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).