$\lambda$-Almost summable spaces
UDC 517.5 In this paper, we investigate some new sequence spaces which arise from the notation of generalized de la Vallée-Poussin means and introduce the spaces of strongly $\lambda$-almost summable sequences. We also consider some topological results, characterization of strongly $\lambda$-almost...
Gespeichert in:
| Datum: | 2020 |
|---|---|
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2020
|
| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/396 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507018696065024 |
|---|---|
| author | Savaş , E. Savaş , EKREM Savaş , E. |
| author_facet | Savaş , E. Savaş , EKREM Savaş , E. |
| author_sort | Savaş , E. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2025-03-31T08:49:43Z |
| description | UDC 517.5
In this paper, we investigate some new sequence spaces which arise from the notation of generalized de la Vallée-Poussin means and introduce the spaces of strongly $\lambda$-almost summable sequences. We also consider some topological results, characterization of strongly $\lambda$-almost regular matrices. |
| doi_str_mv | 10.37863/umzh.v72i10.396 |
| first_indexed | 2026-03-24T02:02:39Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v72i10.396
UDC 517.5
E. Savaş (Uşak Univ., Turkey)
\bfitlambda -ALMOST SUMMABLE SPACES
\bfitlambda -МАЙЖЕ ПIДСУМОВАНI ПРОСТОРИ
In this paper, we investigate some new sequence spaces which arise from the notation of generalized de la Vallée-Poussin
means and introduce the spaces of strongly \lambda -almost summable sequences. We also consider some topological results,
characterization of strongly \lambda -almost regular matrices.
Вивчаються деякi новi простори послiдовностей, що випливають iз позначення узагальнених середнiх Валле Пуссе-
на та продукують простори сильно \lambda -майже пiдсумованих послiдовностей. Також розглядаються деякi топологiчнi
результати та характеризацiя сильно \lambda -майже регулярних матриць.
1. Introduction. Let w denote the set of all complex sequences x = (xk). By l\infty and c, we denote
the Banach spaces of bounded and convergent sequences x = (xk) of w normed by \| x\| = \mathrm{s}\mathrm{u}\mathrm{p}k | xk| ,
respectively. A linear functional L on l\infty is said to be a Banach limit [2] if it has the following
properties:
L(x) \geq 0 if x \geq 0 (i.e., xn \geq 0 for all n),
L(e) = 1 where e = (1, 1, . . .),
L(Dx) = L(x), where D denotes the sift operator on \ell \infty , that is D : \ell \infty \rightarrow \ell \infty defined by
D(x) = D(xn) = \{ xn+1\} .
Let B be the set of all Banach limits on l\infty . A sequence x \in \ell \infty is said to be almost convergent
if all Banach limits of x coincide. Let \^c denote the space of the almost convergent sequences.
It is easy to verify that if x is a convergent sequence, then L(x) = \mathrm{l}\mathrm{i}\mathrm{m}n xn for any Banach
limits L. In the other words, L(x)takes the same value for any Banach limits L. It is notable that
this condition is meaningful not only for convergent sequences, but also for a certain type of bounded
sequences. Lorentz [7] proved that
\^c =
\Biggl\{
x : \mathrm{l}\mathrm{i}\mathrm{m}
m\rightarrow \infty
1
m+ 1
m\sum
i=0
xn+i exists uniformly in n
\Biggr\}
.
Almost convergent sequences were studied by Lorentz [8], King [7], Duran [4], Nanda [12],
Savas [13 – 15] and others.
The strongly summable sequences have been systematically investigated by Hamilton and Hill [5],
Kuttner [6] and some others. The spaces of strongly summable sequences were introduced and stu-
died by Maddox [9, 11].
The goal of this paper is to study the spaces of strongly \lambda -almost summable sequences, which
naturally come up for investigation and which will fill up a gap in the existing literature.
Let \lambda = (\lambda n) be a nondecreasing sequence of positive numbers tending to \infty such that
\lambda n+1 \leq \lambda n + 1, \lambda 1 = 1.
