On the spaces of Nörlund almost null and Nörlund almost convergent sequences

Tuǧ, Orhan; Başar, Feyzı

doi:10.2298/fil1603773t

Cited by 6 publications

(4 citation statements)

References 14 publications

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“…The spaces l ∞ (N t ) and l p (N t ) were studied by Wang [2] while 1 ≤ p < ∞. f 0 (N t ) and f (N t ) were studied by Tug and Başar [35], where f 0 and f are almost-null and almost-convergent sequence spaces, respectively. Tug and Başar [35] have not investigated whether the space was the expansion or the contraction or overlap of the original space.…”

Section: Discussionmentioning

confidence: 99%

“…Sengönül and Başar [34] introduced and investigated the domain of the Cesaro matrix of order one on sequence spaces c and c 0 . Also, f 0 (N t ) and f (N t ) were defined by Tug and Başar [35], where f 0 and f were almost null and almost convergent sequence spaces, respectively. Yeşilkayagil and Başar [36] investigated the paranormed Nörlund sequence space of the non-absolute type.…”

Section: Literature Surveymentioning

confidence: 99%

“…Lemma 2. [35] Let δ and µ be a subspace of w and let the infinite matrices be S = (s nk ) and V = (v nk ). If S and V are connected with the relation:…”

Section: Mathematical Backgroundmentioning

confidence: 99%

“…(30) S = (s nk ) ∈ (l 1 , cs) if, and only if, Equations (12) and (35) hold. (31) S = (s nk ) ∈ (bv, cs) if, and only if, Equations (12),(35) and(37) hold. (32) S = (s nk ) ∈ (bv 0 , cs) if, and only if, Equations(12) and(37) hold.…”

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confidence: 99%

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On Domain of Nörlund Matrix

Kayaduman

Yaşar²

2018

Mathematics

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In 1978, the domain of the Nörlund matrix on the classical sequence spaces lp and l∞ was introduced by Wang, where 1 ≤ p < ∞. Tuğ and Başar studied the matrix domain of Nörlund mean on the sequence spaces f0 and f in 2016. Additionally, Tuğ defined and investigated a new sequence space as the domain of the Nörlund matrix on the space of bounded variation sequences in 2017. In this article, we defined new space and and examined the domain of the Nörlund mean on the bs and cs, which are bounded and convergent series, respectively. We also examined their inclusion relations. We defined the norms over them and investigated whether these new spaces provide conditions of Banach space. Finally, we determined their α, β, γduals, and characterized their matrix transformations on this space and into this space.

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Section: Discussionmentioning

confidence: 99%

Section: Literature Surveymentioning

confidence: 99%

“…Lemma 2. [35] Let δ and µ be a subspace of w and let the infinite matrices be S = (s nk ) and V = (v nk ). If S and V are connected with the relation:…”

Section: Mathematical Backgroundmentioning

confidence: 99%

mentioning

confidence: 99%

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On Domain of Nörlund Matrix

Kayaduman

Yaşar²

2018

Mathematics

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The spaces of B(r,s,t,u) strongly almost convergent double sequences and matrix transformations

Tuǧ

2021

Bulletin des Sciences Mathématiques

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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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On the spaces of Nörlund almost null and Nörlund almost convergent sequences

Cited by 6 publications

References 14 publications

On Domain of Nörlund Matrix

On Domain of Nörlund Matrix

The spaces of B(r,s,t,u) strongly almost convergent double sequences and matrix transformations

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

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