We construct new bases of real functions from L 2 (Br) and from L 2 (Qp). These functions are eigenfunctions of the p-adic pseudo-differential Vladimirov operator, which is defined on a compact set Br ⊂ Qp of the field of p-adic numbers Qp or, respectively, on the entire field Qp. A relation between the basis of functions from L 2 (Qp) and the basis of p-adic wavelets from L 2 (Qp) 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.