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Almost convergence and generalized weighted mean II
Journal of Inequalities and Applications volume 2014, Article number: 93 (2014)
Abstract
In this paper, we investigate some new sequence spaces, which naturally emerge from the concepts of almost convergence and generalized weighted mean. The object of this paper is to introduce the new sequence spaces obtained as the matrix domain of generalized weighted mean in the spaces of almost null and almost convergent sequences. Furthermore, the beta and gamma dual spaces of the new spaces are determined and some classes of matrix transformations are characterized.
MSC:47A15, 46A32, 47D20.
1 Introduction
By a sequence space, we understand a linear subspace of the space {\mathbb{C}}^{\mathbb{N}} of all complex sequences which contains ϕ, the set of all finitely nonzero sequences, where ℂ denotes the complex field and \mathbb{N}=\{0,1,2,\dots \}. We write {\ell}_{\mathrm{\infty}}, c, and {c}_{0} for the classical spaces of all bounded, convergent and null sequences, respectively. Also by bs, cs, {\ell}_{1}, and {\ell}_{p}, we denote the spaces of all bounded, convergent, absolutely, and pabsolutely convergent series, respectively.
A sequence spaces μ with a linear topology is called a Kspace if each of the maps {p}_{i}:\mu \to \mathbb{C} defined by {p}_{i}(x)={x}_{i} is continuous for all i\in \mathbb{N}. A Kspace is called an FKspace if μ is a complete linear metric space; a BKspace is a normed FKspace.
A sequence {({b}^{(n)})}_{n=0}^{\mathrm{\infty}} in a normed space μ is called a Schauder basis if for every x\in \mu, there is a unique sequence {({\beta}_{n})}_{n=0}^{\mathrm{\infty}} of scalars such that
The sequence ({e}^{(n)}) is the Schauder basis for {\ell}_{p} and {c}_{0}, and \{e,{e}^{(n)}\} is the Schauder basis for the space c, while the space {\ell}_{\mathrm{\infty}} has no Schauder basis, where {e}^{(n)} and e denote the sequences whose only nonzero entry is a 1 in the n th place for each n\in \mathbb{N} and e=(1,1,1,\dots ).
A subset M of a metric space (X,d) is said to be dense in X if \overline{M}=X. A metric space (X,d) is said to be separable if it contains a countable subset which is dense in X. Note that a nonseparable space has no Schauder basis.
Let λ and μ be two sequence spaces, and A=({a}_{nk}) be an infinite matrix of complex numbers {a}_{nk}, where k,n\in \mathbb{N}. Then we say that A defines a matrix mapping from λ into μ, and we denote it by writing A:\lambda \to \mu if for every sequence x=({x}_{k})\in \lambda the sequence Ax=\{{(Ax)}_{n}\}, the Atransform of x, is in μ; here
By (\lambda :\mu ), we denote the class of all matrices A such that A:\lambda \to \mu. Thus, A\in (\lambda :\mu ) if and only if the series on the right side of (1.1) converges for each n\in \mathbb{N} and each x\in \lambda and we have Ax={\{{(Ax)}_{n}\}}_{n\in \mathbb{N}}\in \mu for all x\in \lambda. A sequence x is said to be Asummable to α if Ax converges to α, which is called the Alimit of x. Also by (\lambda :\mu ;p), we denote the subset of (\lambda :\mu ) for which limits or sums are preserved whenever there is a limit or sum on the spaces λ and μ. The matrix domain {\lambda}_{A} of an infinite matrix A in a sequence space λ is defined by
which is a sequence space. If A=({a}_{nk}) is triangle, that is to say, {a}_{nn}\ne 0 and {a}_{nk}=0 for all k>n, then one can easily observe that the sequence spaces {\lambda}_{A} and λ are linearly isomorphic, i.e., {\lambda}_{A}\cong \lambda [1]. We write U for the set of all sequences u=({u}_{k}) such that ({u}_{k})\ne 0 for all k\in \mathbb{N}. For u\in U, let 1/u=(1/{u}_{k}). Let u,v\in U and define the generalized weighted mean or factorable matrix G(u,v)=({g}_{nk}) by
