Strongly statistical convergence
UDC 519.21 We introduce $A$-strongly statistical convergence for sequences of complex numbers, where $A=\left(a_{nk}\right)_{n,k\in \mathbb{N}}$ is an infinite matrix with nonnegative entries. A sequence $\left(x_{n}\right)$ is called strongly convergent to $L$ if $\displaystyle{\lim\nolimits_{n\to\...
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/2368 |
| 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_ | 1860508267088707584 |
|---|---|
| author | Kaya, U. Aral , N. D. Kaya, U. Kaya, U. Aral , N. D. |
| author_facet | Kaya, U. Aral , N. D. Kaya, U. Kaya, U. Aral , N. D. |
| author_sort | Kaya, U. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-04-07T12:13:59Z |
| description | UDC 519.21
We introduce $A$-strongly statistical convergence for sequences of complex numbers, where $A=\left(a_{nk}\right)_{n,k\in \mathbb{N}}$ is an infinite matrix with nonnegative entries. A sequence $\left(x_{n}\right)$ is called strongly convergent to $L$ if $\displaystyle{\lim\nolimits_{n\to\infty} \sum\nolimits_{k=1}^{\infty}a_{nk}\left|x_{k}-L\right|=0}$ in the ordinary sense. In the definition of $A$-strongly statistical limit, we use the statistical limit instead of the ordinary limit via a convenient density. We study some densities and show that the $\left(a_{nk}\right)$-strongly statistical limit is a $\left(a_{m_{n}k}\right)$-strong limit, where the density of the set $\left\{m_{n}\in\mathbb{N}\colon n\in\mathbb{N}\right\}$ is equal to 1. We introduce the notion of dense positivity for nonnegative sequences. A nonnegative sequence $\left(r_{n}\right)$ is dense positive provided the limit superior of a subsequence $\left(r_{m_{n}}\right)$ is positive for all $\left(m_{n}\right)$ with density equal to 1. We show that the dense positivity of $\left(r_{n}\right)$ is a necessary and sufficient condition for the uniqueness of $A$-strongly statistical limit, where $A=\left(a_{nk}\right)$ and $\displaystyle{r_{n}=\sum\nolimits_{k=1}^{\infty}a_{nk}}.$ Furthermore, necessary conditions for the regularity, linearity and multiplicativity of $A$-strongly statistical limit are established. |
| first_indexed | 2026-03-24T02:22:29Z |
| format | Article |
| fulltext |
UDC 519.21
U. Kaya, N. D. Aral (Bitlis Eren Univ., Turkey)
STRONGLY STATISTICAL CONVERGENCE
СИЛЬНО СТАТИСТИЧНА ЗБIЖНIСТЬ
We introduce A-strongly statistical convergence for sequences of complex numbers, where A = (ank)n,k\in \BbbN is an infinite
matrix with nonnegative entries. A sequence (xn) is called strongly convergent to L if \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty
\sum \infty
k=1
ank | xk - L| = 0
in the ordinary sense. In the definition of A-strongly statistical limit, we use the statistical limit instead of the ordinary limit
via a convenient density. We study some densities and show that the (ank)-strongly statistical limit is a (amnk)-strong limit,
where the density of the set \{ mn \in \BbbN : n \in \BbbN \} is equal to 1. We introduce the notion of dense positivity for nonnegative
sequences. A nonnegative sequence (rn) is dense positive provided the limit superior of a subsequence (rmn) is positive
for all (mn) with density equal to 1. We show that the dense positivity of (rn) is a necessary and sufficient condition for
the uniqueness of A-strongly statistical limit, where A = (ank) and rn =
\sum \infty
k=1
ank. Furthermore, necessary conditions
for the regularity, linearity and multiplicativity of A-strongly statistical limit are established.
Введено поняття A-сильно статистичної збiжностi для послiдовностей комплексних чисел, де A = (ank)n,k\in \BbbN —
нескiнченна матриця з невiд’ємними елементами. Послiдовнiсть (xn) називається сильно збiжною до L, якщо
\mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty
\sum \infty
k=1
ank | xk - L| = 0 у звичайному сенсi. У визначеннi A-сильно статистичної границi застосовується
поняття статистичної границi замiсть звичайної границi з вiдповiдною щiльнiстю. Вивчено деякi щiльностi i пока-
зано, що (ank)-сильно статистична границя — це (amnk)-сильна границя, де щiльнiсть множини \{ mn \in \BbbN : n \in \BbbN \}
дорiвнює 1. Введено поняття щiльної позитивностi для невiд’ємних послiдовностей. Невiд’ємна послiдовнiсть (rn)
є щiльно позитивною за умови, що верхня границя пiдпослiдовностi (rmn) є додатною для всiх (mn) з щiльнiстю,
що дорiвнює 1. Показано, що щiльна позитивнiсть (rn) є необхiдною та достатньою умовою для єдиностi A-сильно
статистичної границi, де A = (ank) та rn =
\sum \infty
k=1
ank. Крiм того, встановлено необхiднi умови регулярностi, лi-
нiйностi та мультиплiкативностi A-сильно статистичної границi.
