Skip to main content
SpringerLink
Log in

p-Adic wavelets and their applications

  • Published:
Proceedings of the Steklov Institute of Mathematics Aims and scope Submit manuscript

Abstract

The theory of p-adic wavelets is presented. One-dimensional and multidimensional wavelet bases and their relation to the spectral theory of pseudodifferential operators are discussed. For the first time, bases of compactly supported eigenvectors for p-adic pseudodifferential operators were considered by V.S. Vladimirov. In contrast to real wavelets, p-adic wavelets are related to the group representation theory; namely, the frames of p-adic wavelets are the orbits of p-adic transformation groups (systems of coherent states). A p-adic multiresolution analysis is considered and is shown to be a particular case of the construction of a p-adic wavelet frame as an orbit of the action of the affine group.

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

Similar content being viewed by others

References

  1. S. Albeverio, S. Evdokimov, and M. Skopina, “p-Adic nonorthogonal wavelet bases,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 265, 7–18 (2009) [Proc. Steklov Inst. Math. 265, 1–12 (2009)].

    MathSciNet  Google Scholar 

  2. S. Albeverio, S. Evdokimov, and M. Skopina, “p-Adic multiresolution analysis and wavelet frames,” J. Fourier Anal. Appl. 16(5), 693–714 (2010); arXiv: 0802.1079v1 [math.CA].

    MATH  MathSciNet  Google Scholar 

  3. S. Albeverio, A. Yu. Khrennikov, and V. M. Shelkovich, “Harmonic analysis in the p-adic Lizorkin spaces: Fractional operators, pseudo-differential equations, p-adic wavelets, Tauberian theorems,” J. Fourier Anal. Appl. 12(4), 393–425 (2006).

    MATH  MathSciNet  Google Scholar 

  4. S. Albeverio, A. Yu. Khrennikov, and V. M. Shelkovich, “Pseudo-differential operators in the p-adic Lizorkin space,” in p-Adic Mathematical Physics: Proc. 2nd Int. Conf., Belgrade, Sept. 15–21, 2005, Ed. by A. Yu. Khrennikov, Z. Rakić, and I. V. Volovich (Am. Inst. Phys., Melville, NY, 2006), AIP Conf. Proc. 826, pp. 195–205.

    Google Scholar 

  5. S. Albeverio, A. Yu. Khrennikov, and V. M. Shelkovich, “p-Adic semilinear evolutionary pseudodifferential equations in Lizorkin spaces,” Dokl. Akad. Nauk 415(3), 295–299 (2007) [Dokl. Math. 76 (1), 539–543 (2007)].

    Google Scholar 

  6. S. Albeverio, A. Yu. Khrennikov, and V. M. Shelkovich, Theory of p-adic Distributions: Linear and Nonlinear Models (Cambridge Univ. Press, Cambridge, 2010), LMS Lect. Note Ser. 370.

    Google Scholar 

  7. S. Albeverio and S. V. Kozyrev, “Coincidence of the continuous and discrete p-adic wavelet transforms,” arXiv:math-ph/0702010.

  8. S. Albeverio and S. V. Kozyrev, “Multidimensional ultrametric pseudodifferential equations,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 265, 19–35 (2009) [Proc. Steklov Inst. Math. 265, 13–29 (2009)]; arXiv: 0708.2074 [math-ph].

    MathSciNet  Google Scholar 

  9. S. Albeverio and S. V. Kozyrev, “Frames of p-adic wavelets and orbits of the affine group,” p-Adic Numbers Ultrametric Anal. Appl. 1(1), 18–33 (2009); arXiv: 0801.4713 [math-ph].

    MATH  MathSciNet  Google Scholar 

  10. S. Albeverio and S. V. Kozyrev, “Multidimensional basis of p-adic wavelets and representation theory,” p-Adic Numbers Ultrametric Anal. Appl. 1(3), 181–189 (2009); arXiv: 0903.0461 [math-ph].

    MATH  MathSciNet  Google Scholar 

  11. S. Albeverio and S. V. Kozyrev, “Multidimensional p-adic wavelets for the deformed metric,” p-Adic Numbers Ultrametric Anal. Appl. 2(4), 265–277 (2010); arXiv: 1105.1524 [math.FA].

