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$p$-Adic multidimensional wavelets and their application to $p$-adic pseudo-differential operators

A. Yu. Khrennikov·V. M. Shelkovich
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Abstract

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 ${\cL}^2(\bQ_p^n)$. 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 {\it semi-linear} pseudo-differential equations.

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A. Yu. Khrennikov, V. M. Shelkovich. 2006-12-15. $p$-Adic multidimensional wavelets and their application to $p$-adic pseudo-differential operators. https://arxiv.org/abs/math-ph/0612049

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