Almost convergence of double subsequences

Abstract

Almost-convergence of double sequences (subsequences) is equivalent to almost Cauchy condition. If the set of all almost convergent subsequences of a sequence S = S nm is of the second category, then S is convergent in the simple sense. For the sequence S = Snm which almost converges to L, Lebesgue measure of the set of all its subsequences which almost converge to L is either 1 or 0. 1 pq p−1 i=0 q−1 j=0 S n+i,m+j − L < ε for ∀p, q > N and ∀(n, m) ∈ N × N. We write f − lim S = L. We denote by X the set of all double sequences of 0's and 1's,namely, X = {x = (x nm) : x nm ∈ {0, 1}, n, m ∈ N} In [4] F. Moricz dened the concept of a subsequence of a double sequence.

Generalized Almost Convergence and Core Theorems of Double Sequences

“…For more details on almost convergence for single and double sequences, one can refer to [6][7][8][9][10][11][12][13].…”

Characterization of uniform statistical convergence for double sequences

“…For double sequences almost convergence was introduced by Mòricz and Rhoades [8] and it was studied in some detail in [16] and [9].…”

Almost Conservative Four-Dimensional Matrices through de la Vallée-Poussin Mean

“…The authors of [5] introduced the notion of Banach limit for double sequence and characterized the spaces of almost and strong almost convergence for double sequences through some sublinear functionals. For more details on these concepts, one can refer to [6][7][8][9][10][11][12].…”