The generalized de la Vallèe-Poussin mean is defined by
c\bigcirc E. SAVAŞ, 2020
1410 ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 10
\lambda -ALMOST SUMMABLE SPACES 1411
tn(x) =
1
\lambda n
\sum
k\in In
xk,
where In = [n - \lambda n + 1, n]. A sequence x = (xk) is said to be (V, \lambda )-summable to a number L, if
tn(x) \rightarrow L as n \rightarrow \infty .
Let A = (ank) be an infinite matrix of nonnegative real numbers and p = (pk) be a sequence
such that pk > 0. (These assumptions are made throughout.) We write Ax =
\bigl\{
An(x)
\bigr\}
if An(x) =
=
\sum
k
ank\| xk\| pk converges for each n. We write
dmn(A\lambda x) =
1
\lambda m
\sum
i\in Im
An+i(x) =
\sum
k
a(n, k,m)| xk| pk ,
where
a(n, k,m) =
1
\lambda m
\sum
i\in Im
an+i,k.
If we take \lambda m = m,m = 1, 2, 3, . . . , the above reduces to
tmn(Ax) =
1
m+ 1
m\sum
i=0
An+i(x) =
\sum
k
a(n, k,m)| xk| pk ,
where
a(n, k,m) =
1
m+ 1
m\sum
i=0
an+i,k.
We now write \bigl[
\^A\lambda , p
\bigr]
0
=
\bigl\{
x : dmn(A\lambda x) \rightarrow 0 uniformly in n
\bigr\}
,\bigl[
\^A\lambda , p
\bigr]
=
\bigl\{
x : dmn(A\lambda x - l) \rightarrow 0 for some l uniformly in n
\bigr\}
and \bigl[
\^A\lambda , p
\bigr]
\infty =
\Bigl\{
x : \mathrm{s}\mathrm{u}\mathrm{p}
mn
dmn(A\lambda x) < \infty
\Bigr\}
.
The sets
\bigl[
\^A\lambda , p
\bigr]
0
,
\bigl[
\^A\lambda , p
\bigr]
and
\bigl[
\^A\lambda , p
\bigr]
\infty will be respectively called the spaces of strongly \lambda -almost
summable to zero, strongly \lambda -summable and strongly \lambda -bounded sequences.
If x is strongly \lambda -almost summable to l we write xk \rightarrow l
\bigl[
\^A\lambda , p
\bigr]
. A pair (A, p) will be called
strongly \lambda -almost regular if
xk \rightarrow l \Rightarrow xk \rightarrow l
\bigl[
\^A\lambda , p
\bigr]
.
2. Main results. In this section, we give few propositions which are useful in the sequel of this
paper.
Proposition 2.1. If p \in \ell \infty , then
\bigl[
\^A\lambda , p
\bigr]
0
,
\bigl[
\^A\lambda , p
\bigr]
and
\bigl[
\^A\lambda , p
\bigr]
\infty are linear spaces over \BbbC .
Proof. it is easy to prove, so we omit the detail.
We have the following proposition.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 10
1412 E. SAVAŞ
Proposition 2.2.
\bigl[
\^A\lambda , p
\bigr]
\subset
\bigl[
\^A\lambda , p
\bigr]
\infty , if
\| A\| = \mathrm{s}\mathrm{u}\mathrm{p}
m
\sum
k
a(n, k,m) < \infty . (2.1)
Proof. Assume that xk \rightarrow l
\bigl[
\^A\lambda , p
\bigr]
and (2.1) holds. Now we write
dmn(A\lambda x) = dmn(A\lambda x - l + l) \leq
\leq Kdmn(A\lambda x - l) +K
\sum
k
a(n, k,m)| l| pk \leq
\leq Kdmn(A\lambda x - l) +K
\bigl(
\mathrm{s}\mathrm{u}\mathrm{p} | l| pk
\bigr) \sum
k
a(n, k,m).
