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On Certain Sequence Spaces

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In this paper define the spaces l ∞ (Δ), c(Δ), and c 0 (Δ), where for instance l ∞ (Δ) = {x=(x k ):sup k |x k -x k + l |< ∞} , and compute their duals (continuous dual, α-dual, β-dual and γ-dual). We also determine necessary and sufficient conditions for a matrix A to map l ∞ (Δ) or c(Δ) into l ∞ or c , and investigate related questions.

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... We show that there exists a very large class of cbc subsets in those spaces with fixed point property for nonexpansive mappings. Thus, first we will recall the definition of Cesàro sequence spaces introduced by Shiue [36] in 1970, and next we will give Kızmaz's construction in [26] for difference sequence spaces since the dual space we work on is obtained from the generalizations of Kızmaz's idea which are derived differently by many researchers [11,16,17,18,33,34,37]. Moreover, there are other difference operators which can also be used to construct more generalized difference sequence spaces (see, for example, [13], [14], [15]). ...
... After the introduction of Cesàro sequence spaces, Kızmaz [26], denoting by ℓ ∞ (△), c (△), and c 0 (△), introduced difference sequence spaces for ℓ ∞ , c, and c 0 where they are the Banach spaces of bounded, convergent, and null sequences, respectively. Here △ represented the difference operator applied to the sequence x = (x n ) n with the rule given by △ x = (x k − x k+1 ) k . ...
... Here △ represented the difference operator applied to the sequence x = (x n ) n with the rule given by △ x = (x k − x k+1 ) k . In 1981, Kızmaz [26] studied Köthe-Toeplitz duals and topological properties for them. ...
... Kizmaz [21] developed the concept of difference sequence spaces by studying the difference sequence spaces ! = ! ...
... As seen below, Kizmaz [21] defines the difference sequence spaces using the difference matrix. ...
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This paper introduces the neutrosophic -statistical convergent difference sequence spaces defined through a modulus function. Additionally, we establish new topological spaces and examine various topological properties within these neutrosophic -statistical convergent difference sequence spaces.
... Kizmaz [20] developed the concept of difference sequence spaces by studying the difference ...
... Some novel sequence spaces were introduce by means of varius matrix transformation in [19,21,22] and [23][24][25]. As seen below, Kizmaz [20] defines the difference sequence spaces using the difference matrix. X(∆) = {ζ = ζ n : ∆ζ ∈ X} for X = c, l ∞ , c 0 , where ∆ζ n = ζ n − ζ n+1 and ∆ shows the difference matrix ∆ = (∆ nm ) defined by ...
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In this paper, we introduce the neutrosophic I-convergent difference sequence spaces I(Δ)(Y)(f){I^{(\mathcal{Y})}_{(\Delta)}}(f) and I(Δ)0(Y)(f){I^{0(\mathcal{Y})}_{(\Delta)}}(f) defined by modulus function. Also, we define an open ball B(x,ϵ,γ)(f)B(x,\epsilon,\gamma)(f) in neutrosophic norm space defined by modulus function. Furthermore, We construct new topological spaces and look into various topological aspects in neutrosophic I-convergent difference sequence spaces I(Δ)(Y)(f){I^{(\mathcal{Y})}_{(\Delta)}}(f) and I(Δ)0(Y)(f){I^{0(\mathcal{Y})}_{(\Delta)}}(f) defined by modulus function
... Especially in studies in the field of summability theory, topological sequence spaces and difference sequence spaces have contributed to obtaining functional results. The concept of difference sequence space has been introduced by Kızmaz in [3] as follows: Suppose that = ∞ , , 0 . Then, (Δ) = { = ( ) ∈ : Δ = (Δ ) = ( − +1 ) ∈ } will be called the difference sequence space. ...
... As a result, we defined non-absolute type difference sequence spaces 0 (Δ) , (Δ) and (Δ) based on the definitions of 0 , and sequence spaces defined by Kaya and Furkan in 2015, and the difference sequence space defined by Kızmaz (1981), and show that the difference sequence spaces 0 (Δ), (Δ) and (Δ) are BK-spaces. Additionally, it is defined that these spaces are isomorphic to the spaces, s 0 , s and s respectively, and their Schauder basis are given. ...
