Complete Systems of Eigenfunctions of the Vladimirov Operator in L2(Br) and L2(ℚp)

Abstract

We construct new bases of real functions from L2(Br) and from L2(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 L2(ℚp) and the basis of p-adic wavelets from L2(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 that is diagonal in the basis constructed.