$p$-Adic wavelets and their applications

S. V. Kozyrev ; A. Yu. Khrennikov ; V. M. Shelkovich

Informatics and Automation, Selected topics of mathematical physics and analysis, Tome 285 (2014), pp. 166-206.

Résumé

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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