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Examination of spaces in the field of functional analysis, especially revealing their topological and algebraic structures, is very important in terms of forming a basis for studies in the field of pure mathematics and applied sciences. In this context, topology, which was widely used only in the field of geometry at the beginning, gave a solid foundation to the fields in which it was used by causing methodological changes in all branches of mathematics over time. Frechet-Coordinate space (FK space) is a concept that has a functional role in fields such as topological sequence spaces and summability. Topological vector spaces are described as linear spaces defined by a topology that provides continuous vector space operations. If this vector space has a complete metric space structure, it is called Frechet space, and if it has a topology with continuous coordinate functions, it is called Frechet-Coordinate (FK) space. The theory of FK spaces has gained more importance in recent years and has found applications in various fields thanks to the efforts of many researchers. If the topology of an FK space can be derived from the norm, this space is called as a BK space. In this study, cs_0^λ (Δ), cs^λ (Δ), and bs^λ (Δ) difference sequence spaces are defined, and it is revealed that these spaces are BK spaces. In addition, considering the topological properties of these spaces, some spaces that are isomorphic and their duals have been determined.
... A large amount of research work to enrich the theory of sequence spaces is due to the notion of difference spaces, the credit of introduction of which goes to H. Kızmaz [20]. He introduced ...
... Following Kızmaz [20], various mathematicians mainly Altay and Başar [1], Başar and Braha [5], Bhardwaj and Gupta [7], Ç olak [10], Et and Esi [13], Gnanaseelan and Srivastva [14], Mursaleen and Baliarsingh [25], Tripathy and Dutta [30] and many more extended this notion of difference sequence spaces to have various extensions/ generalizations. One may refer to [2, 4, 6-9, 12, 16-19, 23, 24, 27-29, 31-33] and much more references can be found therein. ...
... Kizmaz [13] introduced the concept of difference sequence. Then Esi, Tripathy and Sarma [7] introduced the generalized difference sequence spaces as follows: ...
... For m = 1 and n = 1, these spaces represent the spaces ℓ ∞ (∆), c(∆) and c 0 (∆) introduced and studied by Kizmaz [13]. For n = 1, these spaces represent the spaces ℓ ∞ (∆ m ), c(∆ m ) and c 0 (∆ m ) introduced and studied by Et and Colak [8]. ...
... See [15] for results in connection with infinite products. These two examples provide "sequence spaces" [11] and then "Schauder bases", and then a possibility to generalize these bases [13]. ...
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The usual modulus function of a real number is the maximum of the real number and the additive inverse of the real number. The multiplicative modulus function of a positive real number is the maximum of the positive real number and the multiplicative inverse of the positive real number. This is the beginning of the "multiplicative" mathematics. The possibilities for extensions of various branches of mathematics have already been studied for "multiplicative" branches. This article presents a study on possibilities for extensions of functional analysis to multiplicative functional analysis.-2-World Scientific News 197 (2024) 2-12
... Moreover, composition of a modulus function over itself is also a modulus function. Kizmaz [22] in 1981, introduced the idea of difference sequence space by introducing the following difference sequence spaces : ...
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In the present paper, using the notion of difference sequence spaces, we introduce new kind of Cesàro summable difference sequence spaces of vector valued sequences with the aid of paranorm and modulus function. In addition, we extend the notion of statistical convergence to introduce a new sequence space SC 1 (∆, q) which coincides with C 1 1 (X, ∆, φ, λ, q) (one of the above defined Cesàro summable difference sequence spaces) under the restriction of bounded modulus function.
... Difference sequence spaces, a recent development in summability theory, were first introduced by Kızmaz in the 1980s and have since been extensively studied by mathematicians. The difference sequence spaces ℓ ∞ (∆), c(∆), and c 0 (∆) were introduced by Kızmaz [22] as the domain of the forward difference matrix ∆ F , transforming a sequence, x = (x k ), into the difference sequence ∆ F x = (x k − x k+1 ) in the classical spaces ℓ ∞ , c and c 0 of bounded, convergent, and null sequences, respectively. Quite recently, the difference space bv p was introduced as the domain of the backward difference matrix ∆ B ,, transforming a sequence, ...