for all k,n\in \mathbb{N}; here {u}_{n} depends only on n and {v}_{k} only on k.
We shall write throughout for brevity
for all k,m,n\in \mathbb{N}.
The main purpose of present paper is to introduce the sequence spaces {f}_{0}(G) and f(G) derived as the domain of the generalized weighted mean in the spaces {f}_{0} and f of almost null and almost convergent sequences, and to determine the β and γduals of these spaces. Furthermore, some classes of matrix mappings on/in the space f(G) are characterized.
2 Spaces of almost null and almost convergent sequences
The shift operator P is defined on ω by {(Px)}_{m}={x}_{m+1} for all n\in \mathbb{N}. A Banach limit L is defined on {\ell}_{\mathrm{\infty}}, as a nonnegative linear functional, such that L(Px)=L(x) and L(e)=1. A sequence x=({x}_{k})\in {\ell}_{\mathrm{\infty}} is said to be almost convergent to the generalized limit α if all Banach limits of x are α [2] and is denoted by f\phantom{\rule{0.2em}{0ex}}\text{}\text{lim}\phantom{\rule{0.2em}{0ex}}{x}_{k}=\alpha. Let {P}^{i} be the composition of P with itself i times and let us write for a sequence x=({x}_{k})
Lorentz [2] proved that f\phantom{\rule{0.2em}{0ex}}\text{}\text{lim}\phantom{\rule{0.2em}{0ex}}{x}_{k}=\alpha if and only if {lim}_{n\to \mathrm{\infty}}{t}_{mn}(x)=\alpha uniformly in m. It is well known that a convergent sequence is almost convergent such that its ordinary and generalized limits are equal.
The spaces {f}_{0} and f of almost null and almost convergent sequences are defined as follows:
where {t}_{mn}(x) is defined by (2.1). Also, by fs, we denote the space of all almost convergent series.
3 New sequence spaces and their duals
In this section, we introduce the sequence spaces {f}_{0}(G) and f(G) and give some results concerning them, and we determine their beta and gamma duals.
Malkowsky and Savaş [3] have defined the sequence space Z=(u,v;X), which consists of all sequences whose G(u,v)transforms are in X\in \{{\ell}_{\mathrm{\infty}},c,{c}_{0},{\ell}_{p}\}, where u,v\in U. The space Z(u,v;X) defined by
Altay and Başar [4] constructed the new paranormed sequence spaces \lambda (u,v;p) defined by
where \lambda \in \{{\ell}_{\mathrm{\infty}},c,{c}_{0}\}.
Afterward, Altay and Başar [5] studied the sequence space \ell (u,v;p) as follows:
Şimşek et al. [6] have introduced a modular structure of the sequence spaces defined by Altay and Başar [5] and studied KadecKlee and uniform Opial properties of this sequence space on Köthe sequence spaces.
The new sequence spaces {f}_{0}(G) and f(G) are the set of all sequences whose G(u,v)transforms are in the spaces {f}_{0} and f, that is,
By the notation of (1.2), the sequence spaces {f}_{0}(G) and f(G) are restated as
Define the sequence y=({y}_{k}) by the G(u,v)transform of a sequence x=({x}_{k}),
Theorem 3.1 The sequence spaces f(G) and {f}_{0}(G) are BKspaces with the same norm given by
where
Proof {f}_{0} and f endowed with the norm {\parallel \cdot \parallel}_{\mathrm{\infty}} are BKspaces (Boos [[7], Example 7.3.2(b)]) and G(u,v) is a triangle matrix. Theorem 4.3.2 of Wilansky [[8], p. 61] gives the fact that f(G) and {f}_{0}(G) are BKspaces with the norm {\parallel \cdot \parallel}_{f(G)}. □
Theorem 3.2 The sequence spaces {f}_{0}(G) and f(G) strictly include the spaces {f}_{0} and f, respectively.
Proof By the definition on the sequence spaces {f}_{0} and f, it is immediate that f\subset f(G) and {f}_{0}\subset {f}_{0}(G).
