Tauberian theory provides striking methods to attack hard problems in analysis. The study of Tauberian type theorems has been historically stimulated by their potential applications in diverse fields of mathematics such as number theory, combinatorics, complex analysis, probability theory, and the analysis of differential operators [1, 3, 7, 8, 17, 19]. Even mathematical physics has pushed forw...

1.1. Tauberian type theorems. As is well known, in mathematical physics there are so-called Tauberian theorems which have many applications. Tauberian theorems are usually assumed to connect asymptotical behavior of a function (distribution) at zero with asymptotical behavior of its Fourier (Laplace or other integral transforms) at infinity. The inverse theorems are usually called “Abelian” [4]...

In this paper we prove some Tauberian theorems for the product of Borel and Hölder summability methods which improve the classical Tauberian theorems for the Borel summability method. Mathematics Subject Classification 2010: 40E05, 40G05, 40G10.

We prove several Tauberian theorems for regularizing transforms of vector-valued distributions. The regularizing transform of f is given by the integral transform M φ(x, y) = (f ∗ φy)(x), (x, y) ∈ R n × R+, with kernel φy(t) = yφ(t/y). We apply our results to the analysis of asymptotic stability for a class of Cauchy problems, Tauberian theorems for the Laplace transform, the comparison of quas...

We study Tauberian conditions for the existence of Cesàro limits in terms of the Laplace transform. We also analyze Tauberian theorems for the existence of distributional point values in terms of analytic representations. The development of these theorems is parallel to Tauber’s second theorem on the converse of Abel’s theorem. For Schwartz distributions, we obtain extensions of many classical ...

We prove necessary and sufficient Tauberian conditions for sequences summable by weighted mean methods. The main results of this paper apply to all weighted meanmethods and unify the results known in the literature for particularmethods. Among others, the conditions in our theorems are easy consequences of the slowly decreasing condition for real numbers, or slowly oscillating condition for com...