Therefore, x \in
\bigl[
\^A\lambda , p
\bigr]
\infty and this completes the proof.
Proposition 2.3. Let p \in \ell \infty , then
\bigl[
\^A\lambda , p
\bigr]
0
and
\bigl[
\^A\lambda , p
\bigr]
\infty (\mathrm{i}\mathrm{n}\mathrm{f} pk > 0) are linear topological
spaces paranormed by g (see [11]) defined by
g(x) = \mathrm{s}\mathrm{u}\mathrm{p}
m,n
\bigl[
dm,n(A\lambda x)
\bigr] 1/M
,
where M = \mathrm{m}\mathrm{a}\mathrm{x}(1, H = \mathrm{s}\mathrm{u}\mathrm{p} pk). If (2.1) holds, then
\bigl[
\^A\lambda , p
\bigr]
has the same paranorm.
Proof. Obviously g(0) = 0 and g(x) = g( - x). Since M \geq 1, by Minkowski’s inequality it
follows that g is subadditive. We now show that the scalar multiplication is continuous. It follows
that
g(\alpha x) \leq \mathrm{s}\mathrm{u}\mathrm{p} | \alpha | pk/Mg(x).
Therefore x \rightarrow 0 \Rightarrow \alpha x \rightarrow 0 (for fixed \alpha ). Now let \alpha \rightarrow 0 and x be fixed. Fot given \varepsilon > 0 there
exists N such that
dm,n(A\lambda \alpha x) < \varepsilon /2 (\forall n \forall m > N). (2.2)
Since dm,n(A\lambda x) exists for all m, we write
dm,n(A\lambda x) = K(m), 1 \leq m \leq N,
and
\delta =
\biggl(
\varepsilon
2K(m)
\biggr) 1/pk
.
Then | \alpha | < \delta ,
dm,n(A\lambda \alpha x) <
\varepsilon
2
(\forall n, 1 \leq m \leq N). (2.3)
It follows from (2.2) and (2.3) that
\alpha \rightarrow 0 \Rightarrow \alpha x \rightarrow 0 (x fixed).
This proves the assertion about
\bigl[
\^A\lambda , p
\bigr]
0
. If \mathrm{i}\mathrm{n}\mathrm{f} pk = \theta > 0 and 0 < | \alpha | < 1, then
gM (\alpha x) \leq | \alpha | \theta gM (x) \forall x \in [ \^A\lambda , p]\infty .
Therefore
\bigl[
\^A\lambda , p
\bigr]
\infty has the paranorm g. If (2.1) holds it is clear from Proposition 2.2 that g(x)
exists for each x \in
\bigl[
\^A\lambda , p
\bigr]
.
Proposition 2.3 is proved.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 10
\lambda -ALMOST SUMMABLE SPACES 1413
Remark 2.1. It is evident that g is not a norm in general. But if pk = p for all k, then clearly g
is a norm for 1 \leq p \leq \infty and a p-norm for 0 < p < 1.
Proposition 2.4.
\bigl[
\^A\lambda , p
\bigr]
0
and
\bigl[
\^A\lambda , p
\bigr]
\infty are complete with respect to their paranorm topologies\bigl[
\^A\lambda , p
\bigr]
is complete if (2.1) holds and\sum
k
a(n, k,m) \rightarrow 0 uniformly in n. (2.4)
Proof. Omitted.
Combining the above facts, we obtain the main result.
Theorem 2.1. Let p \in \ell \infty . Then
\bigl[
\^A\lambda , p
\bigr]
0
and
\bigl[
\^A\lambda , p
\bigr]
\infty (\mathrm{i}\mathrm{n}\mathrm{f} pk > 0) are complete linear
topological spaces paranormed by g. If (2.1) and (2.4) hold, then
\bigl[
\^A\lambda , p
\bigr]
has the same property.
If further pk = p for all k, they are Banach spaces for 1 \leq p < \infty and p-normed spaces for
0 < p < 1.
