NON-HAAR p-ADIC WAVELETS AND THEIR APPLICATION TO PSEUDO-DIFFERENTIAL OPERATORS AND EQUATIONS

In this paper a countable family of new compactly supported non-Haar p-adic wavelet bases in L(Q p ) is constructed. We use the wavelet bases in the following applications: in the theory of p-adic pseudo-differential operators and equations. Namely, we study the connections between wavelet analysis and spectral analysis of p-adic pseudo-differential operators. A criterion for a multidimensional p-adic wavelet to be an eigenfunction for a pseudo-differential operator is derived. We prove that these wavelets are eigenfunctions of the fractional operator. In addition, p-adic wavelets are used to construct solutions of linear and semi-linear pseudo-differential equations. Since many p-adic models use pseudo-differential operators (fractional operator), these results can be intensively used in these models.