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

TL;DR

This paper introduces new multidimensional $p$-adic wavelets, establishes their orthonormal basis properties, and demonstrates their role as eigenfunctions for pseudo-differential operators, with applications to solving $p$-adic equations.

Contribution

The paper develops a new class of $p$-adic wavelets, proves their basis properties, and identifies their eigenfunction status for key pseudo-differential operators.

Findings

New $p$-adic wavelets form an orthonormal basis in ${\\cL}^2(Q_p^n)$.

Wavelets are eigenfunctions of the Taibleson fractional operator.

Results facilitate solutions to $p$-adic pseudo-differential equations.

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.