Now, we should show that these inclusions are strict. We consider the sequence t=({t}_{k}) defined by
The sequence is not almost convergent but Gt is almost convergent to 1/2. This step completes the proof. □
Theorem 3.3 The inclusions c\subset f(G) and f(G)\subset {\ell}_{\mathrm{\infty}} strictly hold.
Proof It is clear that c\subset f(G) and f(G)\subset {\ell}_{\mathrm{\infty}} because of Theorem 3.2 and c\subset f\subset {\ell}_{\mathrm{\infty}}. Further, we show that these inclusions are strict.
Consider the sequence x={G}^{1}(u,v)y with the sequence y in the set {\ell}_{\mathrm{\infty}}\setminus f given by Miller and Orhan [9] as y=(0,\dots ,0,1,\dots ,1,0,\dots ,0,1,\dots ,1,\dots ), where the blocks of 0’s are increasing by factors of 100 and blocks of 1’s are increasing by factors of 10. Then the sequence x is not in f(G) but in the space {\ell}_{\mathrm{\infty}}. This shows that the inclusion f(G)\subset {\ell}_{\mathrm{\infty}} is strict.
Since the inclusion f\subset f(G) strictly holds by Theorem 3.2, by combining this fact with the wellknown strict inclusion c\subset f, one can easily see that the inclusion c\subset f(G) also strictly holds, as was desired. □
It is known from Corollary 3.3 of Başar and Kirişçi [10] that the Banach space f has no Schauder basis. It is also known from Theorem 2.3 of Jarrah and Malkowsky [11] that the domain {\mu}_{A} of a matrix A in a normed sequence space μ has a basis if and only if μ has a basis whenever A=({a}_{nk}) is a triangle. Combining these two facts one can immediately conclude that neither the space {f}_{0}(G) nor the space f(G) have a Schauder basis.
Lemma 3.4 [[12], Theorem 2.1]
Let λ, μ be the BKspaces and {B}_{\mu}^{U}=({b}_{nk}) be defined via the sequence \alpha =({\alpha}_{k})\in \mu and triangle matrix U=({u}_{nk}) by
for all k,n\in \mathbb{N}. Then the inclusion \mu {\lambda}_{U}\subset {\lambda}_{U} holds if and only if the matrix {B}_{\mu}^{U}=U{D}_{\alpha}{U}^{1} is in the classes (\lambda :\lambda ), where {D}_{\alpha} is the diagonal matrix defined by {[{D}_{\alpha}]}_{nn}={\alpha}_{n} for all n\in \mathbb{N}.
Lemma 3.5 [[12], Theorem 3.1]
{B}_{\mu}^{U}=({b}_{nk}) be defined via a sequence a=({a}_{k})\in \omega and inverse of the triangle matrix U=({u}_{nk}) by
for all k,n\in \mathbb{N}. Then
and
From Lemma 3.4 and Lemma 3.5, we may give the theorem determining the β and γduals of the sequence space f(G).
Theorem 3.6 Let u,v\in U and z=({z}_{k})\in \omega. Define the matrix E=({e}_{nk}) by
for all k,n\in \mathbb{N}. Then
and
Proof Consider the equality
where E=({e}_{nk}) is defined by (3.4). We therefore observe by (3.5) that zx=({z}_{k}{x}_{k})\in cs or bs whenever x=({x}_{k})\in f(G) if and only if Ey\in c or {\ell}_{\mathrm{\infty}} whenever y=({y}_{k})\in f. We obtain from Lemma 3.4 and Lemma 3.5 the result that z=({z}_{k})\in {\{f(G)\}}^{\beta} or z=({z}_{k})\in {\{f(G)\}}^{\gamma} if and only if E\in (f:c) or E\in (f:{\ell}_{\mathrm{\infty}}), which is what we wished to prove. □
As a direct consequence of Theorem 3.6, we have the following.
Corollary 3.7 Let u,v\in U for all k\in \mathbb{N}. Then
and
4 Some matrix mappings related to the space f(G)
In this section, we give two theorems characterizing the classes of matrix transformations from the sequence space f(G) into any given sequence space μ and from any sequence space μ into the given sequence space f(G).
We write throughout for brevity
for all k,n\in \mathbb{N}.
Lemma 4.1 Let A=({a}_{nk}) be an infinite matrix. Then the following statements hold:

