Skip to main content
SpringerLink
Log in

Discrete orthogonal and Riesz refinable functions on local fields of positive characteristic

  • Research Article
  • Published:
European Journal of Mathematics Aims and scope Submit manuscript

Abstract

Necessary conditions under which shift systems of refinable functions are orthogonal, or Riesz systems on Vilenkin groups and local fields of positive characteristic, are well known. These conditions are expressed in terms of a mask. We indicate a class of refinable functions for which these conditions are necessary and sufficient.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

References

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Behera, B., Jahan, Q.: Biorthogonal wavelets on local fields of positive characteristic. Commun. Math. Anal. 15(2), 52–75 (2013)

    MathSciNet  MATH  Google Scholar 

  4. Berdnikov, G.S., Kruss, Iu.S., Lukomskii, S.F.: On orthogonal systems of shifts of scaling function on local fields of positive characteristic. Turkish J. Math 41(2), 244–253 (2017)

  5. Farkov, Yu.A.: Orthogonal wavelets with compact support on locally compact abelian groups. Izv. Math. 69(3), 623–650 (2005)

  6. Farkov, Yu.A.: Orthogonal wavelets on direct products of cyclic groups. Math. Notes 82(6), 843–859 (2007)

  7. Farkov, Yu.A.: Biorthogonal wavelets on Vilenkin groups. Proc. Steklov Inst. Math. 265(1), 101–114 (2009)

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  11. Lukomskii, S.F.: Riesz multiresolution analysis on zero-dimensional groups. Izv. Math. 79(1), 145–176 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Lukomskii, S.F., Berdnikov, G.S.: \(N\)-valid trees in wavelet theory on Vilenkin groups. Int. J. Wavelets Multiresolut. Inf. Process. 13(5), # 1550037 (2015)

    MathSciNet  MATH  Google Scholar 

  13. Lukomskii, S.F., Vodolazov, A.M.: Non-Haar MRA on local fields of positive characteristic. J. Math. Anal. Appl. 433(2), 1415–1440 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  14. Novikov, I.Ya., Protasov, V.Yu., Skopina, M.A.: Wavelet Theory. Mathematical Monographs, vol. 239. American Mathematical Society, Providence (2011)

  15. Protasov, V.Yu., Farkov, Yu.A: Dyadic wavelets and refinable functions on a half-line. Sb. Math. 197(10), 1529–1558 (2006)

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

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Mechanics, N.G. Chernyshevskii Saratov State University, Astrahanskaya st.,83, 410026, Saratov, Russia

    Gleb S. Berdnikov & Sergei F. Lukomskii

Authors
  1. Gleb S. Berdnikov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sergei F. Lukomskii
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sergei F. Lukomskii.

Additional information

Publisher's Note

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

This work was supported by SEMC “Mathematics of Future Technologies”.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Berdnikov, G.S., Lukomskii, S.F. Discrete orthogonal and Riesz refinable functions on local fields of positive characteristic. European Journal of Mathematics 6, 1505–1522 (2020). https://doi.org/10.1007/s40879-020-00420-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40879-020-00420-1

Keywords

Mathematics Subject Classification

Search

Navigation