3. Topological results. We now study locally boundedness and r-convexity for the spaces of
strongly almost summable sequences. For 0 < r \leq 1 a non-void subset W of a linear space is
said to be absolutely r-convex if x, y \in W and | \gamma | r + | \mu | r \leq 1 together imply that \gamma x + \mu y \in W.
It is obvious that if W is absolutely r-convex, then it is absolutely t-convex for t < r. A linear
topological space E is said to be r-convex if every neighbourhood of 0 \in E contains an absolutely
r-convex neighbourhood of 0 \in E. The r-convexity for r > 1 is of little interest, since E is r-
convex for r > 1 if and only if E is the only neighbourhood of 0 \in E (see [10]). A subset B of
E is said to be bounded if for each neighbourhood W of 0 \in E there exists an integer N > 1 such
that B \subseteq NW. E is called locally bounded if there is a bounded neighbourhood of zero.
We first prove the following theorem.
Theorem 3.1. Let 0 < pk \leq 1. Then [ \^A\lambda , p]0 and [ \^A\lambda , p]\infty are locally bounded if \mathrm{i}\mathrm{n}\mathrm{f} pk > 0.
If (2.1) holds, then [ \^A\lambda , p] has the same property.
Proof. We shall only prove for [ \^A\lambda , p]\infty . Let \mathrm{i}\mathrm{n}\mathrm{f} pk = \theta > 0. If x \in
\bigl[
\^A\lambda , p
\bigr]
\infty , then there exists
a constant K \prime > 0 such that \sum
k
a(n, k,m)| xk| pk \leq K \prime (\forall m,n).
For this K \prime and given \delta > 0 choose an integer N > 1 such that
N \theta \geq K \prime
\delta
.
Since
1
N
< 1 and pk \leq \theta we write
1
Npk
\leq 1
N \theta
(\forall k).
Then, for all m and n, we get\sum
k
a(n, k,m)
\bigm| \bigm| \bigm| xk
N
\bigm| \bigm| \bigm| pk \leq 1
N \theta
\sum
k
a(n, k,m)| xk| pk \leq K \prime
N\theta
\leq \delta .
Therefore, by taking supremum over m and n, we get
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 10
1414 E. SAVAŞ\bigl\{
x : g(x) \leq K \prime \bigr\} \subseteq N
\bigl\{
x : g(x) \leq \delta
\bigr\}
.
For every \delta > 0 exists \bfN > 1, for which the above inclusion holds, and so\bigl\{
x : g(x) \leq K \prime \bigr\}
is bounded.
Theorem 3.1 is proved.
It is known that every locally bounded linear topological space is r-convex for some r such that
0 < r \leq 1. But the following theorem gives exact conditions for r-convexity.
Theorem 3.2. Let 0 < pk \leq 1. Then
\bigl[
\^A\lambda , p
\bigr]
0
and
\bigl[
\^A\lambda , p
\bigr]
\infty are r-convex for all r where
0 < r < \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f} pk. Moreover, if pk = p \leq 1 \forall k, then they are p-convex.
\bigl[
\^A\lambda , p
\bigr]
has the same
properties if (2.1) holds.
Proof. We prove the theorem only for
\bigl[
\^A\lambda , p
\bigr]
\infty . Let
\bigl[
\^A\lambda , p
\bigr]
\infty and r \in
\bigl(
0, \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f} pk
\bigr)
. Then
exists k0 such that r \leq pk (\forall k > k0). Now define
\^g(x) = \mathrm{s}\mathrm{u}\mathrm{p}
m,n
\left[ k0\sum
k=1
a(n, k,m)| xk| r +
\infty \sum
k=k0+1
a(n, k,m)| xk| pk
\right] .
Since r \leq pk \leq 1 (\forall k > k0), \^g is subadditive. Further, for 0 < | \gamma | \leq 1,
| \gamma | pk \leq | \gamma | r (\forall k > k0).
Therefore, for such \gamma , we have
\^g(\gamma x) \leq | \gamma | r\^g(x).
