Four dimensional dual and dual of the new sort summability methods

Yeşılkayagıl, Medıne; Başar, Feyzı

doi:10.18532/caam.72843

Cited by 6 publications

(6 citation statements)

References 9 publications

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“…In the current section, we deal with some 4-dimensional matrix mapping classes related to the double sequence spaces Φ (M u ), Φ (C p ), Φ (C bp ) and Φ (C r ) by using dual summability methods for double sequences which have been presented and examined by Başar [4] and Yeşilkayagil and Başar [37] and which have been applied by Tuḡ [36]. for every i, j ∈ N. In that case, if we take ϑ-limit on equality above as i, j → ∞, we have Bx = Hy.…”

Section: Charactarization Of Some Classes Of 4-dimensional Matricesmentioning

confidence: 99%

4-Dimensional Euler-Totient Matrix Operator and Some Double Sequence Spaces

Erdem

Demiriz

2020

Mathematical Sciences and Applications E-Notes

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Our main purpose in this study is to investigate the matrix domains of the 4-dimensional Euler-totient matrix operator on the classical double sequence spaces M u , C p , C bp and C r. Besides these, we examine their topological and algebraic properties and give inclusion relations about the new spaces. Also, the α−, β(ϑ)− and γ−duals of these spaces are determined and finally, some matrix classes are characterized.

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Section: Charactarization Of Some Classes Of 4-dimensional Matricesmentioning

confidence: 99%

4-Dimensional Euler-Totient Matrix Operator and Some Double Sequence Spaces

Erdem

Demiriz

2020

Mathematical Sciences and Applications E-Notes

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“…Then we complete this section with some significant results of four-dimensional matrix mapping via the dual summability methods for double sequences which has been introduced and studied by Başar [20] and Yeşilkayagil and Başar [21] and has been recently applied in [10,12]. (17) is necessary.…”

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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“…In this section, we characterize some new four-dimensional matrix classes , . Then we complete this section with some significant results of four-dimensional matrix mapping via the dual summability methods for double sequences which have been introduced and studied by Başar [ 22 ] and Yeşilkayagil and Başar [ 23 ], and which have recently been applied in [ 2 ].…”

Section: Matrix Transformations Related To the Sequence Spacementioning

confidence: 99%

“…Tuǧ [ 2 ] has recently applied the dual summability methods for double sequences which has been introduced and studied by Başar [ 22 ], and Yeşilkayagil and Başar [ 23 ]. In this work, we use the relation between the four-dimensional matrices , , and with , which has been proved and studied in [ 2 , Lemma 4.2, Theorem 4.5].…”

Section: Matrix Transformations Related To the Sequence Spacementioning

confidence: 99%

On almost B-summable double sequence spaces

Tuǧ

2018

J Inequal Appl

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The concept of a four-dimensional generalized difference matrix and its domain on some double sequence spaces was recently introduced and studied by Tuğ and Başar (AIP Conference Proceedings, vol. 1759, 2016) and Tuğ (J. Inequal. Appl. 2017(1):149, 2017). In this present paper, as a natural continuation of (J. Inequal. Appl. 2017(1):149, 2017), we introduce new almost null and almost convergent double sequence spaces and as the four-dimensional generalized difference matrix domain in the spaces and , respectively. Firstly, we prove that the spaces and of double sequences are Banach spaces under some certain conditions. Then we give an inclusion relation of these new almost convergent double sequence spaces. Moreover, we identify the α-dual, -dual and γ-dual of the space . Finally, we characterize some new matrix classes , , and we complete this work with some significant results.

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Four dimensional dual and dual of the new sort summability methods

Cited by 6 publications

References 9 publications

4-Dimensional Euler-Totient Matrix Operator and Some Double Sequence Spaces

4-Dimensional Euler-Totient Matrix Operator and Some Double Sequence Spaces

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

On almost B-summable double sequence spaces

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