(i)
(cf. [13]) A\in (f:{\ell}_{\mathrm{\infty}}) if and only if
\underset{n\in \mathbb{N}}{sup}\sum _{k}{a}_{nk}<\mathrm{\infty}.(4.2) 
(ii)
(cf. [13]) A\in (f:c) if and only if (4.2) holds, and there are {\alpha}_{k},\alpha \in \mathbb{C} such that
\underset{n\to \mathrm{\infty}}{lim}{a}_{nk}={\alpha}_{k}\phantom{\rule{1em}{0ex}}\mathit{\text{for each}}k\in \mathbb{N},(4.3)\underset{n\to \mathrm{\infty}}{lim}\sum _{k}{a}_{nk}=0,(4.4)\underset{n\to \mathrm{\infty}}{lim}\sum _{k}\mathrm{\Delta}({a}_{nk}{\alpha}_{k})=\alpha .(4.5) 
(iii)
(cf. [14]) A\in (f:f) if and only if (4.2) holds and
\underset{n\to \mathrm{\infty}}{f\phantom{\rule{0.2em}{0ex}}\text{}\text{lim}\phantom{\rule{0.2em}{0ex}}}{a}_{nk}={\alpha}_{k}\phantom{\rule{1em}{0ex}}\mathit{\text{for each fixed}}k\in \mathbb{N},(4.6)\underset{n\to \mathrm{\infty}}{f\phantom{\rule{0.2em}{0ex}}\text{}\text{lim}\phantom{\rule{0.2em}{0ex}}}\sum _{k}{a}_{nk}=\alpha ,(4.7)\underset{m\to \mathrm{\infty}}{lim}\sum _{k}\left\mathrm{\Delta}[a(n,k,m){\alpha}_{k}]\right=0\phantom{\rule{1em}{0ex}}\mathit{\text{uniformly in}}n.(4.8)
Theorem 4.2 Suppose that the entries of the infinite matrices E=({e}_{nk}) and F=({f}_{nk}) are connected with the relation
for all k,n\in \mathbb{N} and μ be any given sequence space. Then E\in (f(G):\mu ) if and only if {\{{e}_{nk}\}}_{k\in \mathbb{N}}\in {\{f(G)\}}^{\beta} for all n\in \mathbb{N} and F\in (f:\mu ).
Proof Let μ be any given sequence. Suppose that (4.9) holds between the infinite matrices E=({e}_{nk}) and F=({f}_{nk}), and we take into account that the spaces f(G) and f are linearly isomorphic.
Let E\in (f(G):\mu ) and take any y=({y}_{k})\in f. Then FG(u,v) exists and {\{{e}_{nk}\}}_{k\in \mathbb{N}}\in {\{f(G)\}}^{\beta}, which yields the result that (4.9) is necessary and {\{{f}_{nk}\}}_{k\in \mathbb{N}}\in {f}^{\beta} for each n\in \mathbb{N}. Hence, Fy exists for each y\in f and thus by letting m\to \mathrm{\infty} in the equality
we obtain Ex=Fy, which leads to the consequence F\in (f:\mu ).
Conversely, let {\{{e}_{nk}\}}_{k\in \mathbb{N}}\in {\{f(G)\}}^{\beta} for each n\in \mathbb{N} and F\in (f:\mu ), and we take any x=({x}_{k})\in f(G). Then Ex exists. Therefore, we obtain from the equality
as m\to \mathrm{\infty} the result that Fy=Ex and this shows that E\in (f(G):\mu ). This completes the proof. □
By changing the roles of the spaces {f}_{0}(G) and f(G) with μ, we have the following theorem.
Theorem 4.3 Suppose that the entries of the infinite matrices A=({a}_{nk}) and B=({b}_{nk}) are connected with the relation (4.1) and μ be any given sequence space. Then A\in (\mu :f(G)) if and only if B\in (\mu :f).
Proof Let z=({z}_{k})\in \mu and consider the following equality:
Equation (4.10) yields as m\to \mathrm{\infty} the result that {(Bz)}_{n}={\{G(u,v)(Az)\}}_{n}. Therefore, one can immediately observe from this that Az\in f(G) whenever z\in \mu if and only if Bz\in f whenever z\in \mu. This completes the proof. □
It is of course so that Theorem 4.2 and Theorem 4.3 have several consequences depending on the choice of sequence space μ and the sequences u=({u}_{n}) and v=({v}_{k}). Therefore by Theorem 4.2 and Theorem 4.3, necessary and sufficient conditions for (f(G):\mu ) and (\mu :f(G)) may be derived by replacing the entries of E and \tilde{A} by those of the entries of F=E{G}^{1}(u,v) and B=G(u,v)\tilde{A}, respectively, where the necessary and sufficient conditions on the matrices F and B are read from the concerning results in the existing literature.
