Convergence Methods for Double Sequences and Applications by M. Mursaleen, S.A. Mohiuddine

By M. Mursaleen, S.A. Mohiuddine

This publication solely offers with the research of just about convergence and statistical convergence of double sequences. The proposal of “almost convergence” could be the main important inspiration to be able to receive a vulnerable restrict of a bounded non-convergent series. there's one other idea of convergence often called the “statistical convergence”, brought through H. quick, that is an extension of the standard inspiration of sequential limits. this idea arises for instance of “convergence in density” that's additionally studied as a summability strategy. Even unbounded sequences will be handled by utilizing this system. The booklet additionally discusses the functions of those non-matrix equipment in approximation idea. Written in a self-contained variety, the booklet discusses intimately the equipment of virtually convergence and statistical convergence for double sequences in addition to purposes and appropriate examples. The final bankruptcy is dedicated to the research convergence of double sequence and describes a number of convergence exams analogous to these of unmarried sequences. as well as functions in approximation thought, the consequences are anticipated to discover program in lots of different parts of natural and utilized arithmetic comparable to mathematical research, likelihood, mounted aspect idea and statistics.

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However, every bounded convergent double sequence is almost convergent, and every almost convergent double sequence is bounded. Let A = (amnj k ) (j, k = 0, 1, . . ) be a double infinite matrix of real numbers. For all m, n = 0, 1, . . , the sums ∞ ∞ ymn = amnj k xj k j =0 k=0 are called the A-means of the sequence x = (xj k ). We say that a sequence x is Asummable to the limit t if the A-means exist for all m, n = 0, 1, . . in the sense of Pringsheim’s convergence: p q amnj k xj k = ymn P - lim p,q→∞ j =0 k=0 and P - lim ymn = t.

01 vpk and Proof Abel’s transformation for single series is m m vi (ui ∓ ui+1 ) = i=1 ui (vi ∓ vi−1 ) ∓ um+1 vm . 5 Some Basic Lemmas 29 p q + q uj,q+1 vj −1,q − j =1 up+1,k vpk + k=1 q p = up+1,k vp,k−1 + up+1,q+1 vpq k=1 p uj k ( 11 vj k ) − j =1 k=1 q 10 vj q ) − uj,q+1 ( j =1 up+1,k ( 01 vpk ) k=1 + up+1,q+1 vpq . Another form of Abel’s transformation for double series is given by Altay and Ba¸sar [4]. 8 [w2 ]- lim x = L if and only if (i) w2 - lim x = L; u v 1 (ii) uv m=1 n=1 |T1 (m, n, s, t) − L| −→ 0 (u, v −→ ∞) uniformly in s, t; u 1 (iii) uv m=1 vn=1 |T2 (m, n, s, t) − L| −→ 0 (u, v −→ ∞) uniformly in s, t; u v 1 (iv) uv m=1 n=1 |τmnst +dmnst −T1 (m, n, s, t)−T2 (m, n, s, t)| −→ 0 (u, v −→ ∞) uniformly in s, t, where 1 T1 (m, n, s, t) = (m + 1) m τpnst p=0 1 and T2 (m, n, s, t) = (n + 1) n τmqst .

Ii) Mu ∩ w2 = {x = (xj k ) ∈ Mu : φ(x) = −φ(−x)} = F . Proof (i) By the definition of ξ , for given > 0, there exist p0 , q0 such that τpqst (x) < ξ(x) + for p > p0 , q > q0 and all s, t. This implies that φ(x) ≤ ξ(x) + for all x = (xj k ) ∈ Mu . , φ generates Banach limits. Conversely, suppose that L ∈ B2 . , 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.

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