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TMF, 2004, Volume 138, Number 3, Pages 383–394 (Mi tmf31)  

This article is cited in 42 scientific papers (total in 42 papers)

$p$-Adic Pseudodifferential Operators and $p$-Adic Wavelets

S. V. Kozyrev

N. N. Semenov Institute of Chemical Physics, Russian Academy of Sciences

Abstract: We introduce a new wide class of $p$-adic pseudodifferential operators. We show that the basis of $p$-adic wavelets is the basis of eigenvectors for the introduced operators.

Keywords: $p$-adic diffusion

DOI: https://doi.org/10.4213/tmf31

Full text: PDF file (247 kB)
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English version:
Theoretical and Mathematical Physics, 2004, 138:3, 322–332

Bibliographic databases:

Document Type: Article
Received: 15.12.2002
Revised: 15.04.2003

Citation: S. V. Kozyrev, “$p$-Adic Pseudodifferential Operators and $p$-Adic Wavelets”, TMF, 138:3 (2004), 383–394; Theoret. and Math. Phys., 138:3 (2004), 322–332

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. S. V. Kozyrev, “$p$-Adic Pseudodifferential Operators: Methods and Applications”, Proc. Steklov Inst. Math., 245 (2004), 143–153  mathnet  mathscinet  zmath
    2. S. V. Kozyrev, A. Yu. Khrennikov, “Pseudodifferential operators on ultrametric spaces and ultrametric wavelets”, Izv. Math., 69:5 (2005), 989–1003  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    3. A. Yu. Khrennikov, S. V. Kozyrev, “$p$-Adic Pseudodifferential Operators and Analytic Continuation of Replica Matrices”, Theoret. and Math. Phys., 144:2 (2005), 1166–1170  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    4. Kozyrev S.V., Osipov V.A., Avetisov V.A., “Nondegenerate ultrametric diffusion”, J. Math. Phys., 46:6 (2005), 063302, 15 pp.  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    5. Albeverio S., Khrennikov A.Yu., Shelkovich V.M., “Harmonic analysis in the $p$-adic Lizorkin spaces: fractional operators, pseudo-differential equations, $p$-adic wavelets, Tauberian theorems”, J. Fourier Anal. Appl., 12:4 (2006), 393–425  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    6. Khrennikov A.Yu., Kozyrev S.V., “Ultrametric random field”, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 9:2 (2006), 199–213  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    7. Khrennikov A.Yu., Kozyrev S.V., “Replica symmetry breaking related to a general ultrametric space I: Replica matrices and functionals”, Phys. A, 359 (2006), 222–240  crossref  mathscinet  adsnasa  isi  elib  scopus  scopus
    8. Khrennikov A.Yu., Kozyrev S.V., “Replica symmetry breaking related to a general ultrametric space – II: RSB solutions and the $n\to 0$ limit”, Phys. A, 359 (2006), 241–266  crossref  adsnasa  isi  elib  scopus  scopus
    9. S. V. Kozyrev, “Wavelets and spectral analysis of ultrametric pseudodifferential operators”, Sb. Math., 198:1 (2007), 97–116  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    10. Aref'eva I.Yu., Volovich I.V., “Quantization of the Riemann zeta-function and cosmology”, Int. J. Geom. Methods Mod. Phys., 4:5 (2007), 881–895  crossref  mathscinet  zmath  isi  scopus  scopus
    11. Murtagh F., “The Haar wavelet transform of a dendrogram”, J. Classification, 24:1 (2007), 3–32  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    12. Khrennikov A.Yu., Kozyrev S.V., “Replica symmetry breaking related to a general ultrametric space III: The case of general measure”, Phys. A, 378:2 (2007), 283–298  crossref  adsnasa  isi  elib  scopus  scopus
    13. 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
    14. Chuong Nguyen Minh, Co Nguyen Van, “The Cauchy problem for a class of pseudodifferential equations over $p$-adic field”, J. Math. Anal. Appl., 340:1 (2008), 629–645  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    15. Chuong N.M., Van Co N., “P-adic pseudodifferential operators and wavelets”, Frames and Operator Theory in Analysis and Signal Processing, Contemporary Mathematics Series, 451, 2008, 33–45  crossref  mathscinet  zmath  isi
    16. Shelkovich V., Skopina M., “$p$-adic Haar multiresolution analysis and pseudo-differential operators”, Journal of Fourier Analysis and Applications, 15:3 (2009), 366–393  crossref  mathscinet  zmath  isi  scopus  scopus
