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

Sefa Anıl Sezer; İbrahim Çanak

doi:10.3842/umzh.v76i7.7507

Sefa Anıl Sezer Department of Mathematics, Istanbul Medeniyet University, Turkey https://orcid.org/0000-0002-8053-9991
İbrahim Çanak Department of Mathematics, Ege University, Izmir, Turkey http://orcid.org/0000-0002-1754-1685

DOI: https://doi.org/10.3842/umzh.v76i7.7507

Keywords: Summability by deferred weighted means, Tauberian conditions, deferred slow decrease and oscillation, Landau and Hardy type conditions, ordered linear spaces

Abstract

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.

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