    MATH  MathSciNet  Google Scholar 

  12. S. Albeverio, S. Kuzhel, and S. Torba, “p-Adic Schrödinger-type operator with point interactions,” J. Math. Anal. Appl. 338(2), 1267–1281 (2008).

    MATH  MathSciNet  Google Scholar 

  13. I. Ya. Aref’eva, B. Dragovich, P. H. Frampton, and I. V. Volovich, “The wave function of the Universe and p-adic gravity,” Int. J. Mod. Phys. A 6(24), 4341–4358 (1991).

    MATH  MathSciNet  Google Scholar 

  14. I. Ya. Aref’eva, B. G. Dragović, and I. V. Volovich, “On the adelic string amplitudes,” Phys. Lett. B 209(4), 445–450 (1988).

    MathSciNet  Google Scholar 

  15. I. Aref’eva and P. H. Frampton, “Beyond Planck energy to non-Archimedean geometry,” Mod. Phys. Lett. A 6(4), 313–316 (1991).

    MATH  MathSciNet  Google Scholar 

  16. I. Ya. Aref’eva and I. V. Volovich, “Strings, gravity and p-adic space-time,” in Quantum Gravity: Proc. Fourth Seminar, Moscow, May 25–29, 1987, Ed. by M. A. Markov, V. A. Berezin, and V. P. Frolov (World Sci., Singapore, 1988), pp. 409–422.

    Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  18. R. L. Benedetto, “Examples of wavelets for local fields,” in Wavelets, Frames, and Operator Theory: Proc. Workshop, College Park, MD, 2003 (Am. Math. Soc., Providence, RI, 2004), Contemp. Math. 345, pp. 27–47.

    Google Scholar 

  19. P. Cartier, “Harmonic analysis on trees,” in Harmonic Analysis on Homogeneous Spaces: Proc. Symp. Pure Math., Williamstown, MA, 1972 (Am. Math. Soc., Providence, RI, 1973), Proc. Symp. Pure Math. 26, pp. 419–424.

    Google Scholar 

  20. P. Cartier, “Representations of p-adic groups: A survey,” in Automorphic Forms, Representations and L-Functions: Proc. Symp. Pure Math., Corvallis, OR, 1977 (Am. Math. Soc., Providence, RI, 1979), Part 1, Proc. Symp. Pure Math. 33, pp. 111–155.

    Google Scholar 

  21. O. Casas-Sánchez and W. A. Zúñiga-Galindo, “Riesz kernels and pseudodifferential operators attached to quadratic forms over p-adic fields,” p-Adic Numbers Ultrametric Anal. Appl. 5(3), 177–193 (2013).

    MATH  MathSciNet  Google Scholar 

  22. L. F. Chacón-Cortes and W. A. Zúñiga-Galindo, “Nonlocal operators, parabolic-type equations, and ultrametric random walks,” J. Math. Phys. 54(11), 113503 (2013).

    MathSciNet  Google Scholar 

  23. I. Daubechies, Ten Lectures on Wavelets (SIAM, Philadelphia, PA, 1992), CBMS-NSF Reg. Conf. Ser. Appl. Math. 61.

    MATH  Google Scholar 

  24. B. G. Dragovich, “On signature change in p-adic space-times,” Mod. Phys. Lett. A 6(25), 2301–2307 (1991).

    MathSciNet  Google Scholar 

  25. B. Dragovich, A. Yu. Khrennikov, S. V. Kozyrev, and I. V. Volovich, “On p-adic mathematical physics,” p-Adic Numbers Ultrametric Anal. Appl. 1(1), 1–17 (2009).

    MATH  MathSciNet  Google Scholar 

  26. B. Dragovich and Lj. Nesic, “On p-adic numbers in gravity,” Balkan Phys. Lett. 6, 78–81 (1998).

    Google Scholar 

  27. Yu. A. Farkov, “Orthogonal wavelets with compact support on locally compact Abelian groups,” Izv. Ross. Akad. Nauk, Ser. Mat. 69(3), 193–220 (2005) [Izv. Math. 69, 623–650 (2005)].

    MathSciNet  Google Scholar 

  28. Yu. A. Farkov, “Multiresolution analysis and wavelets on Vilenkin groups,” Facta Univ., Ser. Electron. Energ. 21(3), 309–325 (2008).

    Google Scholar 

  29. Yu. A. Farkov, “Biorthogonal wavelets on Vilenkin groups,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 265, 110–124 (2009) [Proc. Steklov Inst. Math. 265, 101–114 (2009)].

