An Alternative Proof of a Tauberian Theorem for the Weighted Mean Method of Summability

Abstract

Let $(p_n)$ be a sequence of nonnegative numbers such that $p_0>0$ and

$$\begin{aligned} P_n:=\sum _{k=0}^{n}p_k\rightarrow \infty, n\rightarrow \infty . \end{aligned}$$

Let $(u_n)$ be a sequence of real or complex numbers. The nth weighted mean of $(u_n)$ is defined by

$$\begin{aligned} \sigma _n:=\frac{1}{P_n}\sum _{k=0}^{n}p_k u_k \quad (n =0,1,2,\ldots ) \end{aligned}$$

We give an alternative proof of a Tauberian theorem stating that the existence of the limit $\lim _{n \rightarrow \infty } u_n=s$ follows from that of $\lim _{n \rightarrow \infty } \sigma _n=s$ and a Tauberian condition.

If $(u_n)$ is a sequence of real numbers, then these Tauberian conditions are one-sided. If $(u_n)$ is a sequence of complex numbers, these Tauberian conditions are two-sided.

Significance Statement: If a sequence converges, then its weighted means converge to the same number. But, the converse of this implication is not true in general and its partial converse might be valid. This manuscript presents an alternative proof of a well-known Tauberian theorem stating that convergence of a slowly decreasing sequence (in case of sequences of real numbers) or a slowly oscillating sequence (in case of sequences of complex numbers) follows from its weighted mean summability. Corollaries of the main results in this manuscript consist of well-known Tauberian theorems for Cesàro and logarithmic summability methods.