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Studies on difference sequences was introduced in the 1980s, and since then, many mathematicians have studied this kind of sequences and obtained some generalized difference sequence spaces. In this paper, using the generalized difference operator, we introduce the concept of the deferred f-statistical convergence of generalized difference sequences of the order α and give some inclusion relations between the deferred f-statistical convergence of generalized difference sequences and deferred f-statistical convergence of generalized difference sequences of the order α. Our results are more general than the corresponding results in the existing literature.
... The difference sequence spaces ℓ ∞ (∆), c(∆), and c 0 (∆) were introduced in [20] and are defined as follows: ...
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This study introduces novel concepts of convergence and summability for numerical sequences, grounded in the newly formulated deferred Nörlund density, and explores their intrinsic connections to symmetry in mathematical structures. By leveraging symmetry principles inherent in sequence behavior and employing two distinct modulus functions under varying conditions, profound links between sequence convergence and summability are established. The study further incorporates lacunary refinements, enhancing the understanding of Nörlund statistical convergence and its symmetric properties. Key theorems, properties, and illustrative examples validate the proposed concepts, providing fresh insights into the role of symmetry in shaping broader convergence theories and advancing the understanding of sequence behavior across diverse mathematical frameworks.
... Let K ⊂ N × N be a two dimentional set of positive integers, and let K(m, n) be the numbers of (p, q) in K such that p ≤ m and q ≤ n. Then we can define the two-dimentional analogue of natural density as follows: Kizmaz [14] introduced the difference sequence space Z(∆) as given below: ...
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In this paper, we explore the idea of lacunary ∆-statistical convergence for double sequences on L-fuzzy normed spaces. Then, we provide a useful characterization of the lacunary ∆-statistical convergence of double sequences with respect to their convergence in the classical sense and show how our method of convergence is weaker than the usual convergence for double sequences on L-fuzzy normed spaces. Towards the end, we give the novel relation between lacunary ∆-statistical cauchy sequence and lacunary ∆-statistical bounded double sequence.
... Kızmaz [21] introduced the difference spaces ℓ ∞ (∆), c (∆) and c 0 (∆), consisting of all real valued sequences x = (x k ) such that ∆x = ∆ 1 x = (x k − x k+1 ) in the sequence spaces ℓ ∞ , c and c 0 . Later, Altinok and Mursaleen [3] generalized this definition by using a difference operator ∆, where (X k ) is a sequence of fuzzy numbers and ∆X = X k − X k+1 . ...
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The main object of this article is to introduce the concepts of ∆f-lacunary statistical convergence of order β and strong ∆f-lacunary summability of order β for sequences of fuzzy numbers and define some sequence classes related to these concepts. We give some inclusion relations between those sequence classes.
... "Kizmaz [23] defined the concept of the difference sequence spaces Z(∆) as follows: ...
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This paper explores the notions of lacunary ∆ m-statistical convergence, lacunary ∆ m-statistical Cauchy sequences, and lacunary strongly ∆ m-convergence within the framework of neutrosophic normed linear spaces (NNLS). We establish the uniqueness of limit and certain algebraic properties of lacunary ∆ m-statistical convergence in NNLS, offering a rigorous foundation for these concepts. Additionally, we introduce the concept of strong Cesàro summability in NNLS and conduct an in-depth analysis of its properties. The connections between lacunary strong ∆ m-convergence and Cesàro summability are also examined, revealing significant interrelationships between these two convergence methods.
... Study of difference sequence spaces is quite new in summability theory. This concept was introduced by Kızmaz [23] and generalized by Et and Ç olak [10]. Afterwards Et and Nuray [12] studied it in order to mainly generalize statistical convergence with respect to ∆ m difference operator as follows ...
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In this work, a new generalization of statistical boundedness is provided for difference sequences in regard to lacunary α−density and lacunary statistical sense. Apart from examining some inclusion theorems on related sequence spaces, we show that ∆m θ (Sαb) does not form a sequence algebra unlike Sαθ(b).
... In this study, since the concept of rough statistical φ-convergence will be worked for difference double sequences, it is important to present some literature knowledge about difference sequences. Kizmaz [19] investigated the concept of difference sequence such that ∆x = (∆x i ) = (x i − x i+1 ). After this study, which can be accepted as a base about difference sequences, Aydın and Başar [1], Başarır [4], Bektaş et al. [5], Demir and Gümüş [8,9], Et [10], Et and Ç olak [11], Et and Nuray [12], Et and Esi [13], Savaş [32] and many others researched significant properties of this concept. ...
