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

Abstract

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.

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References

  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)

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Acknowledgments

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

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Correspondence to Niraj K. Shukla.

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Communicated by Palle Jorgensen.

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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). https://doi.org/10.1007/s11785-014-0396-9

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Keywords

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

Mathematics Subject Classification (2010)

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