Some applications of summability theory. (English) Zbl 1359.40001

Braha, Naim L.

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].

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

Keywords:

sequence spaces; Opial-property; Kadec-Klee property; Rotund property; uniform Opial property; \((\beta)\)-property; kNUC-property; Banach-Saks property of type \(p\); Orlicz function; modular functions; statistical convergence; \(\lambda\)-statistical convergence; statistical summability \((v; \lambda)\); Tauberian theorems; summability methods