1. Introduction. The usual limit concept has many useful applications in several fields of ma-
thematics, statistics, physics, engineering and so on. It is well known that a complex sequence is
convergent to a point if and only if every neighbourhood of the given point includes all the elements
of the sequence but a finite number. If a sequence (xn) is convergent to L, we write
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
| xn - L| = 0. (1.1)
Hamilton and Hill [8] developed this concept by introducing strong summability in 1938. They
generalized equality (1.1) as follows:
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\infty \sum
k=1
ank | xk - L| p = 0, (1.2)
where A = (ank) is an infinite matrix and p > 0. If equality (1.2) holds, (xn) is said to be
strongly summable to L. In the case that A is identity matrix and p = 1 in (1.2), then we get usual
convergence in (1.1).
Whenever a new convergence method is introduced, mathematicians investigate the typical pro-
perties of it, such as uniqueness of limit point, regularity, linearity and so on. Under some conditions,
Hamilton and Hill studied these typical properties of strong convergence.
In 1963, Wlodarski [15] generalized the strong summability into strong continuous summability
method. He considered a sequence of continuous functions (ak (t)) instead of an infinite matrix
c\bigcirc U. KAYA, N. D. ARAL, 2020
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 2 221
222 U. KAYA, N. D. ARAL
(ank) and gave his definition as follows:
\mathrm{l}\mathrm{i}\mathrm{m}
t\rightarrow T
\infty \sum
k=1
ak (t) | xk - L| p = 0.
Besides, he defined some pseudonormed, normed and Banach spaces by using the new convergence
method.
Maddox [10] generalized the strong convergence in 1966 by introducing A-strong convergence
of order (pk)k\in \BbbN for a positive sequence (pk)k\in \BbbN as follows:
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\infty \sum
k=1
ank | xk - L| pk = 0.
Maddox also investigated the uniqueness, regularity, linearity of this concept and studied its applica-
tion to some Cesàro-type space.
The concept of statistical convergence which is an extension of the usual concept of sequential
limit was independently introduced by Fast [4] and Steinhaus [14]. This new method was not as strict
as usual convergence, i.e., it is easy that a sequence is statistical convergent in comparison with usual
convergence. Indeed, a sequence is statistically convergent to a point if and only if it is convergent
to this point in a subset of naturals \BbbN with the asymptotic density of 1. Here, we must define the
asymptotic density. Let K \subset \BbbN . If the limit
\delta (K) = \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
1
n
| \{ k \leq n : k \in K\} |
exists, then \delta (K) is said to be the asymptotic density of K, where the notation | \cdot | denotes the
cardinality of a set. By means of the asymptotic density, the statistical limit can be defined by the
condition
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
1
n
| \{ k \leq n : | xk - L| \geq \varepsilon \} | = 0 (1.3)
for every \varepsilon > 0.
In 1980, Salat [13] showed that the statistical limit can be considered as a linear operator and
every statistical convergent sequence has a unique limit. Also, Salat proved that the space of the
bounded statistical convergent sequences is nowhere dense in the space of bounded sequences and
the space of statistically convergent sequences of real numbers is a dense subset in the first Baire
category in the Fréchet space.
Fridy [6], in 1985, proved that if a sequence is convergent to L, then it is also statistically
convergent to the number L, i.e., statistical convergence is regular. He introduced the concept of
statistically Cauchy sequence and proved that it is equivalent to statistical convergence. Finally, he
proved some Tauberian theorems.
The relation between A-strong convergence and statistical convergence was studied by Connor
[2]. Freedman and Sember [5] investigated densities on natural numbers and showed that the density
used in statistical convergence can be defined by Cesàro matrix. Also, they generalized this concept
for arbitrary regular matrix.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 2
STRONGLY STATISTICAL CONVERGENCE 223
Karakaya and Chishti [9] introduced the concept of weighted statistical convergence in 2009.
