On spliced sequences and the density of points with respect to a matrix constructed by using a weight function - Bose, K.; Das, P.; Sengupta, S. (Ukrains’kyi Matematychnyi Zhurnal, 2019)
UDC 517.5 Following the line of investigation in [Linear Algebra and Appl. – 2015. – 487. – P. 22–42], for yR and a sequence x=(x_n)^∞ we define а new notion of density δ_g with respect to a weight function g of indices of the elements x_n close to y, where gN→[ 0,∞ ) is such that g(n) → ∞ and n / g(n) rightarrow 0. We present the relationships between the densities δ_g of indices of (x_n) and the variation of the Cesàro-limit of (x_n). Our main result states that if the set of limit points of (x_n) is countable and δ_g(y) exists for any yR, then olimits_n→∞ 1g(n)olimits_i=1^n x_i = olimits_yR δ_g(y)· 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. – 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.