    MathSciNet  Google Scholar 

  30. Yu. A. Farkov, “On wavelets related to the Walsh series,” J. Approx. Theory 161(1), 259–279 (2009).

    MATH  MathSciNet  Google Scholar 

  31. Yu. A. Farkov, “Wavelets and frames based on Walsh-Dirichlet type kernels,” Commun. Math. Appl. 1(1), 27–46 (2010).

    MATH  MathSciNet  Google Scholar 

  32. P. G. O. Freund and E. Witten, “Adelic string amplitudes,” Phys. Lett. B 199(2), 191–194 (1987).

    MathSciNet  Google Scholar 

  33. I. M. Gel’fand, M. I. Graev, and I. I. Pyatetski-Shapiro, Representation Theory and Automorphic Functions (Nauka, Moscow, 1966; Academic, Boston, 1990), Generalized Functions 6.

    MATH  Google Scholar 

  34. B. I. Golubov, “A modified strong dyadic integral and derivative,” Mat. Sb. 193(4), 37–60 (2002) [Sb. Math. 193, 507–529 (2002)].

    MathSciNet  Google Scholar 

  35. B. I. Golubov, “A dyadic analogue of Wiener’s Tauberian theorem and some related questions,” Izv. Ross. Akad. Nauk, Ser. Mat. 67(1), 33–58 (2003) [Izv. Math. 67, 29–53 (2003)].

    MathSciNet  Google Scholar 

  36. B. I. Golubov, “Fractional modified dyadic integral and derivative on ℝ+,” Funkts. Anal. Prilozh. 39(2), 64–70 (2005) [Funct. Anal. Appl. 39, 135–139 (2005)].

    MathSciNet  Google Scholar 

  37. B. I. Golubov, “Modified dyadic integral and fractional derivative on ℝ+,” Mat. Zametki 79(2), 213–233 (2006) [Math. Notes 79, 196–214 (2006)].

    MathSciNet  Google Scholar 

  38. B. I. Golubov, A. V. Efimov, and V. A. Skvortsov, Walsh Series and Transforms: Theory and Applications, 2nd ed. (Izd. LKI, Moscow, 2008); Engl. transl. of the 1st ed.: Walsh Series and Transforms: Theory and Applications (Kluwer, Dordrecht, 1991).

    Google Scholar 

  39. K. Gröchenig and W. R. Madych,, “Multiresolution analysis, Haar bases, and self-similar tilings of R n,” IEEE Trans. Inf. Theory 38(2), 556–568 (1992).

    MATH  Google Scholar 

  40. A. Haar, “Zur Theorie der orthogonalen Funktionensysteme,” Math. Ann. 69, 331–371 (1910).

    MATH  MathSciNet  Google Scholar 

  41. G. Kaiser, A Friendly Guide to Wavelets (Birkhäuser, Boston, 1994).

    MATH  Google Scholar 

  42. B. S. Kashin and A. A. Saakyan, Orthogonal Series, 2nd ed. (AFTs, Moscow, 1999); Engl. transl. of the 1st ed.: Orthogonal Series (Am. Math. Soc., Providence, RI, 1989), Transl. Math. Monogr. 75.

    MATH  Google Scholar 

  43. A. Yu. Khrennikov, “Fundamental solutions over the field of p-adic numbers,” Algebra Anal. 4(3), 248–266 (1992) [St. Petersburg Math. J. 4, 613–628 (1993)].

    MathSciNet  Google Scholar 

  44. A. Yu. Khrennikov, p-Adic Valued Distributions in Mathematical Physics (Kluwer, Dordrecht, 1994).

    MATH  Google Scholar 

  45. A. Yu. Khrennikov, Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models (Kluwer, Dordrecht, 1997).

    MATH  Google Scholar 

  46. A. Yu. Khrennikov, Non-Archimedean Analysis and Its Applications (Fizmatlit, Moscow, 2003) [in Russian].

    MATH  Google Scholar 

  47. A. Yu. Khrennikov and S. V. Kozyrev, “Wavelets on ultrametric spaces,” Appl. Comput. Harmon. Anal. 19(1), 61–76 (2005).

