On IA-Density of Points and Some of its Consequences

Abstract

Continuing in the line of recent investigation of [2] we further consider the more general approach of [22, 21] and for y ∈R and a bounded sequence x = (xn)  we define the more general notion of IA-density of indices of those xn’s which are close to y, denoted by IδA(y) where A is a non-negative regular matrix. Connections are drawn between IδA(y) and the IA-limit of (xn). Our main result states that if the set of limit points of (xn) is countable and IδA(y) exists for any y ∈ R where A is a non-negative regular matrix, then lim I−(Ax)n = Sum of all δA(y)·y where y runs over R. which presents a more general version of the different view of Osikiewicz Theorem given in [2].