Quincunx multiresolution analysis for L 2(ℚ 2 2 )

Research Articles


With an eye on applications in quantum mechanics and other areas of science, much work has been done to generalize traditional analytic methods to p-adic systems. In 2002 the first paper on p-adic wavelets was published. Since then p-adic wavelet sets, multiresolution analyses, and wavelet frames have all been introduced. However, so far all constructions have involved dilations by p. This paper presents the first construction of a p-adic wavelet system with a more general matrix dilation, laying the foundation for further work in this direction.

Key words

p-adic MRA quincunx matrix wavelet basis 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    V.A. Avetisov, A. H. Bikulov and S. V. Kozyrev, “Application of p-adic analysis tomodels of breaking of replica symmetry,” J. Phys. A 32(50), 8785–8791 (1999).CrossRefMathSciNetMATHGoogle Scholar
  2. 2.
    V. A. Avetisov, A. H. Bikulov, S. V. Kozyrev and V. A. Osipov, “p-Adic models of ultrametric diffusion constrained by hierarchical energy landscapes,” J. Phys. A 35(2), 177–189 (2002).CrossRefMathSciNetMATHGoogle Scholar
  3. 3.
    S. Albeverio, S. Evdokimov and M. Skopina, “p-Adicmultiresolution analysis and wavelet frames,” J. Fourier Anal. Appl. (2010), Accepted and pre-published online.Google Scholar
  4. 4.
    J. J. Benedetto and R. L. Benedetto, “A wavelet theory for local fields and related groups,” J. Geom. Anal. 14(3), 423–456 (2004).MathSciNetMATHGoogle Scholar
  5. 5.
    R. L. Benedetto, “Examples of wavelets for local fields,” Wavelets, Frames and Operator Theory, Contemp. Math. 345, 27–47 (2004) (Amer.Math. Soc., Providence, RI, 2004).Google Scholar
  6. 6.
    I. Daubechies, Ten Lectures onWavelets, CBMS-NSFRegionalConferenceSeries inAppliedMathematics 61 (Society for Industrial and AppliedMath., SIAM, Philadelphia, PA, 1992).Google Scholar
  7. 7.
    C. de Boor, R. A. DeVore and A. Ron, “On the construction of multivariate (pre)wavelets,” Constr. Approx. 9(2–3), 123–166 (1993).CrossRefMathSciNetMATHGoogle Scholar
  8. 8.
    B. Dragovich and A. Dragovich, “A p-adic model of DNA sequence and genetic code,” p-Adic Numbers, Ultrametric Analysis and Applications 1(1), 34–41 (2009).CrossRefMathSciNetMATHGoogle Scholar
  9. 9.
    K. Gröchenig and W. R. Madych, “Multiresolution analysis, Haar bases, and self-similar tilings of R n, IEEE Trans. Inform. Theory 38(2), 556–568 (1992).CrossRefMathSciNetGoogle Scholar
  10. 10.
    K. Gröchenig and T. Strohmer, “Pseudodifferential operators on locally compact abelian groups and Sjöstrand’s symbol class,” J. Reine Angew.Math. 613, 121–146 (2007).MathSciNetMATHGoogle Scholar
  11. 11.
    K. Hensel, “Über eine neue Begründung der Theorie der algebraischen Zahlen,” Jahresbericht der Deutschen Mathematiker-Vereinigung 6(3), 83–88 (1897).Google Scholar
  12. 12.
    S. Katok, p-Adic Analysis Compared with Real, Student Math. Library 37 (American Math. Society, Providence, RI, 2007).MATHGoogle Scholar
  13. 13.
    A. Khrennikov, p-Adic Valued Distributions in Mathematical Physics, Mathematics and its Appl. 309 (Kluwer Academic Publ. Group, Dordrecht, 1994).MATHGoogle Scholar
  14. 14.
    A. Yu. Khrennikov and S. V. Kozyrev, “Genetic code on the diadic plane,” Physica A: Stat. Mech. Appl. 381, 265–272 (2007).CrossRefGoogle Scholar
  15. 15.
    H. Koch, Number Theory, Graduate Studies in Math. 24 (American Math. Society, Providence, RI, 2000). Algebraic Numbers and Functions Translated from the 1997 German original by D. Kramer.Google Scholar
  16. 16.
    S. V. Kozyrev, “Wavelet theory as p-adic spectral analysis,” Izv. Ross. Akad. Nauk Ser.Mat. 66(2), 149–158 (2002).MathSciNetGoogle Scholar
  17. 17.
    S. V. Kozyrev, “p-Adic pseudodifferential operators and p-adic wavelets, Theor. Math. Phys. 138(3), 383–394 (2004).CrossRefMathSciNetGoogle Scholar
  18. 18.
    A. Yu. Khrennikov and V. M. Shelkovich, “An infinite family of p-adic non-Haar wavelet bases and pseudodifferential operators,” p-Adic Numbers, Ultrametric Analysis and Applications 1(3), 204–216 (2009).CrossRefMathSciNetMATHGoogle Scholar
  19. 19.
    A. Yu. Khrennikov, V. M. Shelkovich and M. Skopina, “p-Adic refinable functions and MRA-based wavelets,” J. Approx. Theory 161(1), 226–238 (2009).CrossRefMathSciNetMATHGoogle Scholar
  20. 20.
    A. Yu. Khrennikov, V. M. Shelkovich and M. Skopina, “p-Adic orthogonal wavelet bases,” p-Adic Numbers, Ultrametric Analysis and Applications 1(2), 145–156 (2009).CrossRefMathSciNetMATHGoogle Scholar
  21. 21.
    H. Reiter and J.D. Stegeman, Classical Harmonic Analysis and Locally CompactGroups, LondonMath. SocietyMonographs. New Series 22 (Clarendon Press Oxford Univ. Press, New York, 2000).Google Scholar
  22. 22.
    V. Shelkovich and M. Skopina, “p-Adic Haar multiresolution analysis and pseudo-differential operators,” J. Fourier Anal. Appl. 15(3), 366–393 (2009).CrossRefMathSciNetMATHGoogle Scholar
  23. 23.
    M. H. Taibleson, Fourier Analysis on Local Fields (Princeton Univ. Press, Princeton, N.J., 1975).MATHGoogle Scholar
  24. 24.
    V. S. Vladimirov, I. V. Volovich and E. I. Zelenov, p-Adic Analysis and Mathematical Physics (World Scientific Publ. Co. Inc., River Edge, NJ, 1994).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2010

Authors and Affiliations

  1. 1.Laboratory for Integrative and Medical Biophysics, National Institute for Child Health and Human DevelopmentNational Institutes of HealthBethesdaUSA
  2. 2.Department of Applied Mathematics and Control ProcessesSt. Petersburg State UniversitySt.PetersburgRussia

Personalised recommendations