    MATH  MathSciNet  Google Scholar 

  48. A. Yu. Khrennikov and S. V. Kozyrev, “Wavelets and the Cauchy problem for the Schrödinger equation on analytic ultrametric space,” in Mathematical Modeling of Wave Phenomena: Proc. 2nd Conf., Växjö, Sweden, Aug. 14–19, 2005, Ed. by B. Nilsson and L. Fishman (Am. Inst. Phys., Melville, NY, 2006), AIP Conf. Proc. 834, pp. 344–350.

    Google Scholar 

  49. A. Yu. Khrennikov and S. V. Kozyrev, “Ultrametric random field,” Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9(2), 199–213 (2006); arXiv:math/0603584 [math.PR].

    MATH  MathSciNet  Google Scholar 

  50. A. Yu. Khrennikov, S. V. Kozyrev, K. Oleschko, A. G. Jaramillo, and M. de Jesús Correa López, “Application of p-adic analysis to time series,” Infin. Dimens. Anal. Quantum Probab. Relat. Top. 16(4), 1350030 (2013).

    MathSciNet  Google Scholar 

  51. A. Yu. Khrennikov and V. M. Shelkovich, “Distributional asymptotics and p-adic Tauberian and Shannon-Kotelnikov theorems,” Asymptotic Anal. 46(2), 163–187 (2006).

    MATH  MathSciNet  Google Scholar 

  52. A. Yu. Khrennikov and V. M. Shelkovich, “p-Adic multidimensional wavelets and their application to p-adic pseudo-differential operators,” arXiv:math-ph/0612049.

  53. A. Yu. Khrennikov and V. M. Shelkovich, “Non-Haar p-adic wavelets and pseudodifferential operators,” Dokl. Akad. Nauk 418(2), 167–170 (2008) [Dokl. Math. 77 (1), 42–45 (2008)].

    MathSciNet  Google Scholar 

  54. A. Yu. Khrennikov and V. M. Shelkovich, “An infinite family of p-adic non-Haar wavelet bases and pseudodifferential operators,” p-Adic Numbers Ultrametric Anal. Appl. 1(3), 204–216 (2009).

    MATH  MathSciNet  Google Scholar 

  55. A. Yu. Khrennikov and V. M. Shelkovich, “Non-Haar p-adic wavelets and their application to pseudo-differential operators and equations,” Appl. Comput. Harmon. Anal. 28(1), 1–23 (2010); arXiv: 0808.3338v1 [math-ph].

    MATH  MathSciNet  Google Scholar 

  56. A. Yu. Khrennikov, V. M. Shelkovich, and M. Skopina, “p-Adic orthogonal wavelet bases,” p-Adic Numbers Ultrametric Anal. Appl. 1(2), 145–156 (2009).

    MATH  MathSciNet  Google Scholar 

  57. 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).

    MATH  MathSciNet  Google Scholar 

  58. A. Yu. Khrennikov, V. M. Shelkovich, and J. H. van der Walt, “Adelic multiresolution analysis, construction of wavelet bases and pseudo-differential operators,” J. Fourier Anal. Appl. 19(6), 1323–1358 (2013).

    MathSciNet  Google Scholar 

  59. E. J. King and M. A. Skopina, “Quincunx multiresolution analysis for \(L^2 (\mathbb{Q}_2^2 )\),” p-Adic Numbers Ultrametric Anal. Appl. 2(3), 222–231 (2010).

    MATH  MathSciNet  Google Scholar 

  60. J. R. Klauder and E. C. G. Sudarshan, Fundamentals of Quantum Optics (Benjamin, New York, 1968).

    Google Scholar 

  61. A. N. Kochubei, “Schrödinger-type operator over p-adic number field,” Teor. Mat. Fiz. 86(3), 323–333 (1991) [Theor. Math. Phys. 86, 221–228 (1991)].

    MathSciNet  Google Scholar 

  62. A. N. Kochubei, “Parabolic equations over the field of p-adic numbers,” Izv. Akad. Nauk. SSSR, Ser. Mat. 55(6), 1312–1330 (1991) [Math. USSR, Izv. 39 (3), 1263–1280 (1992)].

    Google Scholar 

  63. A. N. Kochubei, “The differentiation operator on subsets of the field of p-adic numbers,” Izv. Ross. Akad. Nauk., Ser. Mat. 56(5), 1021–1039 (1992) [Russ. Acad. Sci., Izv. Math. 41 (2), 289–305 (1993)].

