2019
Том 71
№ 11

Das P.

Articles: 2
Article (English)

On spliced sequences and the density of points with respect to a matrix constructed by using a weight function

Ukr. Mat. Zh. - 2019. - 71, № 9. - pp. 1192-1207

UDC 517.5
Following the line of investigation in [Linear Algebra and Appl. -- 2015. -- {\bf 487}. -- P. 22--42], for $y\in\mathbb{R}$ and a sequence $x=(x_n)\in\ell^\infty$ we define а new notion of density $\delta_{g}$ with respect to a weight function $g$ of indices of the elements $x_n$ close to $y,$ where $g\colon \mathbb{N}\to[ {0,\infty })$ is such that $g(n) \to \infty$ and $n / g(n) \nrightarrow 0.$ We present the relationships between the densities $\delta_{g}$ of indices of $(x_n)$ and the variation of the Ces\aro-limit of $(x_n).$ Our main result states that if the set of limit points of $(x_n)$ is countable and $\delta_g(y)$ exists for any $y\in\mathbb{R},$ then $\lim\nolimits_{n\to\infty} \dfrac{1}{g(n)}\displaystyle\sum\nolimits_{i=1}^{n} x_i = \sum\nolimits_{y\in\mathbb{R}}\delta_g(y)\cdot y ,$ which is an extended and much more general form of the `natural density version of the Osikiewicz theorem''. Note that in [Linear Algebra and Appl. -- 2015. -- {\bf 487}. -- P. 22--42], the regularity of the matrix was used in the entire investigation, whereas in the present paper the investigation is actually performed with respect to a special type of matrix, which is not necessarily regular.

Article (English)

On generalized statistical and ideal convergence of metric-valued sequences

Ukr. Mat. Zh. - 2016. - 68, № 12. - pp. 1598-1606

We consider the notion of generalized density, namely, natural density of weight g recently introduced in [Balcerzak M., Das P., Filipczak M., Swaczyna J. Generalized kinds of density and the associated ideals // Acta Math. Hung. – 2015. –147, № 1. – P. 97 – 115] 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. Some results are also obtained in more general form using the notion of ideals. The entire investigation is performed in the setting of general metric spaces extending the recent results of Kucukaslan M., Deger U., Dovgoshey O. On statistical convergence of metric valued sequences, see Ukr. Math. J. – 2014. – 66, № 5. – P. 712 – 720.