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    19. Khrennikov A.Yu., Shelkovich V.M., Skopina M., “$p$-adic refinable functions and MRA-based wavelets”, J. Approx. Theory, 161:1 (2009), 226–238  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    20. S. A. Evdokimov, M. A. Skopina, “2-adic wavelet bases”, Proc. Steklov Inst. Math. (Suppl.), 266, suppl. 1 (2009), S143–S154  mathnet  crossref  mathscinet  isi  elib
    21. Khrennikov A.Yu., Shelkovich V.M., “Non-Haar $p$-adic wavelets and their application to pseudo-differential operators and equations”, Appl. Comput. Harmon. Anal., 28:1 (2010), 1–23  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    22. S. F. Lukomskii, “Multiresolution analysis on zero-dimensional Abelian groups and wavelets bases”, Sb. Math., 201:5 (2010), 669–691  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    23. Albeverio S., Evdokimov S., Skopina M., “p-Adic Multiresolution Analysis and Wavelet Frames”, J Fourier Anal Appl, 16:5 (2010), 693–714  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    24. S. F. Lukomskii, “Neortogonalnyi kratnomasshtabnyi analiz na nul-mernykh lokalno kompaktnykh gruppakh”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 11:3(1) (2011), 25–32  mathnet  elib
    25. S. F. Lukomskii, “Haar System on the Product of Groups of $p$-Adic Integers”, Math. Notes, 90:4 (2011), 517–532  mathnet  crossref  crossref  mathscinet  isi
    26. V. M. Shelkovich, “$p$-adic evolution pseudo-differential equations and $p$-adic wavelets”, Izv. Math., 75:6 (2011), 1249–1278  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    27. Kopaliani T., “Higher rank Haar wavelet bases in spaces L-w(p)(R)”, Georgian Math J, 18:3 (2011), 517–532  mathscinet  zmath  isi  elib
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    30. Chacon-Cortes L.F. Zuniga-Galindo W.A., “Nonlocal Operators, Parabolic-Type Equations, and Ultrametric Random Walks”, J. Math. Phys., 54:11 (2013), 113503  crossref  mathscinet  zmath  adsnasa  isi  scopus
    31. 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
    32. S. F. Lukomskii, “Riesz multiresolution analysis on zero-dimensional groups”, Izv. Math., 79:1 (2015), 145–176  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    33. Evdokimov S., Skopina M., “On Orthogonal P-Adic Wavelet Bases”, J. Math. Anal. Appl., 424:2 (2015), 952–965  crossref  mathscinet  zmath  isi  scopus  scopus
    34. Evdokimov S., “On non-compactly supported p-adic wavelets”, J. Math. Anal. Appl., 443:2 (2016), 1260–1266  crossref  mathscinet  zmath  isi  elib  scopus
    35. 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  scopus
    36. A. Kh. Bikulov, A. P. Zubarev, “Polnye sistemy sobstvennykh funktsii operatora Vladimirova v $L^{2}(B_r)$ i $L^{2}(\mathbb{Q}_{p})$”, Fundament. i prikl. matem., 21:3 (2016), 39–56  mathnet
    37. K. Oleschko, A. Yu. Khrennikov, “Applications of $p$-adics to geophysics: Linear and quasilinear diffusion of water-in-oil and oil-in-water emulsions”, Theoret. and Math. Phys., 190:1 (2017), 154–163  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    38. E. Dzh. King, M. A. Skopina, “O biortogonalnykh $p$-adicheskikh bazisakh vspleskov”, Voprosy teorii predstavlenii algebr i grupp. 31, Zap. nauchn. sem. POMI, 455, POMI, SPb., 2017, 67–83  mathnet
    39. Oleschko K. Khrennikov A., “Transport Through a Network of Capillaries From Ultrametric Diffusion Equation With Quadratic Nonlinearity”, Russ. J. Math. Phys., 24:4 (2017), 505–516  crossref  mathscinet  zmath  isi  scopus  scopus
    40. 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  isi  scopus  scopus
    41. Antoniouk V A., Oleschko K., Kochubei A.N., Khrennikov A.Yu., “A Stochastic P-Adic Model of the Capillary Flow in Porous Random Medium”, Physica A, 505 (2018), 763–777  crossref  mathscinet  isi  scopus  scopus
    42. Pourhadi E., Khrennikov A.Yu., “On the Solutions of Cauchy Problem For Two Classes of Semi-Linear Pseudo-Differential Equations Over P-Adic Field”, P-Adic Numbers Ultrametric Anal. Appl., 10:4 (2018), 322–343  crossref  mathscinet  isi  scopus
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