COMPLETE SYSTEMS OF EIGENFUNCTIONS OF THE VLADIMIROV OPERATOR IN L~2(B_r) AND L~2(Q_p)

摘要

We construct new bases of real functions from L~2(B_r) and from L~2(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 L~2(Q_p) and the basis of p-adic wavelets from L~2(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.