A generalization in integral form of certain Tauberian theorems
Brunk, H. D
Master of Arts
The purpose of this paper is to generalize in integral form certain Tauberian theorems, Tauber, Hardy, Littlewood, and landau were among the first to investigate the problem of finding sufficient conditions for the convergence of series known to be summable with respect to a given method of sustainability. Certain of their theorems, generalized in integral form, will be considered in this paper. The convergence of the partial sums Sn of the series considered may be thought of as the convergence to a limit of a stop function S(x)j where S(x) = Sn for n-1 < x<n, (n -1,2,3, ...) S(0) = S1. In the integral forms of the theorems, we consider functions y(x) corresponding to S(x), with the hypothesis that y(x) be a function of finite variation in every finite interval replacing the more restrictive hypothesis that S(x) be a step-function. The close relationship between series and the Stieltjes integral has been made use of in many connections. Every series can be written as a Stieltjes integral, the integration being performed with respect to a step-function. In this sense the Stieltjes integral may be regarded as a generalization of the concept of series. In the second part of this thesis the Stieltjes integral analogues of theorems concerning Abel's method of summability are considered. The first part of the paper is devoted to the integral analogue of a Tauberian theorem due to Hardy and landau, given by Dr. H.E. Bray in his lectures in 1940-41 on the theory of the functions of a real variable. The proof as it appears here is essentially that given by Dr. Bray.