  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.