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In this paper, we put forward rough statistical φ-convergence of difference sequences as a generalization of rough statistical convergence as well as statistical φ-convergence. We study some of its fundamental properties. We obtain some results for rough statistical φ-convergence for difference double sequences by introducing the rough statistical-φ limit set. So our main objective is to find out the different behaviour of the new convergence concept based on rough statistical-φ limit set.
... Various authors have explored some geometric characteristics of this space. Kızmaz [8] introduced the concept of difference space, which was later generalized by Et. and Çolak [5] into the difference sequence space as follows: ...
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This paper aims to investigate the algebraic and topological properties of a newly constructed difference function space on a rooted tree defined by Musielak-Orlicz function.
... In [9], Kızmaz introduced the notion of difference sequence spaces λ(∆), where λ denotes any one of the classical sequence spaces ℓ ∞ , c, and c 0 . Ç olak and Et [5] further generalized the notion of difference sequence space λ(∆ m ) for λ ∈ {ℓ ∞ , c, c 0 }. ...
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In this paper, the seminormed Cesàro difference sequence space ℓ(F j , q, g, r, µ, ∆ t (s) , C) is defined by using the generalized Orlicz function. Some algebraic and topological properties of the space ℓ(F j , q, g, r, µ, ∆ t (s) , C) are investigated. Various inclusion relations for this sequence space are also studied.
... In all this study we use instead of 0 , ℓ ∞ and . The difference in sequence spaces (∆) = { = ( ): ∆ ∈ } first defined by Kızmaz [1]. Et and Çolak [2] generalized this. ...
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We will define the sequence spaces c_0 (u,∆_v^m )_p, c(u,∆_v^m )_p and l_∞ (u,∆_v^m )_p in this article. Furthermore we give some topological properties and compute their Köthe-Toeplitz duals.
... 05-04-2016 * The corresponding author. 1 Kizmaz [12] introduecd the notion of the difference operator ∆. The operator ∆ denote the matrix ∆ = (∆ nk ) defined by (1.3) ∆ nk = (−1) n−k , n − 1 ≤ k ≤ n 0, 0 ≤ k < n − 1 or k > n. ...
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The aim of the paper is to introduced the spaces c0λ(F^)c_{0}^{\lambda}(\hat{F}) and cλ(F^)c^{\lambda}(\hat{F}) which are the BK-spaces of non-absolute type and also derive some inclusion relations. Further, we determine the α,β,γ\alpha-,\beta-,\gamma-duals of those spaces and also construct their bases. We also characterize some matrix classes on the spaces c0λ(F^)c_{0}^{\lambda}(\hat{F}) and cλ(F^).c^{\lambda}(\hat{F}). Here we characterize the subclasses K(X,Y)\mathcal{K}(X,Y) of compact operators where X is c0λ(F^)c_{0}^{\lambda}(\hat{F}) or cλ(F^)c^{\lambda}(\hat{F}) and Y is one of the spaces c0,c,l,l1,bvc_{0},c, l_{\infty}, l_{1}, bv by applying Hausdorff measure of noncompactness.
... The difference spaces ℓ ∞ (∆), c (∆) and c 0 (∆), consisting of all real valued sequences x = (x k ) such that ∆x = ∆ 1 x = (x k − x k+1 ) in the sequence spaces ℓ ∞ , c and c 0 , were defined by Kızmaz [16]. The idea of difference sequences was generalized by Et and Ç olak [10], Altinok [2], Ç olak et al. [6], Tripathy and Baruah [25] and many others. ...
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In this paper, we define the spaces Nθβ(p,F,Δm),N_{\theta }^{\beta }\left( p,F,\Delta ^{m}\right) , Sθβ(F,Δm),S_{\theta }^{\beta }\left( F,\Delta ^{m}\right) , wpβ(F,Δm)w_{p}^{\beta }\left( F,\Delta ^{m}\right) for sequences of fuzzy numbers using generalized difference operator Δm\Delta ^{m} and a lacunary sequence θ\theta and give some relations between them, where β(0,1] \beta \in \left( 0,1\right] and p>0p>0. Furthermore, in the last section of paper, some inclusion theorems are presented related to the spaces Sθβ(F,Δm) S_{\theta }^{\beta }\left( F,\Delta ^{m}\right) and wpβ(θ,f,F,Δm)w_{p}^{\beta }\left( \theta ,f,F,\Delta ^{m}\right) according to modulus function f.
