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Izv. RAN. Ser. Mat., 2005, Volume 69, Issue 5, Pages 133–148 (Mi izv657)  

This article is cited in 31 scientific papers (total in 31 papers)

Pseudodifferential operators on ultrametric spaces and ultrametric wavelets

S. V. Kozyrev, A. Yu. Khrennikov


Abstract: We construct a wavelet analysis and spectral theory of pseudodifferential operators on general ultrametric spaces. Operators generalizing the Vladimirov operator of $p$-adic fractional differentiation are introduced. We construct a family of ultrametric wavelet bases in spaces of square-integrable complex-valued functions for a wide family of ultrametric spaces. We show that the pseudodifferential operators introduced are diagonal in these wavelet bases and compute the corresponding eigenvalues.

DOI: https://doi.org/10.4213/im657

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English version:
Izvestiya: Mathematics, 2005, 69:5, 989–1003

Bibliographic databases:

Document Type: Article
UDC: 517.58
MSC: 35S05, 47G10, 26E30
Received: 08.04.2004

Citation: S. V. Kozyrev, A. Yu. Khrennikov, “Pseudodifferential operators on ultrametric spaces and ultrametric wavelets”, Izv. RAN. Ser. Mat., 69:5 (2005), 133–148; Izv. Math., 69:5 (2005), 989–1003

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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. 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
    2. Kozyrev S. V., Khrennikov A. Yu., “Localization in space for a free particle in ultrametric quantum mechanics”, Dokl. Math., 74:3 (2006), 906–909  mathnet  crossref  mathscinet  zmath  isi  elib  elib  scopus
    3. 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
    4. Khrennikov, A. Yu., Kozyrev, S. V., “Wavelets and the Cauchy problem for the Schrodinger equation on analytic ultrametric space”, Mathematical Modeling of Wave Phenomena, AIP Conference Proceedings, 834, eds. Nilsson, B, Fishman, L, Amer. Inst. Physics, 2006, 344–350  crossref  mathscinet  adsnasa  isi  scopus
    5. Kozyrev, S. V., “Ultrametric analysis and interbasin kinetics”, p-Adic Mathematical Physics, AIP Conference Proceedings, 826, eds. Khrennikov, AY, Rakic, Z, Volovich, IV, 2006, 121–128  crossref  mathscinet  zmath  adsnasa  isi  scopus
    6. 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  mathscinet  adsnasa  isi  elib  scopus
    7. 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
    8. 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
    9. Chuong Nguyen Minh, Co Nguyen, “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
    10. 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
    11. Nguyen Minh Chuong, Nguyen Van Co, “$p$-adic pseudodifferential operators and wavelets”, Frames and operator theory in analysis and signal processing, Contemp. Math., 451, Amer. Math. Soc., Providence, RI, 2008, 33–45  crossref  mathscinet  zmath  isi
    12. 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
    13. Proc. Steklov Inst. Math., 265 (2009), 13–29  mathnet  crossref  mathscinet  zmath  isi  elib
    14. Shelkovich V., Skopina M., “$p$-adic Haar multiresolution analysis and pseudo-differential operators”, J. Fourier Anal. Appl., 15:3 (2009), 366–393  crossref  mathscinet  zmath  isi  elib  scopus
    15. 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
    16. Albeverio S., Kozyrev S.V., “Pseudodifferential $p$-adic vector fields and pseudodifferentiation of a composite $p$-adic function”, P-Adic Num. Ultrametr. Anal. Appl., 2:1 (2010), 21–34  crossref  mathscinet  zmath
    17. Kozyrev S.V., “Dynamics on rugged landscapes of energy and ultrametric diffusion”, p-Adic Numbers Ultrametric Anal. Appl., 2:2 (2010), 122–132  crossref  mathscinet  zmath
    18. Chuong Nguyen Minh, Hung Ha Duy, “Maximal functions and weighted norm inequalities on local fields”, Appl. Comput. Harmon. Anal., 29:3 (2010), 272–286  crossref  mathscinet  zmath  isi  elib  scopus
    19. Murtagh F., “On Ultrametric Algorithmic Information”, Computer Journal, 53:4 (2010), 405–416  crossref  isi  elib  scopus
    20. 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
    21. L. F. Chacón-Cortes, W. A. Zúñiga-Galindo, “Nonlocal operators, parabolic-type equations, and ultrametric random walks”, J. Math. Phys, 54:11 (2013), 113503  crossref  mathscinet  zmath  isi  scopus
    22. A.YU. KHRENNIKOV, S.V. KOZYREV, K. OLESCHKO, A. G. JARAMILLO, M. d.J.ESÚS.C.ORREA LÓPEZ, “APPLICATION OF p-ADIC ANALYSIS TO TIME SERIES”, Infin. Dimens. Anal. Quantum. Probab. Relat. Top, 2013, 1311190043  crossref  mathscinet  isi  scopus
    23. Chuong N.M., Duong D.V., “Boundedness of the Wavelet Integral Operator on Weighted Function Spaces”, Russ. J. Math. Phys., 20:3 (2013), 268–275  crossref  mathscinet  zmath  isi  elib  scopus
    24. Ha Duy Hung, “The P-Adic Weighted Hardy-Cesaro Operator and an Application to Discrete Hardy Inequalities”, J. Math. Anal. Appl., 409:2 (2014), 868–879  crossref  mathscinet  zmath  isi  scopus
    25. 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
    26. 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
    27. 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
    28. Nguyen Minh Chuong, Ha Duy Hung, Nguyen Thi Hong, “Bounds of P-Adic Weighted Hardy-Cesaro Operators and Their Commutators on P-Adic Weighted Spaces of Morrey Types”, P-Adic Numbers Ultrametric Anal. Appl., 8:1 (2016), 31–44  crossref  mathscinet  zmath  isi  scopus
    29. St. Petersburg Math. J., 29:3 (2018), 529–544  mathnet  crossref  isi  elib
    30. 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
    31. 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
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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