Absolute tauberian conditions for absolute hausdorff and quasi-hausdorff methods

The purpose of this paper is to prove that for a large set of absolute Hausdorff and quasi-Hausdorff methods the condition {Mathematical expression} is a Tauberian condition, i.e., its fulfillment together with the absolute summability of {Mathematical expression} to s implies that {Mathematical expression} and {Mathematical expression} a _n =s.