... The most common type of sets of sequences are probably the sets of difference sequences among the sequence spaces studied. The difference sequence spaces first introduced in Kızmaz's study [15]. Many authors have made efforts to investigate the topological structures of these spaces during the past decade (see [4], [6], [12], [13], [18], [19], [23]). ...
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Fractional difference sequence spaces have been studied in the literature recently. In this work, some identities or estimates for the operator norms and the Hausdorff measures of noncompactness of certain operators on some difference sequence spaces of fractional orders are established. Some classes of compact operators on those spaces are characterized. The results of this work are more general and comprehensive then many other studies in literature.
... Following Kizmaz [13], generalized sequence spaces ∞ (∆ m ), c(∆ m ) and c 0 (∆ m ) were introduced by Et and Ç olak [7]. Based on Et and Ç olak [7], Ç olak and Et [6]. ...
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Objective of this paper is to introduce the generalized geometric difference sequence spaces lG(ΔGm),cG(ΔGm),c0G(ΔGm)l_\infty^{G}(\Delta^m_G), c^G(\Delta^m_G), c_0^{G}(\Delta^m_G) and to prove that these are Banach spaces. Then we prove some inclusion properties. Also we compute their dual spaces.
... The difference sequence spaces are given by Kızmaz [10]. If we choose the absolute summable sequence space and apply the difference operator to this space, we obtain the space of all sequences of bounded variation and denote by bv. ...
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The purpose of this paper is twofold. Firstly, the new matrix domains are constructed with the new infinite matrices and some properties are investigated. Furthermore, dual spaces of new matrix domains are computed and matrix transformations are characterized. Secondly, examples between new spaces with classical sequence spaces and sequence spaces which are derived by an infinite matrix are given in the table form.
... The idea of constructing new sequence spaces via infinite matrices started with Kızmaz's study [20] and then it has been developed by numerous researchers using different triangles [2], [4], [6], [7], [18]. ...
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Among the sets of sequences studied, difference sets of sequences are probably the most common type of sets. This paper considers some p\ell_{p} type fractional difference sequence spaces via Euler gamma function. Although we characterize compactness conditions on those spaces using the main tools of Hausdorff measure of noncompactness, we can only obtain sufficient conditions when the final space is \ell _{\infty }. However, we use some recent results to exactly characterize the classes of compact matrix operators when the final space is the set of bounded sequences.
... In 1981, Kizmaz [11] introduced the notion of difference sequence spaces using forward difference operator ∆ and studied the classical difference sequence spaces ℓ ∞ (∆), c(∆), c 0 (∆). In this section we define the following new geometric sequence space ...
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The main purpose of this paper is to introduce the geometric difference sequence space lG(ΔG)l_\infty^{G} (\Delta_G) and prove that lG(ΔG)l_\infty^{G} ({\Delta}_{G}) is a Banach space with respect to the norm .ΔGG.\left\|.\right\|^G_{{\Delta}_G}. Also we compute the α\alpha-dual, β\beta-dual and γ\gamma-dual spaces. Finally we obtain the Geometric Newton-Gregory interpolation formulae.
... If m = 1, the generalized difference sequence spaces reduced to U(∆ n ) defined and investigated by Tripathy and Esi [33]. While, if n = 1 and m = 1, the generalized difference sequence spaces reduced to U(∆) defined and studied by Kizmaz [34]. ...
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We develop and examine the pre-modular space of null variable exponent-weighted backward generalized difference gai sequences of fuzzy functions in this paper. These sequences of fuzzy functions are important contributions to the concept of modular spaces because they have exponent weighting. Using extended s−fuzzy functions as well as this sequence space of fuzzy functions, it has been possible to accomplish an idealization of the mappings. We have presented some topological and geometric properties of this new space, as well as the ideal mappings that correspond to them.
... Kizmaz [19] discovered the difference sequence spaces conception by considering ...
