General Tauberian theorems for the statistically $(C,1)$ summable integrals

Abstract

Let $f$ be a real- or complex-valued function that is measurable in Lebesgue's sense on some interval $(x_0,\infty)$, where $x_0\geq 0$. It is known that the existence of ordinary limit of function $f$ implies the statistical limit of $f$. However, the converse implication is not always true. In this study we introduce some Tauberian conditions in terms of the general control modulo of integer order $r \geq 1$. Also we consider the Tauberian conditions of slow decrease and slow oscillation. Under these Tauberian conditions, we obtain the ordinary limit of a function from its statistical limit. The main results generalize some classical type Tauberian theorems given for statisctical convergence.