Semi-orthogonal Parseval Wavelets Associated with GMRAs on Local Fields of Positive Characteristic

Abstract

In this article, we establish theory of semi-orthogonal Parseval wavelets associated with generalized multiresolution analysis (GMRA) for local fields of positive characteristics (LFPC) and obtain their characterization in terms of consistency equation. As a consequence, we obtain a characterization of an orthonormal (multi)wavelet associated with an MRA in terms of multiplicity function as well as dimension function. Further, we provide characterizations of Parseval scaling functions, scaling sets and bandlimited wavelets together with a Shannon-type multiwavelet. Some examples of such wavelets are also produced for LFPC.

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

References

  1. 1.

    Albeverio, S., Evdokimov, S., Skopina, M.: \(p\)-adic nonorthogonal wavelet bases. Proc. Steklov Inst. Math. 265(1), 135–146 (2009)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Albeverio, S., Evdokimov, S., Skopina, M.: \(p\)-adic multiresolution analysis and wavelet frames. J. Fourier Anal. Appl. 16, 693–714 (2010)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Baggett, L.W., Medina, H.A., Merrill, K.D.: Generalized multi-resolution analyses and a construction procedure for all wavelet sets in \({\mathbb{R}}^n\). J. Fourier Anal. Appl. 5(6), 563–573 (1999)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Bakić, D.: Semi-orthogonal Parseval frame wavelets and generalized multiresolution analyses. Appl. Comput. Harmon. Anal. 21(3), 281–304 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  5. 5.

    Barbieri, D., Hernández, E., Mayeli, A.: Bracket map for the Heisenberg group and the characterization fo cyclic subspaces. Appl. Comput. Harmon. Anal. 37, 218–234 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    Behera, B.: Shift-invariant subspaces and wavelets on local fields. Acta Math. Hungar. 148(1), 157–173 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  7. 7.

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

    MathSciNet  MATH  Article  Google Scholar 

  8. 8.

    Behera, B., Jahan, Q.: Characterization of wavelets and MRA wavelets on local fields of positive characteristic. Collect. Math. 66(1), 33–53 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  9. 9.

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

    MathSciNet  MATH  Article  Google Scholar 

  10. 10.

    Benedetto, R.L.: Examples of wavelets for local fields, Wavelets, frames and operator theory, 27–47, Contemp. Math., 345, Amer. Math. Soc., Providence, RI (2004)

  11. 11.

    Bownik, M.: The structure of shift-invariant subspaces of \(L^2({\mathbb{R}}^n)\). J. Funct. Anal. 177(2), 282–309 (2000)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Bownik, M.: Baggett’s problem for frame wavelets, Representations, wavelets, and frames, 153–173, Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston, MA (2008)

  13. 13.

    Bownik, M., Ross, K.A.: The structure of translation-invariant spaces on locally compact abelian groups. J. Fourier Anal. Appl. 21(4), 849–884 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  14. 14.

    Bownik, M., Rzeszotnik, Z.: On the existence of multiresolution analysis of framelets. Math. Ann. 332(4), 705–720 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  15. 15.

    Bownik, M., Rzeszotnik, Z., Speegle, D.: A characterization of dimension functions of wavelets. Appl. Comput. Harmon. Anal. 10(1), 71–92 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  16. 16.

    Currey, B., Mayeli, A.: Gabor fields and wavelet sets for the Heisenberg group. Monatsh. Math. 162(2), 119–142 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  17. 17.

    Currey, B., Mayeli, A., Oussa, V.: Characterization of shift-invariant spaces on a class of nilpotent Lie groups with applications. J. Fourier Anal. Appl. 20(2), 384–400 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  18. 18.

    Farkov, Yu A.: Orthogonal wavelets on locally compact abelian groups. Funct. Anal. Appl. 31(4), 294–296 (1997)

    MathSciNet  MATH  Article  Google Scholar 

  19. 19.

    Farkov, Yu A.: Multiresolution analysis and wavelets on Vilenkin groups, Facta Universitatis (NIS) Ser. Electron. Energy 21, 309–325 (2008)

    Google Scholar 

  20. 20.

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

    MathSciNet  MATH  Article  Google Scholar 

  21. 21.

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

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Li, D.F., Jiang, H.K.: The necessary condition and sufficient condition for wavelet frame on local fields. J. Math. Anal. Appl. 345(1), 500–510 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  23. 23.

    Rzeszotnik, Z.: Characterization theorems in the theory of wavelets, Ph.D. thesis, Washington University (2000)

  24. 24.

    Shukla, N.K., Vyas, A.: Multiresolution analysis through low-pass filter on local fields of positive characteristic. Complex Anal. Oper. Theory 9(3), 631–652 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  25. 25.

    Shukla, N.K., Maury, S.C.: Super-wavelets on local fields of positive characteristic. Math. Nachr. 291(4), 704–719 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  26. 26.

    Taibleson, M.H.: Fourier analysis on local fields. Princeton Univ. Press, Princeton (1975)

    MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank all the anonymous reviewers for providing fruitful suggestions to improve the presentation of this article. The first and second authors were supported by NBHM (DAE) grant-14723 and TEQIP-III, NPIU (MHRD), respectively, during the revision of manuscript.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Niraj K. Shukla.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Shukla, N.K., Maury, S.C. & Mittal, S. Semi-orthogonal Parseval Wavelets Associated with GMRAs on Local Fields of Positive Characteristic. Mediterr. J. Math. 16, 120 (2019). https://doi.org/10.1007/s00009-019-1383-1

Download citation

Mathematics Subject Classification

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

Keywords

  • Local fields
  • translation invariant spaces
  • multiplicity function
  • semi-orthogonal Parseval wavelets
  • bandlimited wavelets