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Some Tauberian Theorems for the Weighted Mean Method of Summability of Double Sequences

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Ukrainian Mathematical Journal Aims and scope

Let p = (pj) and q = (qk) be real sequences of nonnegative numbers with the property that

\(\begin{array}{ccccccc}{P}_{m}=\sum_{j=0}^{m}{p}_{j}\ne 0& {\text{and}}& {Q}_{m}=\sum_{k=0}^{n}{q}_{k}\ne 0& \mathrm{for all}& m& {\text{and}}& n.\end{array}\)

Also let (Pm) and (Qn) be regularly varying positive indices. Assume that (umn) is a double sequence of complex (real) numbers, which is (\(\overline{N }\), p, q; α, β)-summable and has a finite limit, where (α, β) = (1, 1), (1, 0), or (0, 1). We present some conditions imposed on the weights under which (umn) converges in Pringsheim’s sense. These results generalize and extend the results obtained by the authors in [Comput. Math. Appl., 62, No. 6, 2609–2615 (2011)].

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Authors and Affiliations

  1. Department of Mathematics, Ege University, Izmir, Turkey

    Ümit Totur & İbrahim Çanak

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  1. Ümit Totur
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  2. İbrahim Çanak
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Correspondence to İbrahim Çanak.

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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 75, No. 9, pp. 1276–1293, September, 2023. Ukrainian DOI: https://doi.org/10.37863/umzh.v75i9.509.

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Totur, Ü., Çanak, İ. Some Tauberian Theorems for the Weighted Mean Method of Summability of Double Sequences. Ukr Math J 75, 1453–1472 (2024). https://doi.org/10.1007/s11253-024-02272-4

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  • DOI: https://doi.org/10.1007/s11253-024-02272-4

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