Later, Mursaleen et al. [12] modified this concept in 2012. Let (pn) be a nonnegative sequence such
that p0 > 0 and Pn =
\sum n
k=1
pk \rightarrow \infty as n \rightarrow \infty . A complex sequence (xn) is said to be weighted
statistically convergent to a number L if
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
1
Pn
| \{ k \leq Pn : pk | xk - L| \geq \varepsilon \} | = 0
for every \varepsilon > 0. Also, they found its relationship with the concept of statistical summability
\bigl(
N, p
\bigr)
and gave its applications to Korovkin-type approximation theorems.
Belen and Mohiuddine [1] introduced weighted \lambda -statistical convergence and statistical summa-
bility
\bigl(
N\lambda , p
\bigr)
in 2013. They determined a Korovkin-type approximation theorem through the statis-
tical summability
\bigl(
N\lambda , p
\bigr)
and showed that their approximation theorem was stronger than classical
Korovkin theorem by using classical Bernstein polynomials.
Edely et al. [3], in 2013, used the weighted statistical convergence to give a Korovkin-type
approximation theorem for 2\pi -periodic functions.
In 2014, Ghosal [7] modified the concept of weighted \lambda -statistical convergence by adding the
condition
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
n\rightarrow \infty
pn > 0, (1.4)
where (pn) is the weight sequence. Ghosal proved that (1.4) is a sufficient condition for the unique-
ness of the limit.
In this work, we introduce a new convergence method which we call strongly statistical conver-
gence. Let
A = (ank) =
\left(
a11 a12 a13 \cdot \cdot \cdot
a21 a22 a23 \cdot \cdot \cdot
a31 a32 a33 \cdot \cdot \cdot
...
...
...
. . .
\right)
be a nonnegative matrix, rn =
\sum \infty
k=1
ank < +\infty for every n \in \BbbN and Sn =
\sum n
i=1
ri \rightarrow \infty as
n \rightarrow \infty . We call that (xn) \subset \BbbC is A-strongly statistically convergent to L \in \BbbC if
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
1
Sn
\bigm| \bigm| \bigm| \bigm| \bigm|
\Biggl\{
k \leq Sn :
\infty \sum
i=1
aki | xi - L| \geq \varepsilon
\Biggr\} \bigm| \bigm| \bigm| \bigm| \bigm| = 0
for all \varepsilon > 0.
This new method is generalizations of the following:
1) statistical convergence,
2) weighted statistical convergence,
3) strong convergence.
Relation (1.4) is a sufficient condition for the uniqueness of the weighted statistical limit point.
In this study, we give a necessary and sufficient condition, which we call dense positivity, for the
uniqueness of the strongly statistical limit point (and naturally for the uniqueness of the weighted
statistical limit point).
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 2
224 U. KAYA, N. D. ARAL
Finally, we investigate the regularity, linearity and multiplicativity of this new concept. The
strongly statistical convergence is a new concept and it can be applied the approximation theory,
Fourier analysis, topology and so on. For example, it can be given a Korovkin-type approximation
theorem. Also, it can be determined whether the space of the strongly statistically convergent se-
quences is a subspace of l\infty or not. It can be described strongly statistically Cauchy sequence and
studied its properties.
2. Preliminaries.
Definition 2.1 [5]. Let \delta is a function from all the subsets of natural numbers to the closed
interval [0, 1] . If the following conditions hold, then \delta is said to be a lower density in the sense of
Freedman and Sember:
1) if the symmetric difference of the sets A and B is finite, then \delta (A) = \delta (B) ,
2) if A \cap B = \varnothing , then \delta (A) + \delta (B) \leq \delta (A \cup B) ,
3) \delta (A) + \delta (B) \leq 1 + \delta (A \cap B) for all A and B,
4) \delta (\BbbN ) = 1.
If \delta is a lower density, then \delta is called an upper density associated with \delta defined by the equality
\delta (A) = 1 - \delta (\BbbN \setminus A) .
Proposition 2.1. Assume that (Sn) \subset \BbbR is a nondecreasing, nonnegative and unbounded se-
quence. Then
\delta Sn
(K) := \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
n\rightarrow \infty
1
Sn
| \{ k \leq Sn : k \in K\} | (2.1)
is a lower density, where | E| denotes the cardinality of a set E, and K \subset \BbbN .
Proof. Let
\delta [[Sn]] (K) := \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
n\rightarrow \infty
1
[[Sn]]
| \{ k \leq [[Sn]] : k \in K\} | , (2.2)
where [[Sn]] denotes the integral part of Sn, and K \subset \BbbN . We now show that \delta Sn
(K) = \delta [[Sn]] (K)
for each K \subset \BbbN .