If we get the space {\ell}_{\mathrm{\infty}} and the spaces {e}_{\mathrm{\infty}}^{r}, {r}_{\mathrm{\infty}}^{t}, and {X}_{\mathrm{\infty}}, which are isomorphic to {\ell}_{\mathrm{\infty}} instead of μ in Theorem 4.2, we obtain the following corollaries.
Corollary 4.4 A\in (f(G):{\ell}_{\mathrm{\infty}}) if and only if {\{{a}_{nk}\}}_{k\in \mathbb{N}}\in {\{f(G)\}}^{\beta} and
Corollary 4.5 Let A=({a}_{nk}) be an infinite matrix and define the matrix C=({c}_{nk}) by
Then the necessary and sufficient conditions in order for A to belong to the class (f(G):{e}_{\mathrm{\infty}}^{r}) are obtained from Theorem 4.2 by replacing the entries of the matrix A by those of the matrix C; here {e}_{\mathrm{\infty}}^{r}=\{x=({x}_{k})\in \omega :{sup}_{n\in \mathbb{N}}{\sum}_{k=0}^{n}\left(\genfrac{}{}{0ex}{}{n}{k}\right){(1r)}^{nk}{r}^{k}{x}_{k}<\mathrm{\infty}\} as defined Altay et al. [15]and Altay and Başar [16].
Corollary 4.6 Let A=({a}_{nk}) be an infinite matrix and define the matrix D=({d}_{nk}) by
where {T}_{n}={\sum}_{k=0}^{n}{t}_{k} for all n\in \mathbb{N}. Then the necessary and sufficient conditions in order for A to belong to the class (f(G):{r}_{\mathrm{\infty}}^{t}) are obtained from in Theorem 4.2 by replacing the entries of the matrix A by those of the matrix D; here {r}_{\mathrm{\infty}}^{t} is the space of all sequences whose {R}^{t}transforms are in the space {\ell}_{\mathrm{\infty}} [17].
Remark 4.7 In the case t=e in the space {r}_{\mathrm{\infty}}^{t}, this space reduces to the Cesàro sequence space of nonabsolute type {X}_{\mathrm{\infty}} [18]. Then Corollary 4.6 also includes the characterization of class (f(G):{X}_{\mathrm{\infty}}), as a special case.
As in Corollaries 4.44.6 and Remark 4.7, the following corollaries are obtained for \mu =\{f,f(E),\stackrel{\u02c6}{f},\tilde{f}\}; here the spaces f(E), \stackrel{\u02c6}{f}, \tilde{f} are isomorphic to the space f.
Corollary 4.8 A\in (f(G):f) if and only if {\{{a}_{nk}\}}_{k\in \mathbb{N}}\in {\{f(G)\}}^{\beta}, (4.11) holds and there are {\alpha}_{k},\alpha \in \mathbb{C} such that
where \tilde{a}(n,k,m)=\frac{1}{n+1}{\sum}_{j=0}^{n}{\tilde{a}}_{m+j,k}.
Corollary 4.9 Let A=({a}_{nk}) be an infinite matrix and the matrix C=({c}_{nk}) be defined by Corollary 4.5. Then the necessary and sufficient conditions in order for A to belong to the class (f(G):f(E)) are obtained from Corollary 4.8 by replacing the entries of the matrix A by those of the matrix C; here
defined by Kirişci [19].
Corollary 4.10 Let A=({a}_{nk}) be an infinite matrix and the matrix H=({h}_{nk}) be defined by {h}_{nk}=s{a}_{n1,k}+r{a}_{nk}. Then the necessary and sufficient conditions in order for A to belong to the class (f(G):\stackrel{\u02c6}{f}) are obtained from Corollary 4.8 by replacing the entries of the matrix A by those of the matrix H; here
defined by Başar and Kirişci [10].
Corollary 4.11 Let A=({a}_{nk}) be an infinite matrix and the matrix M=({m}_{nk}) defined by {m}_{nk}={\sum}_{j=k}^{\mathrm{\infty}}\frac{{a}_{nj}}{j+1}. Then the necessary and sufficient conditions in order for A to belong to the class (f(G):\tilde{f}) are obtained from Corollary 4.8 by replacing the entries of the matrix A by those of the matrix M; here
defined by Kayaduman and Şengönul [20].
Now, we list the following conditions:
Prior to giving some consequences as an application of this idea, we give the following basic lemma, which is the collection of the characterization of matrix transformations related to almost convergence.
Lemma 4.12 Let A=({a}_{nk}) be an infinite matrix. Then,

