Tauberian Conditions of Slowly Decreasing Type for the Logarithmic Power Series Method

Abstract

Let \((s_n)\) be a sequence of real numbers. We say that \((s_n)\) is summable to \(\xi \) by the logarithmic power series method if $$\lim _{x\rightarrow 1^-}f(x)=\xi, \quad {\text{where}}\quad f(x)=-\frac{1}{\log (1-x)}\sum_{n=0}^{\infty }\frac{s_n}{n+1}x^{n+1}. $$ It is well known that if the limit \(\lim_{n \rightarrow \infty }s_n=\xi \) exists, then the limit $$\lim _{x \rightarrow 1^-} f(x)=\xi $$ also exists. In this paper, we determine Tauberian conditions of slowly decreasing type to obtain ordinary convergence of \((s_n)\) from its summability by logarithmic power series method. As a consequence of our result, we give a short proof of an earlier Tauberian theorem due to Kwee (Can J Math 20:1324–1331, 1968).