On matrix summability of spliced sequences and A-density of points

Abstract

For y∈R and a sequence x=(xn)∈ℓ∞ we define the new notion of A-density δA(y) of indices of those xn's which are close to y where A is a non-negative regular matrix. We present connections between A-densities δA(y) of indices of (xn) and the A-limit of (xn). Our main result states that if the set of limit points of (xn) is countable and δA(y) exists for any y∈R where A is a non-negative regular matrix, then limn→∞⁡(Ax)n=∑y∈RδA(y)⋅y, which presents a different view of Osikiewicz Theorem.