By M. Mursaleen, S.A. Mohiuddine

This publication solely bargains with the research of just about convergence and statistical convergence of double sequences. The thought of “almost convergence” might be the main precious proposal so that it will receive a vulnerable restrict of a bounded non-convergent series. there's one other thought of convergence referred to as the “statistical convergence”, brought by way of H. quickly, that's an extension of the standard suggestion of sequential limits. this idea arises to illustrate of “convergence in density” that is additionally studied as a summability approach. Even unbounded sequences may be handled by utilizing this technique. The publication additionally discusses the purposes of those non-matrix equipment in approximation idea. Written in a self-contained variety, the ebook discusses intimately the equipment of virtually convergence and statistical convergence for double sequences besides functions and compatible examples. The final bankruptcy is dedicated to the examine convergence of double sequence and describes numerous convergence checks analogous to these of unmarried sequences. as well as purposes in approximation concept, the implications are anticipated to discover program in lots of different parts of natural and utilized arithmetic corresponding to mathematical research, chance, fastened element thought and statistics.

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**Example text**

L(S11 x) = L(x) = L(S10 x) = L(S01 x), we have L(x) = L 1 (p + 1)(q + 1) p q xj +s,k+t j =0 k=0 = L τpqst (x) =L 1 (m + 1)(n + 1) m n τpqst (x) . 7) p=0 q=0 for m > m0 , n > n0 and all s, t. 8) 26 2 Almost Convergence of Double Sequences for m > m0 , n > n0 and all s, t, where e is defined at the beginning of Sect. 3. 8) that L(x) ≤ φ(x) for all x = (xj k ) ∈ Mu . 9) that is, φ dominates Banach limits. 9), we get {Mu , ξ } = {Mu , φ}, which implies that φ dominates and generates Banach limits and φ(x) = ξ(x) for all x ∈ Mu .

Let x = (xj k ) be defined by xj k = (−1)k for all j, that is, ⎛ −1 ⎜−1 ⎜ ⎜−1 ⎜ ⎝ · · 1 1 1 · · −1 −1 −1 · · · · · · · 1 1 1 · · ⎞ · ·⎟ ⎟ ·⎟ ⎟. ·⎠ · · · · · · Then τpqst (x − 0) = ≤ 1 (p + 1)(q + 1) p q xj +s,k+t j =0 k=0 q +1 1 = (p + 1)(q + 1) p+1 uniformly for s, t. 5, x ∈ F . But x ∈ / [w]2 ∩ / [F]. 7 Exercises 1 Check whether the double sequence x = (xnk ) defined by xnk = 1 if n and k are squares, 0 otherwise is almost convergent or not. If yes, then what is its almost limit? 2 Show that the inclusion [F] ⊂ F is proper.

4. 3 Strongly Regular Matrices The notion of strong regularity for single sequences was introduced by Lorentz [63], and for double sequences, by Móricz and Rhoades [83]. , A ∈ (F, CBP )reg . Necessary and sufficient conditions were also established for a matrix A = (amnj k ) to be strongly regular in [83] as follows. 5) j =0 k=0 where 10 amnj k = amnj k − am,n,j +1,k and (j, k, m, n = 0, 1, . . ). ) Let x be an almost convergent sequence with limit s. We need to show that (ymn ) is convergent and bounded.