(i)
A=({a}_{nk})\in (fs:{\ell}_{\mathrm{\infty}}) if and only if (4.17) and (4.18) hold.

(ii)
A=({a}_{nk})\in (fs:c) if and only if (4.3) and (4.17)(4.19) hold [21].

(iii)
A=({a}_{nk})\in (c:f) if and only if (4.6), (4.7), and (4.15) hold [22].

(iv)
A=({a}_{nk})\in ({\ell}_{\mathrm{\infty}}:f) if and only if (4.6), (4.8), and (4.15) hold [14].

(v)
A=({a}_{nk})\in (bs:f) if and only if (4.6), (4.17),(4.18), and (4.21) hold [23].

(vi)
A=({a}_{nk})\in (fs:f) if and only if (4.6), (4.8), (4.18), and (4.24) hold [24].

(vii)
A=({a}_{nk})\in (cs:f) if and only if (4.6) and (4.17) hold [25].

(viii)
A=({a}_{nk})\in (bs:fs) if and only if (4.18) and (4.21)(4.23) hold [23].

(ix)
A=({a}_{nk})\in (fs:fs) if and only if (4.17) and (4.22)(4.24) hold [24].

(x)
A=({a}_{nk})\in (cs:fs) if and only if (4.22) and (4.23) hold [25].

(xi)
A=({a}_{nk})\in (f:cs) if and only if (4.25)(4.28) hold [24].
Now, we can give the following results.
Corollary 4.13 The following statements hold:

(i)
A=({a}_{nk})\in (f(G):c) if and only if {\{{a}_{nk}\}}_{k\in \mathbb{N}}\in {\{f(G)\}}^{\beta} for all n\in \mathbb{N} and (4.2)(4.5) hold with {\tilde{a}}_{nk} instead of {a}_{nk}.

(ii)
A\in (f(G):{c}_{0}) if and only if {\{{a}_{nk}\}}_{k\in \mathbb{N}}\in {\{f(G)\}}^{\beta} for all n\in \mathbb{N} and (4.2) holds, (4.3) and (4.5) hold with {\alpha}_{k}=0 and (4.4) holds with \alpha =0 as {\overline{a}}_{nk} instead of {a}_{nk}.

(iii)
A=({a}_{nk})\in (f(G):bs) if and only if {\{{a}_{nk}\}}_{k\in \mathbb{N}}\in {\{f(G)\}}^{\beta} for all n\in \mathbb{N} and (4.25) holds with {\tilde{a}}_{nk} instead of {a}_{nk}.

(iv)
A=({a}_{nk})\in (f(G):cs) if and only if {\{{a}_{nk}\}}_{k\in \mathbb{N}}\in {\{f(G)\}}^{\beta} for all n\in \mathbb{N} and (4.25)(4.28) hold with {\tilde{a}}_{nk} instead of {a}_{nk}.
Corollary 4.14 We have:

(i)
A=({a}_{nk})\in ({\ell}_{\mathrm{\infty}}:f(G)) if and only if (4.2), (4.6), and (4.20) hold with {b}_{nk} instead of {a}_{nk}.

(ii)
A=({a}_{nk})\in (f:f(G)) if and only if (4.2), (4.6), (4.7), and (4.8) hold with {b}_{nk} instead of {a}_{nk}.

(iii)
A=({a}_{nk})\in (c:f(G)) if and only if (4.2), (4.6), and (4.7) hold with {b}_{nk} instead of {a}_{nk}.

(iv)
A=({a}_{nk})\in (bs:f(G)) if and only if (4.17), (4.18), (4.6), and (4.21) hold with {b}_{nk} instead of {a}_{nk}.