    Google Scholar 

  64. A. N. Kochubei, “Additive and multiplicative fractional differentiations over the field of p-adic numbers,” in p-Adic Functional Analysis, Ed. by W. H. Schikhof et al. (M. Dekker, New York, 1997), Lect. Notes Pure Appl. Math. 192, pp. 275–280.

    Google Scholar 

  65. A. N. Kochubei, “Fundamental solutions of pseudodifferential equations connected with p-adic quadratic forms,” Izv. Ross. Akad. Nauk., Ser. Mat. 62(6), 103–124 (1998) [Izv. Math. 62, 1169–1188 (1998)].

    MathSciNet  Google Scholar 

  66. A. N. Kochubei, Pseudo-differential Equations and Stochastics over Non-Archimedean Fields (M. Dekker, New York, 2001).

    MATH  Google Scholar 

  67. A. N. Kochubei, “A non-Archimedean wave equation,” Pac. J. Math. 235(2), 245–261 (2008).

    MATH  MathSciNet  Google Scholar 

  68. S. V. Konyagin and I. E. Shparlinski, Character Sums with Exponential Functions and Their Applications (Cambridge Univ. Press, Cambridge, 1999).

    MATH  Google Scholar 

  69. A. V. Kosyak, A. Yu. Khrennikov, and V. M. Shelkovich, “Wavelet bases on adele rings,” Dokl. Akad. Nauk 442(4), 446–450 (2012) [Dokl. Math. 85 (1), 75–79 (2012)].

    MathSciNet  Google Scholar 

  70. A. V. Kosyak, A. Yu. Khrennikov, and V. M. Shelkovich, “Pseudodifferential operators on adele rings and wavelet bases,” Dokl. Akad. Nauk 444(3), 253–257 (2012) [Dokl. Math. 85 (3), 358–362 (2012)].

    MathSciNet  Google Scholar 

  71. S. V. Kozyrev, “Wavelet theory as p-adic spectral analysis,” Izv. Ross. Akad. Nauk., Ser. Mat. 66(2), 149–158 (2002) [Izv. Math. 66, 367–376 (2002)]; arXiv:math-ph/0012019.

    MathSciNet  Google Scholar 

  72. S. V. Kozyrev, “p-Adic pseudodifferential operators: Methods and applications,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 245, 154–165 (2004) [Proc. Steklov Inst. Math. 245, 143–153 (2004)].

    MathSciNet  Google Scholar 

  73. S. V. Kozyrev, “p-Adic pseudodifferential operators and p-adic wavelets,” Teor. Mat. Fiz. 138(3), 383–394 (2004) [Theor. Math. Phys. 138, 322–332 (2004)]; arXiv:math-ph/0303045.

    MathSciNet  Google Scholar 

  74. S. V. Kozyrev, “Wavelets and spectral analysis of ultrametric pseudodifferential operators,” Mat. Sb. 198(1), 103–126 (2007) [Sb. Math. 198, 97–116 (2007)]; “Ultrametric pseudodifferential operators and wavelets for the case of non-homogeneous measure,” arXiv:math-ph/0412082.

    MathSciNet  Google Scholar 

  75. S. V. Kozyrev, Methods and Applications of Ultrametric and p-adic Analysis: From Wavelet Theory to Biophysics (MIAN, Moscow, 2008), Sovrem. Probl. Mat. 12, http://www.mi.ras.ru/spm/pdf/012.pdf; Engl. transl. in Proc. Steklov Inst. Math. 274 (Suppl. 1), 1–84 (2011).

    Google Scholar 

  76. S. V. Kozyrev, “Toward an ultrametric theory of turbulence,” Teor. Mat. Fiz. 157(3), 413–424 (2008) [Theor. Math. Phys. 157, 1713–1722 (2008)]; arXiv: 0803.2719 [math-ph].

    MathSciNet  Google Scholar 

  77. S. V. Kozyrev and A. Yu. Khrennikov, “Pseudodifferential operators on ultrametric spaces and ultrametric wavelets,” Izv. Ross. Akad. Nauk., Ser. Mat. 69(5), 133–148 (2005) [Izv. Math. 69, 989–1003 (2005)]; arXiv:math-ph/0412062.

