Explicit formula for MRA-wavelets on local fields


We provide an explicit formula for the wavelets associated with a multiresolution analysis of \(L^2(K)\), where K is a local field of positive characteristic. We also give several examples to illustrate this result.

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

Data availability

Not applicable.

Code availability

Not applicable.


  1. 1.

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

    MathSciNet  Article  Google Scholar 

  2. 2.

    Behera, B.: Haar wavelets on the Lebesgue spaces of local fields of positive characteristic. Colloq. Math. 136(2), 149–168 (2014)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Behera, B.: Wavelet sets and scaling sets in local fields (preprint)

  4. 4.

    Behera, B., Jahan, Q.: Wavelet packets and wavelet frame packets on local fields of positive characteristic. J. Math. Anal. Appl. 395, 1–14 (2012)

    MathSciNet  Article  Google Scholar 

  5. 5.

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

    MathSciNet  Article  Google Scholar 

  6. 6.

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

    MathSciNet  Article  Google Scholar 

  7. 7.

    Ciarlet, P.G.: Introduction to Numerical Linear Algebra and Optimisation. Cambridge University Press, Cambridge (1988)

    Google Scholar 

  8. 8.

    Jia, R.Q., Shen, Z.: Multiresolution and wavelets. Proc. Edinb. Math. Soc. 37, 271–300 (1994)

    MathSciNet  Article  Google Scholar 

  9. 9.

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

    MathSciNet  Article  Google Scholar 

  10. 10.

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

    MathSciNet  Article  Google Scholar 

  11. 11.

    Lukomskii, S.F.: Multiresolution analysis on zero-dimensional Abelian groups and wavelets bases. Sb. Math. 201, 669–691 (2010)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Lukomskii, S.F.: Multiresolution analysis on product of zero-dimensional abelian groups. J. Math. Anal. Appl. 385, 1162–1178 (2012)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Ramakrishnan, D., Valenza, R.: Fourier Analysis on Number Fields. Springer, New York (1999)

    Book  Google Scholar 

  14. 14.

    Taibleson, M.: Fourier Analysis on Local Fields. Princeton University Press, Princeton (1975)

    MATH  Google Scholar 

  15. 15.

    Weil, A.: Basic Number Theory. Springer, New York (1974)

    Book  Google Scholar 

  16. 16.

    Zalik, R.A.: Bases of translates and multiresolution analyses. Appl. Comput. Harmon. Anal. 24, 41–57 (2008) [Corrigendum: Appl. Comput. Harmon. Anal. 29, 121 (2010)]

  17. 17.

    Zalik, R.A.: On orthonormal wavelet bases. J. Comput. Anal. Appl. 27, 790–797 (2019)

    Google Scholar 

  18. 18.

    Zalik, R.A.: On multiresolution analyses of multiplicity \(n\). J. Comput. Anal. Appl. 29, 1055–1062 (2021)

    Google Scholar 

  19. 19.

    Zheng, W.X., Su, W.Y., Jiang, H.K.: A note to the concept of derivatives on local fields. Approx. Theory Appl. 6, 48–58 (1990)

    MathSciNet  MATH  Google Scholar 

Download references


Not applicable.

Author information



Corresponding author

Correspondence to Biswaranjan Behera.

Ethics declarations

Conflict of interest

Not applicable.

Additional information

Communicated by Ferenc Weisz.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Behera, B. Explicit formula for MRA-wavelets on local fields. Adv. Oper. Theory 6, 55 (2021). https://doi.org/10.1007/s43036-021-00152-3

Download citation


  • Local field
  • MRA-wavelet
  • p-Adic field
  • p-Series field

Mathematics Subject Classification

  • 43A70
  • 42C15
  • 42C40