Multiresolution Analysis Through Low-Pass Filter on Local Fields of Positive Characteristic


The concept of wavelet basis on the integers can be generalized to a countable subset of a local field having positive characteristic by using a prime element of such a field. In this paper, we provide a characterization of first-stage discrete wavelet system on a countable subset of a local field of positive characteristic. Further, we obtain some results on refinement equation and refinement coefficients which provide sufficient conditions for a function to be a solution of the refinement equation and generate a multiresolution analysis on the local fields.

This is a preview of subscription content, access via your institution.


  1. 1.

    Albeverio, S., Kozyrev, S.: Multidimensional basis of \(p\)-adic wavelets and representation theory. p Adic Numbers Ultrametric Anal. Appl. 1(3), 181–189 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  2. 2.

    Albeverio, S., Skopina, M.: Haar bases for \(L^2 (\mathbb{Q}^2_2)\) generated by one wavelet function. Int. J. Wavelets Multiresolut. Inf. Process. 10(5), 1250042 (2012)

    Article  MathSciNet  Google Scholar 

  3. 3.

    Behera, B., Jahan, Q.: Multiresolution analysis on local fields and characterization of scaling functions. Adv. Pure Appl. Math. 3(2), 181–202 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  4. 4.

    Benedetto, R.L.: Examples of wavelets for local fields, In: Wavelets, frames and operator theory, Contemporary Mathematics, vol. 345, pp. 27–47, American Mathematical Society, Providence (2004)

  5. 5.

    Benedetto, J.J., Benedetto, R.L.: A wavelet theory for local fields and related groups. J. Geom. Anal. 14(3), 423–456 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  6. 6.

    Cabrelli, C.A., Heil, C., Molter, U.M.: Self-similarity and multiwavelets in higher dimensions. Mem. Am. Math. Soc. 170(807), vii+82 (2004)

    MathSciNet  Google Scholar 

  7. 7.

    Dahlke, S.: Multiresolution analysis and wavelets on locally compact abelian groups. In: Wavelets, images, and surface fitting (Chamonix-Mont-Blanc, 1993), pp. 141–156. A K Peters, Wellesley (1994)

  8. 8.

    Farkov, Y.A.: Orthogonal wavelets on locally compact abelian groups (Russian). Funktsional. Anal. i Prilozhen. 31(4), 86–88 (1997). [translation in Funct. Anal. Appl. 31(4), 294–296 (1997)]

    Article  MathSciNet  Google Scholar 

  9. 9.

    Farkov, Y.A.: Periodic wavelets on the p-adic Vilenkin group. p Adic Numbers Ultrametr. Anal. Appl. 3(4), 281–287 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  10. 10.

    Farkov, Y.A.: Orthogonal wavelets with compact support on locally compact abelian groups (Russian). Izv. Ross. Akad. Nauk Ser. Mat. 69(3), 193–220 (2005). [translation in Izv. Math. 69(3), 623650 (2005)]

    Article  MathSciNet  Google Scholar 

  11. 11.

    Farkov, Y.A.: Orthogonal wavelets on direct products of cyclic groups (Russian). Mat. Zametki 82(6), 934–952 (2007). [translation in Math. Notes 82(5–6), 843–859 (2007)]

    Article  MathSciNet  Google Scholar 

  12. 12.

    Farkov, Y.A.: Multiresolution analysis and wavelets on Vilenkin groups. Facta Universitatis Ser. Elec. Enerd. 21(3), 309–325 (2008)

    Article  Google Scholar 

  13. 13.

    Farkov, Y.A., Rodionov, E.A.: Algorithms for wavelet construction on Vilenkin groups, \(p\)-Adic Numbers. Ultrametr. Anal. Appl. 3(3), 181–195 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  14. 14.

    Frazier, M.: An introduction to wavelets through linear algebra. Undergraduate Texts in Mathematics, p. xvi+501. Springer, New York (1999)

    MATH  Google Scholar 

  15. 15.

    Gressman, P.: Wavelets on the integers. Collect. Math. 52(3), 257–288 (2001)

    MATH  MathSciNet  Google Scholar 

  16. 16.

    Han, D., Larson, D.R., Papadakis, M., Stavropoulos, T.H.: Multiresolution analyses of abstract Hilbert spaces and wandering subspaces. In: The functional and harmonic analysis of wavelets and frames (San Antonio, TX, 1999), Contemporary Mathematics, vol. 247, pp. 259–284. American Mathematical Society, Providence (1999)

  17. 17.

    Hernandez, E., Weiss, G.: A first course on wavelets, p. xx+489. CRC Press, Boca Raton (1996)

    Book  MATH  Google Scholar 

  18. 18.

