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Mat. Sb., 2007, Volume 198, Number 1, Pages 103–126 (Mi msb1432)  

This article is cited in 28 scientific papers (total in 28 papers)

Wavelets and spectral analysis of ultrametric pseudodifferential operators

S. V. Kozyrev

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: The spectral theory of pseudodifferential operators on ultrametric spaces of general form is investigated with the use of the analysis of ultrametric wavelets. Bases of ultrametric wavelets are constructed on ultrametric spaces of analytic type; it is proved that bases of ultrametric wavelets are bases of eigenvectors for the introduced pseudodifferential operators and the corresponding eigenvalues are calculated. A generalization of the Vladimirov operator of $p$-adic fractional derivation is introduced for general ultrametric spaces.
Bibliography: 32 titles.


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English version:
Sbornik: Mathematics, 2007, 198:1, 97–116

Bibliographic databases:

UDC: 517.983.37+517.984.57+517.518.34
MSC: 54E45, 42C40
Received: 24.10.2005

Citation: S. V. Kozyrev, “Wavelets and spectral analysis of ultrametric pseudodifferential operators”, Mat. Sb., 198:1 (2007), 103–126; Sb. Math., 198:1 (2007), 97–116

Citation in format AMSBIB
\by S.~V.~Kozyrev
\paper Wavelets and spectral analysis
of~ultrametric pseudodifferential operators
\jour Mat. Sb.
\yr 2007
\vol 198
\issue 1
\pages 103--126
\jour Sb. Math.
\yr 2007
\vol 198
\issue 1
\pages 97--116

