Almost Conservative Four-Dimensional Matrices through de la Vallée-Poussin Mean

Mohiuddine, S. A.; Alotaibi, Abdullah

doi:10.1155/2014/412974

Cited by 5 publications

(7 citation statements)

References 23 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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“…The idea of -convergence for double sequences was introduced in (ÇAKAN et al, 2006) and further studied recently in MOHIUDDINE, 2007). Çakan et al (2009), Mohiuddine and Alotaibi (2013;2014), Mursaleen and Mohiuddine (2008;2009b;2010a;2010b;2010c;2012), studied various classes of four dimensional matrices, e.g.…”

Section: Definitions Notations and Preliminariesmentioning

confidence: 99%

<b>Matrix transformations of paranormed sequence spaces through de la Vallée-Pousin mean

Mohiuddine

2015

Acta Sci. Technol.

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ABSTRACT. In this paper we define the sequence space related to the concept of invariant mean and de la Vallée-Pousin mean. We also determine the necessary and sufficient conditions to characterize the matrices which transform paranormed sequence spaces into the spaces and , where denotes the space of all , -convergent sequences.Keywords: ınvariant mean, de la Vallée-Pousin mean, -convergence, matrix transformation.Transformações matriciais de espaços sequenciais paranormatizados pela média de la Vallée-Pousin RESUMO. Definiu-se a sequência espacial ∞ relacionada ao conceito de média invariável e à media de de la Vallée-Pousin. Determinaram-se também as condições necessárias e suficiente para caracterizar as matrizes que transformam os espaços sequenciais paranormatizados nos espaços and ∞ , onde é o espaço de todas as sequências convergentes , .Palavras-chave: média invariável, media de de la Vallée-Pousin, convergência , transformação matricial.

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“…Later on, it was studied by various authors, for example, Móricz [3], Móricz and Rhoades [4], Başarır and Sonalcan [5], Mursaleen and Mohiuddine [6][7][8], and many others. Mursaleen [9] has defined and characterized the notion of almost strong regularity of four-dimensional matrices and applied these matrices to establish a core theorem (also see [10,11]). Altay and Başar [12] have recently introduced the double sequence spaces BS, BS( ), CS , CS , CS , and BV consisting of all double series whose sequence of partial sums are in the spaces M , M ( ), C , C , C , and L , respectively.…”

Section: Introduction Notations and Preliminariesmentioning

confidence: 99%

“…A Musielak-Orlicz function F = ( ) is said to satisfy Δ 2 -if there exist constants , > 0 and a sequence = ( ) ∞ =1 ∈ 1 + (the positive cone of 1 ) such that the inequality (2 ) ≤ ( ) + (11) holds for all ∈ N and ∈ R + , whenever ( ) ≤ . A double sequence = ( ) is said to be bounded if ‖ ‖ (∞,2) = sup , | | < ∞.…”

Section: Introduction Notations and Preliminariesmentioning

confidence: 99%

Some Paranormed Double Difference Sequence Spaces for Orlicz Functions and Bounded-Regular Matrices

Mohiuddine

Raj

Alotaibi

2014

Abstract and Applied Analysis

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Almost Conservative Four-Dimensional Matrices through de la Vallée-Poussin Mean

Cited by 5 publications

References 23 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

<b>Matrix transformations of paranormed sequence spaces through de la Vallée-Pousin mean

Some Paranormed Double Difference Sequence Spaces for Orlicz Functions and Bounded-Regular Matrices

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