12 - Distributional asymptotics and p-adic Tauberian theorems

Summary

Introduction

Tauberian theorems is a generic name used to indicate results connecting the asymptotic behavior of a function (distribution) at zero with the asymptotic behavior of its Fourier, Laplace or other integral transforms at infinity; the inverse theorems are usually called abelian. In the real setting Tauberian theorems have numerous applications, in particular, in mathematical physics (for example, see Drozzinov and Zavyalov [80], [81], Korevaar [160], Nikolić-Despotović, Pilipović [191], Vladimirov, Drozzinov and Zavyalov [240], Yakymiv [248] and the references cited therein). Multidimensional Tauberian theorems for distributions are treated in the fundamental book [240]. Some of them are connected with the fractional operator. In [240], as a rule, theorems of this type are proved for distributions whose supports belong to a cone in ℝⁿ (semi-axis for n = 1). This is related to the fact that such distributions form a convolution algebra. In this case the kernel of the fractional operator is a distribution whose support belongs to the cone in ℝⁿ or a semi-axis for n = 1 [240, §2.8.]

p-adic analogs of Tauberian theorems do not seem to have been discussed so far except for [140], [141], [21]. In this chapter, we present a first study of them based on the above papers.

In the beginning, in Sections 12.2 and 12.3, we introduce the notion of the p-adic distributional asymptotics [140], [141]. In Section 12.2, the definition of distributional (stabilized) asymptotic estimate at infinity is introduced.