S. V. Kozyrev, A. Yu. Khrennikov, V. M. Shelkovich, “$p$-Adic wavelets and their applications”, Selected topics of mathematical physics and analysis, Collected papers. In commemoration of the 90th anniversary of Academician Vasilii Sergeevich Vladimirov's birth, Trudy Mat. Inst. Steklova, 285, MAIK Nauka/Interperiodica, Moscow, 2014, 166–206; Proc. Steklov Inst. Math., 285 (2014), 157

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Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2014, Volume 285, Pages 166–206
DOI: https://doi.org/10.1134/S0371968514020125 (Mi tm3546)

This article is cited in 13 scientific papers (total in 13 papers)

$p$ -Adic wavelets and their applications

S. V. Kozyrev^a, A. Yu. Khrennikov^b, V. M. Shelkovich^cd

^a Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia
^b International Center for Mathematical Modeling in Physics, Engineering and Cognitive Sciences, Linnaeus University, Växjö, Sweden
^c St. Petersburg State University, Faculty of Physics, St. Petersburg, Russia
^d St. Petersburg State University of Architecture and Civil Engineering, St. Petersburg, Russia

Full-text PDF (448 kB) Citations (13)

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DOI: https://doi.org/10.1134/S0371968514020125

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.

Funding agency	Grant number
Russian Academy of Sciences - Federal Agency for Scientific Organizations
Linnaeus University
This work was supported in part by the grants "Mathematical Modeling and System Collaboration" and "Mathematical Modeling of Complex Hierarchic Systems" from the Faculty of Natural Science and Engineering, Linnaeus University. The first author was also supported in part by the Russian Academy of Sciences within the program "Modern Problems of Theoretical Mathematics."

Received in October 2013

English version:
Proceedings of the Steklov Institute of Mathematics, 2014, Volume 285, Pages 157–196
DOI: https://doi.org/10.1134/S0081543814040129

Bibliographic databases:

Document Type: Article

UDC: 517.5+517.984.5

Language: Russian

Citation: S. V. Kozyrev, A. Yu. Khrennikov, V. M. Shelkovich, “

$p$ -Adic wavelets and their applications”, Selected topics of mathematical physics and analysis, Collected papers. In commemoration of the 90th anniversary of Academician Vasilii Sergeevich Vladimirov's birth, Trudy Mat. Inst. Steklova, 285, MAIK Nauka/Interperiodica, Moscow, 2014, 166–206; Proc. Steklov Inst. Math., 285 (2014), 157–196

Citation in format AMSBIB

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Linking options:

https://www.mathnet.ru/eng/tm3546

https://doi.org/10.1134/S0371968514020125

https://www.mathnet.ru/eng/tm/v285/p166

This publication is cited in the following 13 articles:

N. Athira, M. C. Lineesh, “Linear and nonlinear pseudo-differential operators on p-adic fields”, J. Pseudo-Differ. Oper. Appl., 15:3 (2024)
N. Athira, M. C. Lineesh, “Vanishing Moments of Wavelets on p-adic Fields”, Complex Anal. Oper. Theory, 19:1 (2024)
Owais Ahmad, Neyaz Ahmad Sheikh, “Generalized Multiresolution Structures in Reducing Subspaces of Local Fields”, Acta. Math. Sin.-English Ser., 38:12 (2022), 2163
Owais Ahmad, Neyaz Ahmad, Mobin Ahmad, “Wavelet bi-frames on local fields”, J. Numer. Anal. Approx. Theory, 51:2 (2022), 124
Ahmad O. Wani A.H. Sheikh N.A. Ahmad M., “Vector Valued Nonuniform Nonstationary Wavelets and Associated Mra on Local Fields”, J. Appl. Math. Stat. Inform., 17:2 (2021), 19–46
Ahmad O., Ahmad N., “Construction of Nonuniform Wavelet Frames on Non-Archimedean Fields”, Math. Phys. Anal. Geom., 23:4 (2020), 47
Yu. A. Farkov, “Discrete wavelet transforms in Walsh analysis”, J. Math. Sci. (N. Y.), 257:1 (2021), 127–137
P. Dutta, D. Ghoshal, A. Lala, “Enhanced symmetry of the $p$-adic wavelets”, Phys. Lett. B, 783 (2018), 421–427
B. Dragovich, A. Yu. Khrennikov, S. V. Kozyrev, I. V. Volovich, E. I. Zelenov, “$p$-Adic mathematical physics: the first 30 years”, P-Adic Numbers Ultrametric Anal. Appl., 9:2 (2017), 87–121
B. Behera, Q. Jahan, “Affine, quasi-affine and co-affine frames on local fields of positive characteristic”, Math. Nachr., 290:14-15 (2017), 2154–2169
S. Evdokimov, “On non-compactly supported $p$-adic wavelets”, J. Math. Anal. Appl., 443:2 (2016), 1260–1266
V. Al Osipov, “Wavelet analysis on symbolic sequences and two-fold de Bruijn sequences”, J. Stat. Phys., 164:1 (2016), 142–165
S. Albeverio, A. Yu. Khrennikov, S. V. Kozyrev, S. A. Vakulenko, I. V. Volovich, “In memory of Vladimir M. Shelkovich (1949–2013)”, P-Adic Num Ultrametr Anal Appl, 5:3 (2013), 242

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

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