Now, for 0 < \delta < 1,
U =
\bigl\{
x : \^g(x) \leq \delta
\bigr\}
is an absolutely r-convex set, for | \gamma | r + | \mu | r \leq 1 and x, y \in W imply that
\^g(\gamma x+ \mu y) \leq \^g(\gamma x) + \^g(\mu y) \leq | \gamma | r\^g(x) + | \mu | r\^g(y) \leq
\bigl(
| \gamma | r + | \mu | r
\bigr)
\delta \leq \delta .
If pk = p (\forall k), then, for 0 < \delta < 1,
U =
\bigl\{
x : g(x) \leq \delta
\bigr\}
is an absolutely p-convex set. This can be obtained by a similar analysis and therefore we omit the
details.
Theorem 3.2 is proved.
4. Some further results. Let E and F be two nonempty subsets of the space w of sequences.
If x = \{ xk\} \in E implies that
\Bigl\{ \sum
k
ankxk
\Bigr\}
\in F, we say that A defines a (matrix) transformation
from E into F, and we write A : E \rightarrow F. (E,F.) denotes the class of matrices A such that A :
E \rightarrow F.
Let c0 and (V, \lambda )0 respectively denote the linear spaces of null sequences and sequences \lambda -almost
convergent to zero.
We now characterize the class of strongly \lambda -almost regular matrices.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 10
\lambda -ALMOST SUMMABLE SPACES 1415
Theorem 4.1. Let 0 < \theta \leq pk \leq H < \infty . Then (A, p) is strongly \lambda -almost regular if and only
if A \in
\bigl(
c0, ( \^V , \lambda )0
\bigr)
, where
( \^V , \lambda )0 =
\Biggl\{
x : \mathrm{l}\mathrm{i}\mathrm{m}
m\rightarrow \infty
1
\lambda m
\sum
i\in In
xn+i = 0 uniformly in n
\Biggr\}
.
It is known that (see [1]) A \in (c0, ( \^V , \lambda )0) if and only if
\| A\| < \infty ;
\mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty a(n, k,m) = 0 uniformly in n (\forall k).
To prove Theorem 4.1 we need the following result.
Lemma 4.1 [9, p. 347]. If pk, qk > 0, then c0(q) \subset c0(p) \leftrightarrow \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
pk
qk
> 0.
Proof of Theorem 4.1. Necessity. Suppose that (A, p) is strongly \lambda -almost regular. Therefore
| xk - l| 1/pk \rightarrow 0 \Rightarrow
\sum
k
a(n, k,m)| xk - l| \rightarrow 0
uniformly in n. Since
1
pk
\geq 1
H
> 0, by Lemma 4.1,
xk \rightarrow l \Rightarrow | xk - l| 1/pk \rightarrow 0.
Thus
xk \rightarrow l \Rightarrow
\sum
k
a(n, k,m)(xk - l) \rightarrow 0
uniformly in n and, therefore, A \in
\bigl(
c0, ( \^V , \lambda )0
\bigr)
.
Sufficiency. Since pk \geq \theta > 0, by Lemma 4.1,
xk \rightarrow l \Rightarrow | xk - l| pk \rightarrow 0.
Again we have A \in
\bigl(
c0, ( \^V , \lambda )0
\bigr)
. Therefore xk \rightarrow l
\bigl[
\^A\lambda , p
\bigr]
and this concludes the proof. Note that
pk \leq H superfluous in the sufficiency and \theta \leq pk is superfluous in the necessity.
Theorem 4.1 is proved.
We next consider the uniqueness of generalized limits.
Theorem 4.2. Suppose that A \in
\bigl(
c0, ( \^V , \lambda )0
\bigr)
and p = \{ pk\} converges to a positive limit. Then
x = \{ xk\} \rightarrow l \Rightarrow xk \rightarrow l[ \^A\lambda , p] uniquely if and only\sum
k
a(n, k,m) \nrightarrow 0 uniformly in n. (4.1)
Proof. Necessity. Suppose that A \in
\bigl(
c0, ( \^V , \lambda )0
\bigr)
and \{ pk\} be bounded. Let xk \rightarrow l imply
that xk \rightarrow l
\bigl[
\^A\lambda , p
\bigr]
uniquely. We have e \rightarrow 1
\bigl[
\^A\lambda , p
\bigr]
. Therefore the condition (4.1) must hold. For
otherwise e \rightarrow 0[ \^A, p] which contradicts the uniqueness of l.
