Tauberian Conditions Under Which Convergence Follows from Statistical Summability by Weighted Means

Abstract

Let \((p_n)\) be a sequence of nonnegative numbers such that \(p_0>0\) and $$ P_n:=\sum _{k=0}^{n}p_k\rightarrow \infty \,\,\,\,\text {as}\,\,\,\,n\rightarrow \infty . $$ Let \((s_n)\) be a sequence of real and complex numbers. The weighted mean of \((s_n)\) is defined by $$ t_n:=\frac{1}{P_n}\sum _{k=0}^{n}p_k s_k\,\,\,\,\text {for}\,\,\,\,n =0,1,2,\ldots $$ We obtain some sufficient conditions, under which the existence of the limit \(\lim s_n=\mu \) follows from that of st-\(\lim t_n=\mu \), where \(\mu \) is a finite number. If \((s_n)\) is a sequence of real numbers, then these Tauberian conditions are one-sided. If \((s_n)\) is a sequence of complex numbers, these Tauberian conditions are two-sided. These Tauberian conditions are satisfied if \((s_n)\) satisfies the one-sided condition of Landau type relative to \((P_n)\) in the case of real sequences or if \((s_n)\) satisfies the two-sided condition of Hardy type relative to \((P_n)\) in the case of complex numbers.