A Tauberian Theorem Concerning Borel-Type and Cesàro Methods of Summability

Abstract

Suppose throughout that r ≧ 0, α > 0, αq + β > 0 where q is a non-negative integer. Let {sn} be a sequence of real numbers,The Borel-type summability method (B, α, β) is defined as follows:The method (B, α, β) is regular [5]; and (B, 1, 1) is the standard Borel exponential method B. For a real sequence {sn} we consider the slowly decreasing-type Tauberian conditionWe shall also be concerned with the Cesàro summability method Cp(p > —1), the Valiron method Vα, and the Meyer-König method Sa (0 < a < 1) defined as follows: