Generalized Almost Convergence and Core Theorems of Double Sequences

Mohiuddine, S. A.; Alotaibi, Abdullah

doi:10.1155/2014/152910

Cited by 3 publications

(3 citation statements)

References 33 publications

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“…Then many of the mathematicians have studied the matrix domain on almost null and almost convergent sequences spaces (see [24][25][26][27]). The almost convergence for double sequence was introduced by Moricz and Rhoades [14] and studied on by many researchers (see [16,[28][29][30][31][32][33][34][35][36]). Yeşilkayagil and Basar [37] recently studied the topological properties of the spaces of almost null and almost convergent double sequences.…”

Section: Resultsmentioning

confidence: 99%

On the Characterization of Some Classes of Four-Dimensional Matrices and AlmostB-Summable Double Sequences

Tuǧ

2018

Journal of Mathematics

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We firstly summarize the related literature about ( , , , )-summability of double sequence spaces and almost ( , , , )-summable double sequence spaces. Then we characterize some new matrix classes of (L : C ), ( (L ) : C ), and (L : (C )) of four-dimensional matrices in both cases of 0 < ≤ 1 and 1 < < ∞, and we complete this work with some significant results. Preliminaries, Background, and NotationsWe denote the set of all complex valued double sequence by Ω which is a vector space with coordinatewise addition and scalar multiplication. Any subspace of Ω is called a double sequence space. A double sequence = ( ) of complex numbers is called bounded if ‖ ‖ ∞ = sup , ∈N | | < ∞, where N = {0, 1, 2, . . .}. The space of all bounded double sequences is denoted by M which is a Banach space with the norm ‖ ⋅ ‖ ∞ . Consider the double sequence = ( ) ∈ Ω. If for every > 0 there exists a natural number 0 = 0 ( ) and ∈ C such that | − | < for all , > 0 , then the double sequence is said to be convergent in Pringsheim's sense to the limit point ; say that − lim , →∞ = , where C indicates the complex field. The space C denotes the set of all convergent double sequences in Pringsheim's sense. Although every convergent single sequence is bounded, this is not hold for double sequences in general. That is, there are such double sequences which are convergent in Pringsheim's sense but not bounded. Actually, Boos [1, p. 16] defined the sequence = ( ) byThen it is clearly seen that − lim , →∞ = 0 but ‖ ‖ ∞ = sup , ∈N | | = ∞, so ∈ C −M . The set C denotes the space of both bounded and convergent double sequences; that is, C = C ∩ M . Hardy [2] showed that a double sequence = ( ) is said to converge regularly to if ∈ C and the limits := lim , ( ∈ N) and := lim , ( ∈ N) exist, and the limits lim lim and lim lim exist and are equal to the -limit of . Moreover, by C 0 and C 0 , we may denote the spaces of all null double sequences contained in the sequence spaces C and C , respectively. Móricz [3] proved that the double sequence spaces C , C 0 , C , and C 0 are Banach spaces with the norm ‖ ⋅ ‖ ∞ . The space L of all absolutely −summable double sequences corresponding to the space ℓ of −summable single sequences was defined by Başar and Sever [4]; that is,which is a Banach space with the norm ‖ ⋅ ‖ . Then, the space L which is a special case of the space L with = 1 is introduced by Zeltser [5].Let be a double sequence space and converging with respect to some linear convergence rule is − lim : → C. Then, the sum of a double series ∑ , relating to this rule is defined by − ∑ , = − lim , →∞ ∑ , , =0. Throughout the paper, the summations from 0 to ∞ without limits, that is, ∑ , , mean that ∑ ∞ , =0. Here and after, unless otherwise stated, we consider that denotes any of the symbols , , or .

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Section: Resultsmentioning

confidence: 99%

On the Characterization of Some Classes of Four-Dimensional Matrices and AlmostB-Summable Double Sequences

Tuǧ

2018

Journal of Mathematics

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“…Using Banach limits, in 1948, Lorentz [2] introduced the notion of almost convergence, which is a generalization of usual convergence of real sequences. Moricz and Rhoades [18] extended the idea of almost convergence for double sequences and later it was studied in [22]. Again in 1951 Fast [3] and Steinhaus [4] introduced independently the notion of statistical convergence by a rigorous use of natural density of subsets of N, which is another generalization of usual convergence.…”

Section: Introductionmentioning

confidence: 99%

Generalized almost statistical convergence

Shaikh¹,

Datta²

2019

Preprint

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The objective of this paper is to introduce the notion of generalized almost statistical (briefly, GAS) convergence of bounded real sequences, which generalizes the notion of almost convergence as well as statistical convergence of bounded real sequences. As a special kind of Banach limit functional, we also introduce the concept of Banach statistical limit functional and the notion of GAS convergence mainly depends on the existence of Banach statistical limit functional. We prove the existence of Banach statistical limit functional. Then we have shown the existence of a GAS convergent sequence, which is neither statistical convergent nor almost convergent. Also, some topological properties of the space of all GAS convergent sequences are investigated.

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“…As an extension of the notion of almost convergence, Kayaduman and Şengönül [ 12 , 13 ] defined Cesàro and Riesz almost convergence and established related core theorems. The almost strongly regular matrices for single sequences were introduced and characterized [ 14 ], and for double sequences they were studied by Mursaleen [ 15 ] (also refer to [ 16 – 19 ]). As an application of almost convergence, Mohiuddine [ 20 ] proved a Korovkin-type approximation theorem for a sequence of linear positive operators and also obtained some of its generalizations.…”

Section: Introductionmentioning

confidence: 99%

Weighted almost convergence and related infinite matrices

Mohiuddine

Alotaibi

2018

J Inequal Appl

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The purpose of this paper is to introduce the notion of weighted almost convergence of a sequence and prove that this sequence endowed with the sup-norm is a BK-space. We also define the notions of weighted almost conservative and regular matrices and obtain necessary and sufficient conditions for these matrix classes. Moreover, we define a weighted almost A-summable sequence and prove the related interesting result.

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Generalized Almost Convergence and Core Theorems of Double Sequences

Cited by 3 publications

References 33 publications

On the Characterization of Some Classes of Four-Dimensional Matrices and AlmostB-Summable Double Sequences

On the Characterization of Some Classes of Four-Dimensional Matrices and AlmostB-Summable Double Sequences

Generalized almost statistical convergence

Weighted almost convergence and related infinite matrices

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