ec 2 00 6 p - ADIC MULTIDIMENSIONAL WAVELETS AND THEIR APPLICATION TO p - ADIC PSEUDO - DIFFERENTIAL OPERATORS

In this paper we study some problems related with the theory of multidimensional p-adic wavelets in connection with the theory of multidimensional p-adic pseudo-differential operators (in the p-adic Lizorkin space). We introduce a new class of n-dimensional p-adic compactly supported wavelets. In one-dimensional case this class includes the Kozyrev p-adic wavelets. These wavelets (and their Fourier transforms) form an orthonormal complete basis in L(Q p ). 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 Taibleson fractional operator. Since many p-adic models use pseudo-differential operators (fractional operator), these results can be intensively used in applications. Moreover, p-adic wavelets are used to construct solutions of linear and semi-linear pseudo-differential equations.