    MathSciNet  Google Scholar 

  78. S. V. Kozyrev and A. Yu. Khrennikov, “Localization in space for a free particle in ultrametric quantum mechanics,” Dokl. Akad. Nauk 411(3), 319–322 (2006) [Dokl. Math. 74 (3), 906–909 (2006)].

    MathSciNet  Google Scholar 

  79. S. V. Kozyrev and A. Yu. Khrennikov, “p-Adic integral operators in wavelet bases,” Dokl. Akad. Nauk 437(4), 457–461 (2011) [Dokl. Math. 83 (2), 209–212 (2011)].

    MathSciNet  Google Scholar 

  80. S. V. Kozyrev, V. Al. Osipov, and V. A. Avetisov, “Nondegenerate ultrametric diffusion,” J. Math. Phys. 46(6), 063302 (2005).

    MathSciNet  Google Scholar 

  81. S. Kuzhel and S. Torba, “p-Adic fractional differential operators with point interactions,” Methods Funct. Anal. Topol. 13(2), 169–180 (2007).

    MATH  MathSciNet  Google Scholar 

  82. W. C. Lang, “Orthogonal wavelets on the Cantor dyadic group,” SIAM J. Math. Anal. 27, 305–312 (1996).

    MATH  MathSciNet  Google Scholar 

  83. W. C. Lang, “Wavelet analysis on the Cantor dyadic group,” Houston J. Math. 24, 533–544 (1998).

    MATH  MathSciNet  Google Scholar 

  84. P. I. Lizorkin, “Generalized Liouville differentiation and function spaces L r p (E n). Embedding theorems,” Mat. Sb. 60(3), 325–353 (1963).

    MathSciNet  Google Scholar 

  85. P. I. Lizorkin, “Operators connected with fractional differentiation, and classes of differentiable functions,” Tr. Mat. Inst. im. V.A. Steklova, Akad. Nauk SSSR 117, 212–243 (1972) [Proc. Steklov Inst. Math. 117, 251–286 (1974)].

    MATH  MathSciNet  Google Scholar 

  86. S. Mallat, “An efficient image representation for multiscale analysis,” in Topical Meeting on Machine Vision, Incline Village, Nevada, 1987 (Opt. Soc. Am., Washington D.C., 1987), pp. 172–175.

    Google Scholar 

  87. S. Mallat, “Multiresolution representation and wavelets,” PhD Thesis (Univ. Pennsylvania, Philadelphia, PA, 1988).

    Google Scholar 

  88. Y. Meyer, “Principe d’incertitude, bases hilbertiennes et algèbres d’opérateurs,” in Séminaire Bourbaki 1985/1986 (Soc. Math. France, Paris, 1987), Exp. 662, Astérisque 145/146, pp. 209–223.

    Google Scholar 

  89. Y. Meyer, Ondelettes et fonctions splines (École Polytech., Palaiseau, 1987), Sémin. équations dériv. partielles 1986–1987; Exp. 6.

    Google Scholar 

  90. Y. Meyer, Wavelets and Operators (Cambridge Univ. Press, Cambridge, 1992).

    MATH  Google Scholar 

  91. Yu. A. Neretin, “On combinatorial analogs of the group of diffeomorphisms of the circle,” Izv. Ross. Akad. Nauk., Ser. Mat. 56(5), 1072–1085 (1992) [Russ. Acad. Sci., Izv. Math. 41 (2), 337–349 (1993)].

    Google Scholar 

  92. I. Ya. Novikov, V. Yu. Protasov, and M. A. Skopina, Wavelet Theory (Fizmatlit, Moscow, 2005; Am. Math. Soc., Providence, RI, 2011).

    MATH  Google Scholar 

  93. I. Ya. Novikov and M. A. Skopina, “Why are Haar bases in various structures the same?,” Mat. Zametki 91(6), 950–953 (2012) [Math. Notes 91, 895–898 (2012)].

    MathSciNet  Google Scholar 

  94. G. I. Ol’shanskii, “Classification of irreducible representations of groups of automorphisms of Bruhat-Tits trees,” Funkts. Anal. Prilozh. 11(1), 32–42 (1977) [Funct. Anal. Appl. 11, 26–34 (1977)].

    MATH  Google Scholar 

  95. A. M. Perelomov, Generalized Coherent States and Their Applications (Nauka, Moscow, 1987; Springer, Berlin, 1986).

    MATH  Google Scholar 

  96. V. Yu. Protasov and Yu. A. Farkov, “Dyadic wavelets and refinable functions on a half-line,” Mat. Sb. 197(10), 129–160 (2006) [Sb. Math. 197, 1529–1558 (2006)].

