On one real basis for $L^2(Q_p)$

Abstract

We construct new bases of real functions from $L^2\left(B_r\right)$ and from $L^2\left({\mathbb{Q}}_p\right)$. These functions are eigenfunctions of the $p$-adic pseudo-differential Vladimirov operator, which is defined on a compact set $B_r\subset {\mathbb{Q}}_p$ of the field of $p$-adic numbers ${\mathbb{Q}}_p$ or, respectively, on the entire field ${\mathbb{Q}}_p$. A relation between the basis of functions from $L^2\left({\mathbb{Q}}_p\right)$ and the basis of $p$-adic wavelets from $L^2\left({\mathbb{Q}}_p\right)$ is found. As an application, we consider the solution of the Cauchy problem with the initial condition on a compact set for a pseudo-differential equation with a general pseudo-differential operator, which is diagonal in the basis constructed.

15 pages

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