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p-Adic wavelets and their applications |
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| Authors: | S V Kozyrev A Yu Khrennikov V M Shelkovich |
| Institution: | 1. Steklov Mathematical Institute, Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991, Russia 2. International Center for Mathematical Modeling in Physics, Engineering and Cognitive Sciences, Linnaeus University, SE-351 95, V?xj?, Sweden 3. Saint-Petersburg State University of Architecture and Civil Engineering, Vtoraya Krasnoarmeiskaya ul. 4, St. Petersburg, 190005, Russia 4. Department of Higher Mathematics and Mathematical Physics, Faculty of Physics, St. Petersburg State University, ul. Ul’yanovskaya 3, St. Petersburg, 198904, 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. |
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