Wavelets Multiresolut. Inf. Process., 13:5 (2015), 1550037  crossref  mathscinet  zmath  isi  elib  scopus
    42. Lukomskii S.F., Vodolazov A.M., “Non-Haar Mra on Local Fields of Positive Characteristic”, J. Math. Anal. Appl., 433:2 (2016), 1415–1440  crossref  mathscinet  zmath  isi  elib  scopus
    43. A. M. Vodolazov, S. F. Lukomskii, “Ortogonalnye sistemy sdvigov v pole $p$-adicheskikh chisel”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 16:3 (2016), 256–262  mathnet  crossref  mathscinet  elib
    44. G. S. Berdnikov, “Grafy s konturami v kratnomasshtabnom analize na gruppakh Vilenkina”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 16:4 (2016), 377–388  mathnet  crossref  mathscinet  elib
    45. Shah F.A., Bhat M.Y., “Construction of biorthogonal wavelet packets on local fields of positive characteristic”, Turk. J. Math., 40:2 (2016), 292–309  crossref  mathscinet  isi  elib  scopus
    46. Dragovich B. Khrennikov A.Yu. Kozyrev S.V. Volovich I.V. Zelenov E.I., “P-Adic Mathematical Physics: the First 30 Years”, P-Adic Numbers Ultrametric Anal. Appl., 9:2 (2017), 87–121  crossref  mathscinet  zmath  isi  scopus
    47. Shah F.A., Bhat M.Y., “Nonuniform Wavelet Packets on Local Fields of Positive Characteristic”, Filomat, 31:6 (2017), 1491–1505  crossref  mathscinet  isi  scopus
    48. Berdnikov G., Kruss I., Lukomskii S., “How to Construct Wavelets on Local Fields of Positive Characteristic”, Lobachevskii J. Math., 38:4, SI (2017), 615–621  crossref  mathscinet  zmath  isi  scopus
    49. G. S. Berdnikov, “Necessary and sufficient condition for an orthogonal scaling function on Vilenkin groups”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 19:1 (2019), 24–33  mathnet  crossref
    50. Farkov Yu.A., “Wavelet Frames Related to Walsh Functions”, Eur. J. Math., 5:1, SI (2019), 250–267  crossref  mathscinet  isi  scopus
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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