Some Tauberian theorems for the weighted mean method of summability of double sequence

Ümit Totur; İbrahim  Çanak

doi:10.3842/umzh.v75i9.509

Ümit Totur Department of Mathematics Ege University, Izmir, Turkey
İbrahim Çanak Department of Mathematics Ege University, Izmir, Turkey https://orcid.org/0000-0002-1754-1685

DOI: https://doi.org/10.3842/umzh.v75i9.509

Keywords: TAUBERIAN THEOREMS

Abstract

UDC 517.5

Let $p=(p_j)$ and $q=(q_k)$ be real sequences of nonnegative numbers with the property that $P_m=\sum _{j=0}^{m} p_j \neq 0$ and $Q_n=\sum _{k=0}^{n} q_k \neq 0$ for all $m$ and $n.$ Let $(P_m)$ and $(Q_n)$ be regulary varying positive indices. Assume that $(u_{mn})$ is a double sequence of complex (real) numbers, which is $(\overline{N},p,q; \alpha,\beta)$ summable with a finite limit, where $(\alpha,\beta)=(1,1)$, $(1,0)$, or $(0,1)$. We present some conditions imposed on the weights under which $(u_{mn})$ converges in Pringsheim's sense. These results generalize and extend the results obtained by authors in [Comput. Math. Appl., 62, No. 6, 2609–2615 (2011)].

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