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    This publication is cited in the following articles:
    1. Khrennikov A.Yu., Mukhamedov F.M., Mendes J.F.F., “On $p$-adic Gibbs measures of the countable state Potts model on the Cayley tree”, Nonlinearity, 20:12 (2007), 2923–2937  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    2. O. G. Smolyanov, N. N. Shamarov, “Feynman and Feynman-Kac formulas for evolution equations with Vladimirov operator”, Dokl. Math., 77:3 (2008), 345–349  mathnet  crossref  mathscinet  zmath  isi  elib  elib  scopus
    3. O. G. Smolyanov, N. N. Shamarov, “Representation of solutions to a heat conduction equation with Vladimirov's operator by functional integrals”, Moscow University Mathematics Bulletin, 63:4 (2008), 138-143  mathnet  crossref  mathscinet  mathscinet  zmath  elib  elib  scopus
    4. S. V. Kozyrev, “Methods and Applications of Ultrametric and $p$-Adic Analysis: From Wavelet Theory to Biophysics”, Proc. Steklov Inst. Math., 274, suppl. 1 (2011), S1–S84  mathnet  crossref  crossref  zmath  isi  elib
    5. S. V. Kozyrev, “Toward an ultrametric theory of turbulence”, Theoret. and Math. Phys., 157:3 (2008), 1713–1722  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    6. Dragovich B., Khrennikov A.Yu., Kozyrev S.V., Volovich I.V., “On $p$-adic mathematical physics”, P-Adic Num. Ultrametr. Anal. Appl., 1:1 (2009), 1–17  crossref  mathscinet  zmath  scopus
    7. Proc. Steklov Inst. Math., 265 (2009), 13–29  mathnet  crossref  mathscinet  zmath  isi  elib
    8. Proc. Steklov Inst. Math., 265 (2009), 165–176  mathnet  crossref  mathscinet  zmath  isi  elib
    9. Proc. Steklov Inst. Math., 265 (2009), 177–198  mathnet  crossref  mathscinet  zmath  isi  elib
    10. O. G. Smolyanov, N. N. Shamarov, “Representation of solutions to evolution equations with Vladimirov operator in terms of Feynman path integrals”, Dokl. Math., 79:2 (2009), 270–274  mathnet  crossref  mathscinet  mathscinet  zmath  isi  elib  elib  scopus
    11. S.V.. Kozyrev, “Dynamics on rugged landscapes of energy and ultrametric diffusion”, P-Adic Num Ultrametr Anal Appl, 2:2 (2010), 122  crossref  mathscinet  zmath
    12. Mukhamedov F., “A dynamical system approach to phase transitions for $p$-adic potts model on the cayley tree of order two”, Rep. Math. Phys., 70:3 (2012), 385–406  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    13. Mukhamedov F., “On dynamical systems and phase transitions for $q+1$-state $p$-adic Potts model on the Cayley tree”, Math Phys Anal Geom, 2013  crossref  mathscinet  zmath  isi  elib  scopus
    14. Farrukh Mukhamedov, Hasan Ak{\i}n, “Phase transitions forp-adic Potts model on the Cayley tree of order three”, J. Stat. Mech, 2013:07 (2013), P07014  crossref  mathscinet  isi  scopus
    15. F. M. Mukhamedov, H. Akin, “The $p$-adic Potts model on the Cayley tree of order three”, Theoret. and Math. Phys., 176:3 (2013), 1267–1279  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    16. S. V. Kozyrev, A. Yu. Khrennikov, V. M. Shelkovich, “$p$-Adic wavelets and their applications”, Proc. Steklov Inst. Math., 285 (2014), 157–196  mathnet  crossref  crossref  isi  elib  elib
    17. Farrukh Mukhamedov, “Renormalization Method in p-Adic λ-Model on the Cayley Tree”, Int J Theor Phys, 2015  crossref  mathscinet  scopus
    18. Khrennikov A., Oleschko K., Correa Lopez Maria de Jesus, “Modeling Fluid's Dynamics with Master Equations in Ultrametric Spaces Representing the Treelike Structure of Capillary Networks”, Entropy, 18:7 (2016), 249  crossref  mathscinet  isi  scopus
    19. Oleschko K., Khrennikov A., de Jesus Correa Lopez M., “P-Adic Analog of Navier–Stokes Equations: Dynamics of Fluid'S Flow in Percolation Networks (From Discrete Dynamics With Hierarchic Interactions to Continuous Universal Scaling Model)”, Entropy, 19:4 (2017), 161  crossref  mathscinet  isi  scopus
    20. Dragovich B., Khrennikov A.Yu., Kozyrev S.V., Volovich I.V., Zelenov E.I., “P-Adic Mathematical Physics: the First 30 Years”, P-Adic Numbers Ultrametric Anal. Appl., 9:2 (2017), 87–121  crossref  mathscinet  zmath  scopus
    21. Theory Probab. Appl., 63:1 (2018), 94–116  mathnet  crossref  crossref  isi  elib
    22. Bendikov A., “Heat Kernels For Isotropic-Like Markov Generators on Ultrametric Spaces: a Survey”, P-Adic Numbers Ultrametric Anal. Appl., 10:1 (2018), 1–11  crossref  mathscinet  zmath  isi  scopus
    23. Bendikov A., Cygan W., Woess W., “Oscillating Heat Kernels on Ultrametric Spaces”, J. Spectr. Theory, 9:1 (2019), 195–226  crossref  mathscinet  zmath  isi  scopus
    24. Pourhadi E., Khrennikov A., Saadati R., Oleschko K., Correa Lopez Maria de Jesus, “Solvability of the P-Adic Analogue of Navier-Stokes Equation Via the Wavelet Theory”, Entropy, 21:11 (2019), 1129  crossref  mathscinet  isi
    25. Wu B., Khrennikov A., “P-Adic Analogue of the Wave Equation”, J. Fourier Anal. Appl., 25:5 (2019), 2447–2462  crossref  mathscinet  zmath  isi
    26. Bendikov A., Cygan W., “Poisson Approximation Related to Spectra of Hierarchical Laplacians”, Stoch. Dyn., 20:5 (2020), 2050035  crossref  mathscinet  isi
    27. Pourhadi E., Khrennikov A.Yu., Oleschko K., Lopez Maria de Jesus Correa, “Solving Nonlinearp-Adic Pseudo-Differential Equations: Combining the Wavelet Basis With the Schauder Fixed Point Theorem”, J. Fourier Anal. Appl., 26:4 (2020), 70  crossref  mathscinet  isi
    28. Zuniga-Galindo W.A., “Reaction-Diffusion Equations on Complex Networks and Turing Patterns, Via P-Adic Analysis”, J. Math. Anal. Appl., 491:1 (2020), 124239  crossref  mathscinet  isi
  • Математический сборник Sbornik: Mathematics (from 1967)
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