Note that the restriction on \{ pk\} (except boundedness) is superfluous for the necessity.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 10
1416 E. SAVAŞ
Sufficiency. Suppose that the condition (4.1) holds and A \in (c0, ( \^V , \lambda )0) and that pk \rightarrow r > 0.
Further assume that xk \rightarrow l imply that xk \rightarrow l
\bigl[
\^A\lambda , p
\bigr]
and xk \rightarrow l\'[ \^A, p] where
\bigm| \bigm| l - l\'
\bigm| \bigm| = a > 0.
Then we get
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\sum
k
a(n, k,m)uk = 0 (uniformly in n), (4.2)
where
uk = | xk - l| pk +
\bigm| \bigm| xk - l\'
\bigm| \bigm| pk .
By the assumption we have uk \rightarrow ar. Since A \in
\bigl(
c0, ( \^V , \lambda )0
\bigr)
, uk \rightarrow ar implies that\sum
k
a(n, k,m)
\bigm| \bigm| uk - ar
\bigm| \bigm| \rightarrow 0 (uniformly in n). (4.3)
But we have
ar
\sum
k
(n, k,m) \leq
\sum
k
a(n, k,m)uk +
\sum
k
a(n, k,m)
\bigm| \bigm| uk - ar
\bigm| \bigm| . (4.4)
Now by (4.2), (4.3) and (4.4) it follows that
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\sum
k
a(n, k,m) = 0 (uniformly in n).
Since this contradicts (4.1), we must have l = l\'.
Theorem 4.2 is proved.
Suppose that 0 < pk \leq qk. We conclude this note by showing that
\bigl[
\^A\lambda , q
\bigr]
\subset
\bigl[
\^A\lambda , p
\bigr]
is not true
in general. However the inclusion holds for a special class. We prove the following theorem.
Theorem 4.3. Suppose that \| A\| < \infty and
qk
pk
is bounded, then
\bigl[
\^A\lambda , q
\bigr]
\subset
\bigl[
A\lambda , p
\bigr]
.
Proof. Write wk = | xk - l| qk and pk/qk = \gamma k. So that 0 < \gamma \leq \gamma k \leq 1 (\gamma is constant). Let
x \in
\bigl[
\^A\lambda , q
\bigr]
. Then \sum
k
a(n, k,m)wk \rightarrow 0 (uniformly in n).
Define uk = wk (wk \geq 1) = 0 (wk < 1) and \upsilon k = 0 (wk \geq 1) = wk (wk < 1). So that
wk = uk + \upsilon k, w
\gamma k
k = u\gamma kk + \upsilon \gamma kk . Hence it follows that u\gamma kk \leq uk \leq wk, \upsilon
\gamma k
k < \upsilon \gamma k . We have the
inequality
\sum
k
a(n, k,m)w\gamma k
k \leq
\sum
k
a(n, k,m)wk +
\Biggl( \sum
k
a(n, k,m)\upsilon k
\Biggr) \gamma
\| A\| 1 - \gamma .
Hence, x \in
\bigl[
\^A\lambda , p
\bigr]
and this completes the proof.
References
1. S. Bala, A study on characterization of some classes of matrix transformations between some sequence spaces, Ph.
D. Thesis, A. M. U. Aligarh (1994).
2. S. Banach, Théorie des opérations linéaires, Warsaw (1932).
3. G. Das, B. Kuttner, S. Nanda, Some sequence spaces and absolute almost convergence, Trans. Amer. Math. Soc.,
283, № 2, 729 – 739 (1984).
