1 Introduction and preliminaries
Let
ω denote the space of all complex sequences
\(s=(s_{j})_{j=0}^{\infty}\) (or simply write
\(s=(s_{j})\)). Any vector subspace of
ω is called a sequence space. By
\(\mathbb {N}\) we denote the set of natural numbers, and by
\(\mathbb {R}\) the set of real numbers. We use the standard notation
\(\ell_{\infty}\),
c and
\(c_{0}\) to denote the sets of all bounded, convergent and null sequences of real numbers, respectively, where each of the sets is a Banach space with the sup-norm
\(\Vert . \Vert _{\infty}\) defined by
\(\Vert s \Vert _{\infty}=\sup_{j\in\mathbb {N}} \vert s_{j} \vert \). We write the space
\(\ell_{p}\) of all absolutely
p-summable series by
$$\ell_{p}= \Biggl\{ s\in\omega:\sum_{j=0}^{\infty} \vert s_{j} \vert ^{p}< \infty\ (1\leq p< \infty) \Biggr\} . $$
Clearly,
\(\ell_{p}\) is a Banach space with the following norm:
$$\Vert s \Vert _{p}= \Biggl(\sum_{j=0}^{\infty} \vert s_{j} \vert ^{p} \Biggr)^{1/p}. $$
For
\(p=1\), we obtain the set
\(l_{1}\) of all absolutely summable sequences. For any sequence
\(s=(s_{j})\), let
\(s^{[n]}=\sum_{j=0}^{n}s_{j}e_{j}\) be its
n-section, where
\(e_{j}\) is the sequence with 1 in place
j and 0 elsewhere and
\(e=(1,1,1,\dots)\).
A sequence space X is called a BK-space if it is a Banach space with continuous coordinates \(p_{j}:X\to\mathbb {C}\), the set of complex fields, and \(p_{j}(s)=s_{j}\) for all \(s=(s_{j})\in X\) and every \(j\in \mathbb {N}\). A BK-space \(X\supset\psi\), the set of all finite sequences that terminate in zeros, is said to have AK if every sequence \(s=(s_{j})\in X\) has a unique representation \(s=\sum_{j=0}^{\infty }s_{j}e_{j}\).
Let
X and
Y be two sequence spaces, and let
\(A=(a_{n,k})\) be an infinite matrix. If, for each
\(s=(s_{k})\) in
X, the series
$$ A_{n}s=\sum_{k}a_{n,k}s_{k}= \sum_{k=0}^{\infty}a_{n,k}s_{k} $$
(1)
converges for each
\(n\in\mathbb {N}\) and the sequence
\(As=(A_{n}s)\) belongs to
Y, then we say that matrix
A maps
X into
Y. By the symbol
\((X,Y)\) we denote the set of all such matrices which map
X into
Y. The series in (
1) is called
A-
transform of
s whenever the series converges for
\(n=0,1,\ldots \) . We say that
\(s=(s_{k})\) is
A-
summable to the limit
λ if
\(A_{n}s\) converges to
λ (
\(n\to\infty\)).
The sequence
\(s=(s_{k})\) of
\(\ell_{\infty}\) is said to be almost convergent, denoted by
f, if all of its Banach limits [
1] are equal. We denote such a class by the symbol
f, and one writes
\(f\mbox{-}\lim s =\lambda\) if
λ is the common value of all Banach limits of the sequence
\(s=(s_{k})\). For a bounded sequence
\(s=(s_{k})\), Lorentz [
2] proved that
\(f\mbox{-}\lim s =\lambda \) if and only if
$$\lim_{k\to\infty}\frac{s_{m}+s_{m+1}+\cdots+s_{m+k}}{k+1}=\lambda $$
uniformly in
m. This notion was later used to (i) define and study conservative and regular matrices [
3]; (ii) introduce related sequence spaces derived by the domain of matrices [
4‐
6]; (iii) study some related matrix transformations [
7‐
9]; (iv) define related sequence spaces derived as the domain of the generalized weighted mean and determine duals of these spaces [
10,
11]. As an extension of the notion of almost convergence, Kayaduman and Şengönül [
12,
13] defined Cesàro and Riesz almost convergence and established related core theorems. The almost strongly regular matrices for single sequences were introduced and characterized [
14], and for double sequences they were studied by Mursaleen [
15] (also refer to [
16‐
19]). As an application of almost convergence, Mohiuddine [
20] proved a Korovkin-type approximation theorem for a sequence of linear positive operators and also obtained some of its generalizations. Başar and Kirişçi [
21] determined the duals of the sequence space
f and other related spaces/series and investigated some useful characterizations.
We now recall the following result.
2 Weighted almost convergence
We prove the following characterization of weighted almost conservative matrices.
In the following theorem, we obtain the characterization of weighted almost regular matrices.
We now obtain necessary and sufficient conditions for the matrix A which transform the absolutely convergent series into the space of weighted almost convergence.
Acknowledgements
This work was supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant No. (130-694-D1435). The authors, therefore, gratefully acknowledge the DSR technical and financial support.
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