Complete systems of eigenfunctions of the Vladimirov operator in $L^{2}(B_r)$ and $L^{2}(\mathbb{Q}_{p})$

We construct new bases of real functions from $L^{2}(B_{r})$ and from $L^{2}(\mathbb{Q}_{p})$. 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 $L^{2}(\mathbb{Q}_{p})$ and the basis of $p$-adic wavelets from $L^{2}(\mathbb{Q}_{p})$ 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 that is diagonal in the basis constructed.