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

Abstract

We construct new bases of real functions from L²(B_r) and from L²(ℚ_p). These functions are eigenfunctions of the p-adic pseudo-differential Vladimirov operator, which is defined on a compact set B_r ⊂ ℚ_p of the field of p-adic numbers ℚ_p or, respectively, on the entire field ℚ_p. A relation between the basis of functions from L²(ℚ_p) and the basis of p-adic wavelets from L²(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.