$p$-Adic wavelets and their applications

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.