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Bitimkhan, S.; Alibiyeva, D. T.

Absolute Cesàro summability conditions for double trigonometric series. (English) Zbl 1500.40008

Lobachevskii J. Math. 43, No. 2, 337-344 (2022).
Summary: This article is devoted to the topic of absolute summation of series or Cesàro summation. The relevance of this article lies in the fact that a type of absolute summation with vector index, which has not been previously studied, is considered. The necessary condition of absolute summation with vector index of double trigonometric series is obtained and several sufficient conditions have been proved. In order to prove this necessary condition, the lemma providing the necessary condition of absolute summation with vector index of the double number series is first time proved. Another lemma, which is auxiliary to the main result, has been proven for the special case of absolute summation with vector index. With the help of these two lemmas the necessary condition is obtained. The theorem that gives a sufficient condition proves the conditions that are sufficient in different cases, which may depend on the parameters.

MSC:

40F05 Absolute and strong summability
40D05 General theorems on summability
40G05 Cesàro, Euler, Nörlund and Hausdorff methods
42A24 Summability and absolute summability of Fourier and trigonometric series

Keywords:

trigonometric series; absolute summability; measuring set; coefficient; function
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References:

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