(v)
A=({a}_{nk})\in (fs:f(G)) if and only if (4.18), (4.6), (4.8), and (4.21) hold with {b}_{nk} instead of {a}_{nk}.

(vi)
A=({a}_{nk})\in (cs:f(G)) if and only if (4.17) and (4.6) hold with {b}_{nk} instead of {a}_{nk}.

(vii)
A=({a}_{nk})\in (bs:fs(G)) if and only if (4.18), (4.21), and (4.23) hold with {b}_{nk} instead of {a}_{nk}.

(viii)
A=({a}_{nk})\in (fs:fs(G)) if and only if (4.21)(4.24) hold with {b}_{nk} instead of {a}_{nk}.

(ix)
A=({a}_{nk})\in (cs:fs(G)) if and only if (4.22) and (4.23) hold with {b}_{nk} instead of {a}_{nk}.
Here fs(G) denotes the domain of the G(u,v)generalized weighted mean in the sequence space fs.
5 Conclusion
As an essential work on the algebraic and topological properties of the spaces {f}_{0} and f, Başar and Kirişçi [10] have recently introduced the sequence spaces {\stackrel{\u02c6}{f}}_{0} and \stackrel{\u02c6}{f} derived by the domain of the generalized difference matrix B(r,s) in the sequence spaces {f}_{0} and f, respectively. Following Başar and Kirişçi [10], Kayaduman and Şengönül have studied the domain {\tilde{f}}_{0} and \tilde{f} of the Cesàro mean of order one in the spaces {f}_{0} and f, in [20]. They have determined the β and γduals of the new spaces {\tilde{f}}_{0} and \tilde{f}, and they characterize some classes of matrix transformations on/in the new sequence spaces. They complete the paper by a nice section including some core theorems related to the matrix classes on/in the new sequence space \tilde{f}. Quite recently, in [26], Sönmez has introduced the domain f(B) of the triple band matrix B(r,s,t) in the sequence space f. In this paper, the β and γduals of the space f(B) are determined. Furthermore, the classes (f(B):\mu ) and (\mu :f(B)) of infinite matrices are characterized together with some other classes, where μ is any given sequence space. Finally, in [27] Candan has studied the sequence spaces {f}_{0}(\tilde{B}) and f(\tilde{B}) as the domain of the double sequential band matrix \tilde{B}(\tilde{r},\tilde{s}) in the sequence spaces {f}_{0} and f.
Since Kirişçi and Başar [28], Başar and Kirişçi [10], Kayaduman and Şengönül [20], Sönmez [26, 29], and Candan [27, 30] are recent works on the domain of certain triangle matrices in the spaces {f}_{0}, f, and in the classical sequence spaces, the present paper is their natural continuation. Also these spaces are special cases of the notion of Aalmost convergence and {\mathcal{F}}_{B}convergence ([31, 32]) as well as analogous to the definition introduced in [33].
Finally, we should note that the investigation of the domain of some particular limitation matrices, namely Cesàro means of order m, Nörlund means, etc., in the spaces {f}_{0} and f will lead to new results which are not comparable with the present results.
Author’s contributions
MK defined the new almost sequence spaces derived by generalized weighted mean and studied some properties. MK computed the duals of new spaces and characterized the matrix classes. In last section, it was summarized to studies in manuscripts and given some open problems by MK. The author read and approved the final manuscript.
Article’s information
Some of the results of this study presented in First International Conference on Analysis and Applied Mathematics (ICAAM 2012, Gumushane University, Turkey) [34].
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Acknowledgements
I would like to express thanks to Professor Feyzi Başar, Department of Mathematics, Fatih University, Buyukcekmece, Istanbul34500, Turkey, for his valuable help on some results and the useful comments which improved the presentation paper. I have benefited much from the constructive reports of the anonymous referees, and I am grateful for their valuable comments on the first draft on this paper, which improved the presentation and readability. This work was supported by Scientific Projects Coordination Unit of Istanbul University. Project number 27008.
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Kirişci, M. Almost convergence and generalized weighted mean II. J Inequal Appl 2014, 93 (2014). https://doi.org/10.1186/1029242X201493
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DOI: https://doi.org/10.1186/1029242X201493
Keywords
 almost convergence
 matrix domain
 sequence spaces
 generalized weighted mean
 β and γduals
 matrix transformations