Some applications of summability theory.

*(English)*Zbl 1359.40001
Dutta, Hemen (ed.) et al., Current topics in summability theory and applications. Singapore: Springer (ISBN 978-981-10-0912-9/hbk; 978-981-10-0913-6/ebook). 357-411 (2016).

Summary: We present some recent developments in summability theory and its applications. Concretely, we discuss some applications of summability theory in sequence spaces defined by modulus functions, the Orlicz function, and summability methods, which are related to statistical convergence and their applications. Also, we discuss topological and geometric properties of the sequence spaces, such as the \((\beta)\)-property, the Banach-Saks property, the Kadec-Klee property, the Opial property, etc. In the next section, some applications of summability theory to Tauberian theorems, both in an ordinary sense and in a statistic sense, are discussed. In the last section, we show some results related to the Tauberian theory characterized by weighted summability methods such as the generalized de la Vallée-Poussin method, the generalized Nörlund-Cesàro method, etc.

For the entire collection see [Zbl 1348.40001].

For the entire collection see [Zbl 1348.40001].

##### MSC:

40A05 | Convergence and divergence of series and sequences |

40A35 | Ideal and statistical convergence |

46A45 | Sequence spaces (including Köthe sequence spaces) |

46B45 | Banach sequence spaces |

46B20 | Geometry and structure of normed linear spaces |

40E05 | Tauberian theorems, general |

40-02 | Research exposition (monographs, survey articles) pertaining to sequences, series, summability |