    MathSciNet  Google Scholar 

  97. E. A. Rodionov and Yu. A. Farkov, “Estimates of the smoothness of dyadic orthogonal wavelets of Daubechies type,” Mat. Zametki 86(3), 429–444 (2009) [Math. Notes 86, 407–421 (2009)].

    MathSciNet  Google Scholar 

  98. J. J. Rodríguez-Vega and W. A. Zúñiga-Galindo, “Taibleson operators, p-adic parabolic equations and ultrametric diffusion,” Pac. J. Math. 237(2), 327–347 (2008).

    MATH  Google Scholar 

  99. J.-P. Serre, Arbres, amalgames, SL2 (Soc. Math. France, Paris, 1977), Astérisque 46.

    Google Scholar 

  100. J.-P. Serre, Trees (Springer, Berlin, 1980, 2003).

    MATH  Google Scholar 

  101. V. Shelkovich and M. Skopina, “p-Adic Haar multiresolution analysis and pseudo-differential operators,” J. Fourier Anal. Appl. 15(3), 366–393 (2009).

    MATH  MathSciNet  Google Scholar 

  102. M. Taibleson, “Harmonic analysis on n-dimensional vector spaces over local fields. I: Basic results on fractional integration,” Math. Ann. 176, 191–207 (1968).

    MathSciNet  Google Scholar 

  103. M. H. Taibleson, Fourier Analysis on Local Fields (Princeton Univ. Press, Princeton, NJ, 1975).

    MATH  Google Scholar 

  104. S. M. Torba and W. A. Zúñiga-Galindo, “Parabolic type equations and Markov stochastic processes on adeles,” J. Fourier Anal. Appl. 19(4), 792–835 (2013).

    MathSciNet  Google Scholar 

  105. V. S. Vladimirov, “Generalized functions over the field of p-adic numbers,” Usp. Mat. Nauk 43(5), 17–53 (1988) [Russ. Math. Surv. 43 (5), 19–64 (1988)].

    Google Scholar 

  106. V. S. Vladimirov, “On the spectrum of some pseudodifferential operators over the field of p-adic numbers,” Algebra Anal. 2(6), 107–124 (1990) [Leningrad Math. J. 2, 1261–1278 (1991)].

    MATH  Google Scholar 

  107. V. S. Vladimirov, “On spectral properties of p-adic pseudodifferential operators of Schrödinger type,” Izv. Ross. Akad. Nauk., Ser. Mat. 56(4), 770–789 (1992) [Russ. Acad. Sci., Izv. Math. 41 (1), 55–73 (1993)].

    Google Scholar 

  108. V. S. Vladimirov, “Derivation of Freund-Witten adelic formula for four-point Veneziano amplitudes,” Teor. Mat. Fiz. 94(3), 355–367 (1993) [Theor. Math. Phys. 94, 251–259 (1993)].

    Google Scholar 

  109. V. S. Vladimirov, “Freund-Witten adelic formulae for Veneziano and Virasoro-Shapiro amplitudes,” Usp. Mat. Nauk 48(6), 3–38 (1993) [Russ. Math. Surv. 48 (6), 1–39 (1993)].

    Google Scholar 

  110. V. S. Vladimirov, “On the Freund-Witten adelic formula for Veneziano amplitudes,” Lett. Math. Phys. 27(2), 123–131 (1993).

    MATH  MathSciNet  Google Scholar 

  111. V. S. Vladimirov, “Adelic formulas for the four-point Veneziano and Virasoro-Shapiro amplitudes,” Dokl. Akad. Nauk 333(6), 717–721 (1993) [Phys. Dokl. 38 (12), 486–489 (1993)].

    Google Scholar 

  112. V. S. Vladimirov and I. V. Volovich, “Superanalysis. I: Differential calculus,” Teor. Mat. Fiz. 59(1), 3–27 (1984) [Theor. Math. Phys. 59, 317–335 (1984)].

    MathSciNet  Google Scholar 

  113. V. S. Vladimirov and I. V. Volovich, “Superanalysis. II: Integral calculus,” Teor. Mat. Fiz. 60(2), 169–198 (1984) [Theor. Math. Phys. 60, 743–765 (1984)].

