Conditions under which the convergence of a sequence or its certain subsequences follows from the summability by deferred weighted means

• Date: 2024
• Main Authors: Sezer, Sefa Anıl, Çanak, İbrahim
• Format: Article
• Language: English
• Published: Institute of Mathematics, NAS of Ukraine 2024
• Online Access: https://umj.imath.kiev.ua/index.php/umj/article/view/7507
• Journal Title: Ukrains’kyi Matematychnyi Zhurnal

UDC 517.5 Let $(u_k)$ be a sequence of real or complex numbers. First, we consider a real sequence $(u_k)$ and formulate one-sided Tauberian conditions, which are necessary and sufficient for the  convergence of certain subsequences of $(u_k)$ to follow from its  deferred weighted summability. These conditions are satisfied if $(u_k)$ is deferred slowly decreasing or if $(u_k)$ obeys a Landau-type Tauberian condition. Second, we consider a complex sequence $(u_k)$ and present a two-sided Tauberian condition which is necessary and sufficient in order that the convergence of certain subsequences of $(u_k)$ follow from its deferred weighted summability.  This condition is satisfied either if $(u_k)$ is deferred slowly oscillating or if $(u_k)$ obeys a Hardy-type Tauberian condition. Finally, we extend these results to sequences in ordered linear spaces over the real numbers.