    Holschneider, M.: Wavelet analysis over abelian groups. Appl. Comput. Harmon. Anal. 2(1), 52–60 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  19. 19.

    Jiang, H., Li, D., Jin, N.: Multiresolution analysis on local fields. J. Math. Anal. Appl. 294(2), 523–532 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  20. 20.

    Khrennikov, A., Shelkovich, V., van der Walt, J.H.: Measure-free viewpoint on \(p\)-adic and adelic wavelets. p-Adic Numbers Ultrametr. Anal. Appl. 5(3), 204–217 (2013)

    Article  MATH  Google Scholar 

  21. 21.

    Khrennikov, A.Y., Shelkovich, V.M., Skopina, M.: \(p\)-adic refinable functions and MRA-based wavelets. J. Approx. Theory 161(1), 226–238 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  22. 22.

    Kosyak, A.V., Khrennikov, A., Yu, Shelkovich, V.M.: Wavelet bases on adéles (Russian). Dokl. Akad. Nauk 442(4), 446–450 (2012). [translation in Dokl. Math. 85(1), 75–79 (2012)]

    Google Scholar 

  23. 23.

    Kozyrev, S.: Wavelet theory as \(p\)-adic spectral analysis (Russian). Izv. Ross. Akad. Nauk Ser. Mat. 66(2), 149–158 (2002). [translation in Izv. Math. 66(2), 367–376 (2002)]

    Article  MathSciNet  Google Scholar 

  24. 24.

    Kozyrev, S.V., Khrennikov, A., Yu, : \(p\)-adic integral operators in wavelet bases (Russian). Dokl. Akad. Nauk 437(4), 457–461 (2011). [translation in Dokl. Math. 83(2), 209–212 (2011)]

    Google Scholar 

  25. 25.

    Lang, W.C.: Orthogonal wavelets on the Cantor dyadic group. SIAM J. Math. Anal. 27(1), 305–312 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  26. 26.

    Lang, W.C.: Fractal multiwavelets related to the Cantor dyadic group. Int. J. Math. Math. Sci. 21(2), 307–314 (1998)

    Article  MATH  Google Scholar 

  27. 27.

    Lang, W.C.: Wavelets analysis on the Cantor dyadic group. Houston J. Math. 24(3), 533–544 (1998)

    MATH  MathSciNet  Google Scholar 

  28. 28.

    Lukomskii, S.F.: Step refinable functions and orthogonal MRA on Vilenkin groups. J. Fourier Anal. Appl. 20(1), 42–65 (2014)

  29. 29.

    Packer, J.A.: A survey of projective multiresolution analyses and a projective multiresolution analysis corresponding to the quincunx lattice. In: Representations, wavelets, and frames, Appl. Numer. Harmon. Anal., pp. 239–272 Birkhäuser, Boston (2008)

  30. 30.

    Stavropoulos, T., Papadakis, M.: On the multiresolution analysis of abstract Hilbert spaces. Bull. Greek Math. Soc. 40, 79–92 (1998)

    MATH  MathSciNet  Google Scholar 

  31. 31.

    Shukla, N.K., Mittal, S.: Wavelets on the spectrum. Numer. Funct. Anal. Optim. 35(4), 461–486 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  32. 32.

    Strang, G.: Wavelets and dilation equations: a brief introduction. SIAM Rev. 31(4), 614–627 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  33. 33.

    Taibleson, M.H.: Fourier analysis on local fields, pp. xii+294. Princeton University Press, Princeton. University of Tokyo Press, Tokyo (1975)

  34. 34.

    Trimeche, K.: Wavelets on hypergroups. In: Harmonic analysis and hypergroups (Delhi 1995), Trends Math., pp. 183–213. Birkhäuser, Boston (1998)

  35. 35.

    Wojtaszczyk, P.: A mathematical introduction to wavelets. In: London Mathematical Society Student Texts, vol. 37, pp. xii+261. Cambridge University Press, Cambridge (1997)

Download references


The authors would like to thank the anonymous reviewer for his valuable comments and suggestions to improve the quality of the paper.

Author information



Corresponding author

Correspondence to Niraj K. Shukla.

Additional information

Communicated by Palle Jorgensen.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Shukla, N.K., Vyas, A. Multiresolution Analysis Through Low-Pass Filter on Local Fields of Positive Characteristic. Complex Anal. Oper. Theory 9, 631–652 (2015).

Download citation


  • Local field
  • Wavelet
  • Multiresolution analysis
  • Low-pass filter
  • Wavelets on the integers

Mathematics Subject Classification (2010)

  • Primary 42C40
  • Secondary 42C15
  • 43A70
  • 11S85