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Neutrosophication is a useful tool for handling real-world problems with partially dependent, partially independent, and even independent components. By examining some properties related to λ-statistical convergence on neutrosophic normed spaces, we provide some functional tools that are helpful in situations of inconsistency and indeterminacy. Additionally, we establish some related results on λ-△ m-statistical Cauchy sequences on neutrosophic normed spaces.
... Kizmaz [14] has discovered the difference sequence spaces conception by considering the (−1) r m r x k+r . The ∆ m -statistical convergence concept was studied and considered by Et and Nuray [7] with the help of statistical convergence. ...
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The basic purpose of our work is to define λ-statistical convergence for the generalized difference sequences (i.e. λ-∆ m-statistical convergence) on Intuitionistic Fuzzy Normed space (IFN space). We have proven topological results about this generalized method of sequence convergence. Also, we have given the λ-∆ m-statistical Cauchy sequences along with its Cauchy criteria of convergence on these spaces.
... In [22], the difference sequence spaces c 0 (Δ), c(Δ), and ℓ 1 (Δ) were introduced, defined as follows: where DT ¼ ðDT r Þ ¼ ðT r À T kþ1 Þ and the symbols c 0 , c, and ℓ 1 represent the spaces of null sequences, convergent sequences and bounded sequences, respectively. ...
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This research introduces novel concepts in sequence theory, including Bessel convergence, Bessel boundedness, Bessel statistical convergence, and Bessel statistical Cauchy sequences. These concepts establish new inclusion relations and related results within mathematical analysis. Additionally, we extend the first and second Korovkin-type approximation theorems by incorporating Bessel statistical convergence, providing a more robust and comprehensive framework than existing results. The practical implications of these theorems are demonstrated through examples involving the classical Bernstein operator and Fejér convolution operators. This work contributes to the foundational understanding of sequence behavior, with potential applications across various scientific disciplines.
... where λ h = 0 for h < 0, is the backward difference defined by Kizmaz [7]. The concept of variable exponent function spaces has been carefully developed, drawing upon the boundedness of the Hardy-Littlewood maximal mapping. ...
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This article presents the domain of general quantum difference in Nakano sequence space. Some topological and geometric behavior, the multiplication mappings defined on it, and the spectrum of mapping ideals constructed by this space and s−numbers have been introduced. Existing results are constructed by controlling the general quantum difference and power of this new space, which is a major strength.
... The difference sequence spaces (Δ) = {( ) ∈ : Δ( ) ∈ } where is any of the classical sequence spaces. This concept was first introduced by Kızmaz (1981). Following this, quite a lot of work was done with some generalizations by using the difference operator in some way. ...
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Statistical convergence and summability represent a significant generalization of traditional convergence for sequences of real or complex values, allowing for a broader interpretation of convergence phenomena. This concept has been extensively examined by numerous researchers using various mathematical tools and applied to different mathematical structures over time, revealing its relevance across multiple disciplines. In the present study, a generalized definition of the concepts of statistical convergence and summability, termed (△_v^m )_u-generalized weighted statistical convergence and (△_v^m )_u-generalized weighted by [¯N_t ]-summability for real sequences, is introduced using the weighted density and generalized difference operator. Based on this definition, several fundamental properties and inclusion results, obtained by differentiating the components used in the definitions, are provided.
... In [12], the spaces T(△) were studied and is dened as follows: ...
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The focus of the study in the this paper is to introduce the space L ϑ s p, ∆ w g. The corresponding completeness property will be determined. Also, various topological properties will be enlightened. Mathematics Subject Classication (2010): 46B45 46A45 46B99.
... Kizmaz introduced the notion of difference for single sequence spaces [8] as follows: ...
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In this article, we have introduced the notion of convergence of difference double sequences in Pringsheim's sense, difference null in Pringsheim's sense, bounded difference, bounded convergence difference, bounded null difference, regular convergence difference and regular null difference double sequences of bi-complex numbers. We have proved that these are linear spaces. With the help of the Euclidean norm defined on bi-complex numbers, we have established their different algebraic and topological properties, as well as some of their geometric properties. Suitable examples have been discussed to support the introduction of these classes of sequences and during the investigation of their properties for failure cases.