    MathSciNet  Google Scholar 

  114. V. S. Vladimirov and I. V. Volovich, “p-Adic quantum mechanics,” Dokl. Akad. Nauk SSSR 302(2), 320–323 (1988) [Sov. Phys., Dokl. 33 (9), 669–670 (1988)].

    Google Scholar 

  115. V. S. Vladimirov and I. V. Volovich, “A vacuum state in p-adic quantum mechanics,” Phys. Lett. B 217(4), 411–415 (1989).

    MathSciNet  Google Scholar 

  116. V. S. Vladimirov and I. V. Volovich, “p-Adic quantum mechanics,” Commun. Math. Phys. 123(4), 659–676 (1989).

    MATH  MathSciNet  Google Scholar 

  117. V. S. Vladimirov and I. V. Volovich, “p-Adic Schrödinger-type equation,” Lett. Math. Phys. 18(1), 43–53 (1989).

    MATH  MathSciNet  Google Scholar 

  118. V. S. Vladimirov, I. V. Volovich, and E. I. Zelenov, “Spectral theory in p-adic quantum mechanics and representation theory,” Dokl. Akad. Nauk SSSR 310(2), 272–276 (1990) [Sov. Math., Dokl. 41 (1), 40–44 (1990)].

    MathSciNet  Google Scholar 

  119. V. S. Vladimirov, I. V. Volovich, and E. I. Zelenov, “Spectral theory in p-adic quantum mechanics, and representation theory,” Izv. Akad. Nauk. SSSR, Ser. Mat. 54(2), 275–302 (1990) [Math. USSR, Izv. 36, 281–309 (1991)].

    MATH  MathSciNet  Google Scholar 

  120. V. S. Vladimirov, I. V. Volovich, and E. I. Zelenov, p-Adic Analysis and Mathematical Physics (Nauka, Moscow, 1994; World Sci., Singapore, 1994).

    Google Scholar 

  121. I. V. Volovich, “p-Adic space-time and string theory,” Teor. Mat. Fiz. 71(3), 337–340 (1987) [Theor. Math. Phys. 71, 574–576 (1987)].

    MathSciNet  Google Scholar 

  122. I. V. Volovich, “p-Adic string,” Classical Quantum Gravity 4, L83–L87 (1987).

    MathSciNet  Google Scholar 

  123. W. A. Zuniga-Galindo, “Fundamental solutions of pseudo-differential operators over p-adic fields,” Rend. Semin. Mat. Univ. Padova 109, 241–245 (2003).

    MATH  MathSciNet  Google Scholar 

  124. W. A. Zuniga-Galindo, “Pseudo-differential equations connected with p-adic forms and local zeta functions,” Bull. Aust. Math. Soc. 70(1), 73–86 (2004).

    MATH  MathSciNet  Google Scholar 

  125. W. A. Zúñiga-Galindo, “Local zeta functions and fundamental solutions for pseudo-differential operators over p-adic fields,” p-Adic Numbers Ultrametric Anal. Appl. 3(4), 344–358 (2011).

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Steklov Mathematical Institute, Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991, Russia

    S. V. Kozyrev

  2. International Center for Mathematical Modeling in Physics, Engineering and Cognitive Sciences, Linnaeus University, SE-351 95, Växjö, Sweden

    A. Yu. Khrennikov

  3. Saint-Petersburg State University of Architecture and Civil Engineering, Vtoraya Krasnoarmeiskaya ul. 4, St. Petersburg, 190005, Russia

    V. M. Shelkovich

  4. Department of Higher Mathematics and Mathematical Physics, Faculty of Physics, St. Petersburg State University, ul. Ul’yanovskaya 3, St. Petersburg, 198904, Russia

    V. M. Shelkovich

Authors
  1. S. V. Kozyrev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Yu. Khrennikov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. V. M. Shelkovich
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. V. Kozyrev.

Additional information

Original Russian Text © S.V. Kozyrev, A.Yu. Khrennikov, V.M. Shelkovich, 2014, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2014, Vol. 285, pp. 166–206.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kozyrev, S.V., Khrennikov, A.Y. & Shelkovich, V.M. p-Adic wavelets and their applications. Proc. Steklov Inst. Math. 285, 157–196 (2014). https://doi.org/10.1134/S0081543814040129

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0081543814040129

Keywords

Search

Navigation