A TAUBERIAN THEOREM FOR THE GENERAL EULER-BOREL SUMMABILITY METHOD

Our main result is a Tauberian theorem for the general Euler-Borel summability method. Examples of the method include the discrete Borel, Euler, Meyer-Konig, Taylor and Karamata methods. Every function f analytic on the closed unit disk and satisfying some general conditions generates such a method, denoted by (Ef). For instance the function f(z) = exp(z - 1) generates the discrete Borel method. To each such function f corresponds an even positive integer p = p(f). We show that if a sequence (s(n)) is summable (E,f) and (*) s(m) - s(n) --> 0 as n --> infinity, m > n, (m - n)n-1/p(f) --> 0, then (s(n)) is convergent. If the Maclaurin coefficients off are nonnegative, then p(f) = 2. In this case we may replace the condition (*) by lim(s(m) - s(n)) greater-than-or-equal-to 0. This generalizes the Tauberian theorems for Borel summability due to Hardy and Littlewood, and R. Schmidt. We have also found new examples of the method and proved that the exponent -p(f) in die Tauberian condition (*) is the best possible.