Non-Haar $p$-adic wavelets and their application to pseudo-differential operators and equations

Details Non-Haar $p$-adic wavelets and their application to pseudo-differential operators and equations Non-Haar $p$-adic wavelets and their application to pseudo-differential operators and equations

2008-08-25

arxiv.org/abs/0808.3338v1

Physics

Mathematical Physics

Scientific paper

In this paper a countable family of new compactly supported {\em non-Haar} $p$-adic wavelet bases in ${\cL}^2(\bQ_p^n)$ 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.

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