Tauberian conditions under which convergence follows from the weighted mean summability and its statistical extension for sequences of fuzzy number - Önder, Z.; Çanak, İ. (Ukrains’kyi Matematychnyi Zhurnal, 2021)
UDC 517.5Let $(p_n)$ be a sequence of nonnegative numbers such that $p_0>0$ and$$P_n:=\sum_{k=0}^{n}p_k→∞\qquad \text{as}\qquad n→∞.$$Let $(u_n)$ be a sequence of fuzzy numbers.The weighted mean of $(u_n)$ is defined by$$t_n:=\frac{1}{P_n}\sum_{k=0}^{n}p_ku_k\qquad \text{for}\qquad n =0,1,2,… \,. $$It is known that the existence of the limit $\lim u_n=μ_{0}$ implies that of $\lim t_n=μ_{0}.$ For the the existence of the limit $st$-$\lim t_n=μ_{0},$ we require the boundedness of $(u_n)$ in addition to the existence of the limit $\lim u_n=μ_{0}.$ But, in general, the converse of this implication is not true. In this paper, we obtain Tauberian conditions, under which the existence of the limit $\lim u_n=μ_{0}$ follows from that of $\lim t_n=μ_{0}$ or $st$-$\lim t_n=μ_{0}.$ These Tauberian conditions are satisfied if $(u_n)$ satisfies the two-sided condition of Hardy type relative to $(P_n).$