A note on Tauberian conditions for Abel and Cesàro summability

Abstract

A conjecture has recently been made by P. L. Butzerl that there exist Tauberian conditions for the Cesaro method of summability which are not also Tauberian conditions for Abel summability. It was pointed out2 by G. Lorentz that the conjecture is true in a trivial sense if we define the Tauberian condition as membership of a class T consisting of all series satisfying one of the classical Tauberian conditions for Cesaro summation (nan > K, for example) together with one particular series which is Abel, but not Cesaro, summable. It does not seem possible to restrict the class of admissible Tauberian conditions in a natural way so that examples of this rather artificial kind are excluded. However, even if Butzer's question does not admit a nontrivial yes or no answer, it still has substance if we look for a solution with some intrinsic formal interest. The condition (r) defined below applies to the slightly more general summation methods defined by integral transforms of functions of a continuous real variable. These reduce to the classical methods of summation of series when the functions are step functions with jumps at integral values. Thus, if s(x) is defined for x_0 and is integrable over any finite interval, it tends to A in the Abel sense if