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 Tr. Mat. Inst. Steklova, 2014, Volume 285, Pages 166–206 (Mi tm3546)

$p$-Adic wavelets and their applications

S. V. Kozyreva, A. Yu. Khrennikovb, V. M. Shelkovichcd

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

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."

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

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English version:
Proceedings of the Steklov Institute of Mathematics, 2014, 285, 157–196

Bibliographic databases:

UDC: 517.5+517.984.5

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, Tr. 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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• http://mi.mathnet.ru/eng/tm3546
• https://doi.org/10.1134/S0371968514020125
• http://mi.mathnet.ru/eng/tm/v285/p166

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This publication is cited in the following articles:
1. S. Evdokimov, “On non-compactly supported $p$-adic wavelets”, J. Math. Anal. Appl., 443:2 (2016), 1260–1266
2. V. Al Osipov, “Wavelet analysis on symbolic sequences and two-fold de Bruijn sequences”, J. Stat. Phys., 164:1 (2016), 142–165
3. 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
4. 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
5. P. Dutta, D. Ghoshal, A. Lala, “Enhanced symmetry of the $p$-adic wavelets”, Phys. Lett. B, 783 (2018), 421–427
6. Yu. A. Farkov, “Diskretnye veivlet-preobrazovaniya v analize Uolsha”, Materialy mezhdunarodnoi konferentsii «International Conference on Mathematical Modelling in Applied Sciences, ICMMAS-17», Sankt-Peterburgskii politekhnicheskii universitet, 24–28 iyulya 2017 g., Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 160, VINITI RAN, M., 2019, 126–136
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