Tauberian theorems for absolute summability

Ohne ZusammenfassungHerrn A.Ostrowski zum 60. Geburtstag gewidmetThis work has been completed while the author held a Fellowship at the Summer Research Institute of 1952 of the Canadian Mathematical Congress.     

Tauberian theorems for Abel limitability method

This paper addresses conditions for the Abel method of limitability to imply convergence and subsequential convergence.      

Tauberian theorems in quantum field theory

A survey is given of the application of Tauberian theory to problems of quantum field theory. In particular, the question of the self-similar asymptotics of the form factors of electron-nucleon scattering and its connection with the singular structure of the commutators of electromagnetic currents in a neighborhood of the light cone is considered in detail.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Vol. 15, pp. 95–130, 1980.     

Tauberian theorems for weak almost convergent functions

The almost convergent function which was introduced by Raimi [6] and discussed by Ho [4], Das and Nanda [2, 3], is the continuous analogue of almost convergent sequences (see [5]). In this paper, we establish the Tauberian conditions and the Cauchy criteria for weak almost convergent functions on R~2+ .     

Statistical gap Tauberian theorems in metric spaces

By using the concept of statistical convergence we present statistical Tauberian theorems of gap type for the Cesàro, Euler-Borel family and the Hausdorff families applicable in arbitrary metric spaces. In contrast to the classical gap Tauberian theorems, we show that such theorems exist in the statistical sense for the convolution methods which include the Taylor and the Borel matrix methods. We further provide statistical analogs of the gap Tauberian theorems for the Hausdorff methods and provide an explanation as to how the Tauberian rates over the gaps may differ from those of the classical Tauberian theorems.     

Statistical extensions of some classical Tauberian theorems

Hardy's well-known Tauberian theorem for Cesàro means says that if the sequence satisfies and , then . In this paper it is shown that the hypothesis can be replaced by the weaker assumption of the statistical limit: st-lim , i.e., for every , . Similarly, the ``one-sided' Tauberian theorem of Landau and Schmidt's Tauberian theorem for the Abel method are extended by replacing and with st-lim and st-lim , respectively. The Hardy-Littlewood Tauberian theorem for Borel summability is also extended by replacing , where is a continuous parameter, with , and further replacing it by -st-lim , where is the Borel matrix method.