ON GENERALIZED STATISTICAL CONVERGENCE AND BOUNDEDNESS OF RIESZ SPACE-VALUED SEQUENCES

Abstract

We consider the notion of generalized density, namely, the natural density
of weight g recently introduced in [1] and primarily study some sufficient and
almost converse necessary conditions for the generalized statistically convergent sequence under which the subsequence is also generalized statistically convergent. Also we consider similar types of results for the case of generalized statistically bonded sequence. Some results are further obtained in a more general form by using the notion of ideals. The entire investigation is performed in the setting of Riesz spaces extending the recent results in [2].

[1] M. balcerzak, P.Das, M. Filipczak and J. Swaczyna, Generalized kinds of density and the associated ideals, Acta.Math.Hungar., 147 (2015), No.1, 97-115.

[2] P. Das and E. Savas, On generalized statistical and ideal convergence of metric valued sequences, Ukr. Math. Zh. 68 (2017), No.12, 1849-1859.