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 10
\lambda -ALMOST SUMMABLE SPACES 1417
4. J. P. Duran, Infinite matrices and almost convergence Math. Z., 128, 75 – 83 (1972).
5. H. J. Hamilton, J. D. Hill, On strong summability, Amer. J. Math., 60, 588 – 594 (1938).
6. B. Kuttner, Note on strong summability, J. London Math. Soc., 21, 118 – 222 (1946).
7. J. P. King, Almost summable sequences, Proc. Amer. Math. Soc., 17, 1219 – 1225 (1966).
8. G. G. Lorentz, A contribution to the theory of divergent sequences, Acta Math., 80, 167 – 190 (1948).
9. I. J. Maddox, Spaces of strongly summable sequences, Quart. J. Math. Oxford Ser. (2), 18, 345 – 355 (1967).
10. I. J. Maddox, J. W. Roles, Absolute convexity in certain topological linear spaces, Proc. Cambridge Phil. Soc., 66,
541 – 545 (1969).
11. I. J. Maddox, Elements of functional analysis, Cambridge Univ. Press (1970).
12. S. Nanda, Some sequence spaces and almost convergence, J. Austral. Math. Soc. Ser. A, 22, 446 – 455 (1976).
13. E. Savaş, Matrix transformations and absolute almost convergence, Bull. Inst. Math. Acad. Sin., 15, № 3 (1987).
14. E. Savaş, Almost convergence and almost summability, Tamkang J. Math., 21, № 4 (1990).
15. E. Savaş, Some sequence spaces and almost convergence, Ann. Univ. Timiora, 30, № 2-3 (1992).
16. P. Schaefer, Almost convergent and almost summable sequences, Proc. Amer. Math. Soc., 20, 51 – 54 (1969).
Received 08.10.18
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 10
|
| id | umjimathkievua-article-396 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:02:39Z |
| publishDate | 2020 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/94/1d8a23ad51cc320bf1fa43c355a6c394.pdf |
| spelling | umjimathkievua-article-3962025-03-31T08:49:43Z $\lambda$-Almost summable spaces almost $\lambda$ -Almost summable spaces Savaş , E. Savaş , EKREM Savaş , E. UDC 517.5 In this paper, we investigate some new sequence spaces which arise from the notation of generalized de la Vallée-Poussin means and introduce the spaces of strongly $\lambda$-almost summable sequences. We also consider some topological results, characterization of strongly $\lambda$-almost regular matrices. УДК 517.5 $\lambda$ -майже пiдсумованi простори Вивчаються деякi новi простори послiдовностей, що випливають iз позначення узагальнених середнiх Валле Пуссена та продукують простори сильно $\lambda$ -майже пiдсумованих послiдовностей. Також розглядаються деякi топологiчнi результати та характеризацiя сильно $\lambda$ -майже регулярних матриць. Institute of Mathematics, NAS of Ukraine 2020-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/396 10.37863/umzh.v72i10.396 Ukrains’kyi Matematychnyi Zhurnal; Vol. 72 No. 10 (2020); 1410 - 1417 Український математичний журнал; Том 72 № 10 (2020); 1410 - 1417 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/396/8763 Copyright (c) 2020 EKREM SAVAS |
| spellingShingle | Savaş , E. Savaş , EKREM Savaş , E. $\lambda$-Almost summable spaces |
| title | $\lambda$-Almost summable spaces |
| title_alt | almost $\lambda$ -Almost summable spaces |
| title_full | $\lambda$-Almost summable spaces |
| title_fullStr | $\lambda$-Almost summable spaces |
| title_full_unstemmed | $\lambda$-Almost summable spaces |
| title_short | $\lambda$-Almost summable spaces |
| title_sort | $\lambda$-almost summable spaces |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/396 |
| work_keys_str_mv | AT savase lambdaalmostsummablespaces AT savasekrem lambdaalmostsummablespaces AT savase lambdaalmostsummablespaces AT savase almost AT savasekrem almost AT savase almost |