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

Let ? be a real-or complex-valued function that is measurable in Lebesgue?s sense on some interval (x0, ?), where x0 ? 0. It is known that the existence of ordinary limit of function ? implies the statistical limit of ? . 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 ? 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 statistical convergence.