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

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.