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

  • K. Bose
  • P. Das
  • S. Sengupta

Abstract

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.
Published
25.09.2019
How to Cite
Bose, K., P. Das, and S. Sengupta. “On Spliced Sequences and the Density of Points With respect to a Matrix Constructed by Using a Weight Function”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 71, no. 9, Sept. 2019, pp. 1192-07, http://umj.imath.kiev.ua/index.php/umj/article/view/1509.
Section
Research articles