Almost convergence of double sequences and strong regularity of summability matrices

Abstract

A double sequence x = {xjk: j, k = 0, 1, …} of real numbers is called almost convergent to a limit s ifthat is, the average value of {xjk} taken over any rectangle {(j, k): m ≤ j ≤ m + p − 1, n ≤ k ≤ n + q − 1} tends to s as both p and q tend to ∞, and this convergence is uniform in m and n. The notion of almost convergence for single sequences was introduced by Lorentz [1].