Distributional asymptotics and p-adic Tauberian and Shannon–Kotelnikov theorems

The notion of the quasi-asymptotics adapted to the case of p-adic distributions (generalized functions) is introduced. p-Adic analogs of Tauberian theorems for distributions are proved. We show that some properties of Vladimirov's pseudo-differential operator D^α are connected with a prototype of the Tauberian type theorem (with respect to distributional asymptotic). We also prove the p-adic version of the Shannon–Kotelnikov theorem.