... Initially, Kizmaz [19] proposed the concept of difference sequence spaces as Z(∆) = {y = (y p ) : (∆y p ) ∈ Z} for Z = l ∞ , c, c 0 i.e. spaces of all bounded sequences, convergent sequences and null sequences respectively, where ∆ y = (∆y p ) = (y p − y p+1 ). In particular, l(∆), c(∆) and c 0 (∆) are also Banach spaces, relative to a norm induced by ∥y p ∥ ∆ = |y 1 | + sup k |∆y p | and the generalized difference sequence spaces was defined as (see, Et and Çolak [14]): Z(∆ m y p ) = {y = (y p ) : (∆ m y p ) ∈ Z} for Z = l ∞ , c, c 0 , where ∆ m y = (∆ m y p ) = (∆ m−1 y p − ∆ m−1 y p+1 ) so that ∆ m y p = m r=0 (−1) r m r y k+r . ...
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This study focuses on investigating the concept of rough ideal statistical convergence for generalized difference sequences in intuitionistic fuzzy normed spaces. We have studied the algebraic and topological properties of rough ideal statistical limit points for generalized difference sequence. Apart from this, we also investigated rough ideal statistical cluster points, the relation between rough I-statistical limit points and rough I-statistical cluster points for generalized difference sequence in intuitionistic fuzzy normed spaces.
... The principle of matrix transformation has an abundant reputation in summability theory given by Cesàro, Nörlund, Borel, etc. For more details on sequence spaces defined by matrix domain of infinite matrices see ( [21], [17], [14], [10], [1], [2], [3] and [18], [11], [12], [6] and [7]). ...
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The present paper is emphasis on introducing Orlicz extension of new sequence spaces (i.e br,s0(M,G), br,sc(M,G) and br,s∞(M,G)) by way of the composition of binomial matrix and double band matrix, which are BK-spaces, moreover we prove that these spaces are linearly isomorphic to the spaces l∞, c0 and c. We also derive some inclusion relations. Additionally, we find the Schauder basis for these spaces and finally we also determine the α−, β− and γ− duals of these spaces.
... (Şengönül & Eryilmaz, 2010) introduced and studied the concept of bounded and convergent sequence spaces of interval numbers and showed that these spaces are complete metric space. Furthermore, the concept has been applied by some researcher namely (Dutta & Tripathy, 2016;Devnath, Dutta, & Saha, 2014;Esi, 2014a;Esi, 2011;Esi, 2014b;Esi, & Braha, 2013;Kizmaz, 1981;Mikail, Lee, & Tripathy, 2006;Tripathy & Mahanta, 2007;Tripathy & Borgogain, 2011;Tripathy & Dutta, 2012;Tripathy, Braha, & Dutta, 2014), to study different properties of class of sequences. Recently (Baruah & Dutta, 2020) studied on Quasi-Cauchy sequence spaces of interval numbers and established some important results. ...
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In this article we have introduced some classes generalized difference () Gai sequences of interval numbers with Orlicz functions M. We have studied some algebraic and topological properties of the classes of sequences like, linearity, completeness, solid, symmetric and convergence free.
... In [15], the author introduced the difference sequence spaces ∞ (∆), c(∆), and c 0 (∆) as follows: is the Fibonacci sequence [16], and it is denoted by ( f n ) ∞ n=0 or ( f n ). Mathematicians continue to be attracted by the Fibonacci sequence, which is one of the most well-known number sequences in the world. ...
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The Fibonacci sequence has broad applications in mathematics, where its inherent patterns and properties are utilized to solve various problems. The sequence often emerges in areas involving growth patterns, series, and recursive relationships. It is known for its connection to the golden ratio, which appears in numerous natural phenomena and mathematical constructs. In this research paper, we introduce new concepts of convergence and summability for sequences of real and complex numbers by using Fibonacci sequences, called Δ-Fibonacci statistical convergence, strong Δ-Fibonacci summability, and Δ-Fibonacci statistical summability. And, these new concepts are supported by several significant theorems, properties, and relations in the study. Furthermore, for this type of convergence, we introduce one-sided Tauberian conditions for sequences of real numbers and two-sided Tauberian conditions for sequences of complex numbers.
... Kizmaz [19] discovered the difference sequence spaces conception by considering ...
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... Penelitian terkait dual Köthe-Toeplitz dari suatu ruang barisan dapat dilihat dalam Kizmaz [1], Et [5], Et dan Ç olak [8], serta A. H. Ganie dkk. [9]. ...
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