Obviously, the relations Sn - 1 < [[Sn]] \leq Sn and \{ k \leq Sn : k \in K\} = \{ k \leq [[Sn]] : k \in K\}
hold for each n \in \BbbN . By these relations, we have the inequality
1
Sn
| \{ k \leq Sn : k \in K\} | \leq 1
[[Sn]]
| \{ k \leq [[Sn]] : k \in K\} | <
<
1
Sn - 1
| \{ k \leq Sn : k \in K\} | . (2.3)
Since \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty
Sn - 1
Sn
= 1, the limit inferiors of
1
Sn
| \{ k \leq Sn : k \in K\} | and
1
Sn - 1
| \{ k \leq Sn : k \in K\} | (2.4)
coincide. Therefore, we get \delta Sn
(K) = \delta [[Sn]] (K) for each K \subset \BbbN .
Now, we will prove that \delta [[Sn]] is a lower density. Let
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 2
STRONGLY STATISTICAL CONVERGENCE 225
M = (ank) =
\left\{
1, if 0 \leq Sn < 1 and k = 1,
1
[[Sn]]
, if Sn \geq 1 and k \leq Sn,
0, otherwise.
Note that M is a nonnegative Toeplitz matrix (see [11], Chapter 7.1, Theorem 3). Moreover, the
equality
\delta Sn
(K) = \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
n\rightarrow \infty
(M\chi K)n
holds for every K \subset \BbbN , where \chi K is the characteristic function of the set K, i.e.,
\chi K (j) =
\Biggl\{
1, if j \in K,
0, if j \in \BbbN \setminus K.
Hence, \delta Sn
is a lower density by [5] (Proposition 3.1).
Proposition 2.1 is proved.
Remark 2.1. In (2.1), (2.2), (2.3), and (2.4), some of the numbers Sn, [[Sn]] and Sn - 1 may
be zero for some n. In this case, since \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty Sn = +\infty , there exists n0 \in \BbbN such that Sn > 0,
[[Sn]] > 0 and Sn - 1 > 0 for each n \geq n0. Thus, we will assume n \geq n0.
By Proposition 2.1,
\delta Sn (K) := \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n\rightarrow \infty
(M\chi K)n
is an upper density. If \delta Sn (K) = \delta Sn
(K) , then we say that K has density (see [5]). In this case,
\delta Sn
(K) = \delta Sn (K) will be denoted by \delta Sn (K) and we say that \delta Sn (K) is the density of K with
respect to (Sn) . When Sn = n, we will say that \delta Sn is Cesàro density and write \delta instead of \delta Sn .
K is said to be an Sn-null set in case \delta Sn (K) = 0. K is called a Cesàro-null set when Sn = n and
\delta (K) = 0.
It is easy to observe the following:
0 \leq \delta Sn
(K) \leq \delta Sn (K) \leq 1.
In addition, if \delta Sn (K) exists, then 0 \leq \delta Sn (K) \leq 1. We say that K is dense with respect to (Sn)
provided \delta Sn (K) = 1.
Now, we give a proposition about the intersection of dense subsets of natural numbers.
Proposition 2.2. The intersection of two dense subsets of \BbbN is dense, i.e., if \delta Sn (K1) =
= \delta Sn (K2) = 1, then \delta Sn (K1 \cap K2) = 1.
By using [5] (Propositions 2.1 – 2.3), one can easily prove this assertion.
We now give an example for an Sn-null set that fails to be Cesàro-null set.
Example 2.1. Let Kn =
\Bigl(
2n
2
, 2n
2+1
\Bigr]
\cap \BbbN and K =
\bigcup \infty
n=0
Kn. Since\bigm| \bigm| \bigm| \Bigl\{ k \leq 2n
2+1 : k \in K
\Bigr\} \bigm| \bigm| \bigm|
2n2+1
=
\sum n
k=0
\Bigl(
2k
2+1 - 2k
2
\Bigr)
2n2+1
=
\sum n
k=0
2k
2
2n2+1
\geq 2n
2
2n2+1
=
1
2
,
then \delta (K) \geq 1
2
. Similarly, by the following:
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 2
226 U. KAYA, N. D. ARAL\bigm| \bigm| \bigm| \Bigl\{ k \leq 2n
2
: k \in K
\Bigr\} \bigm| \bigm| \bigm|
2n2 =
\sum n - 1
k=0
\Bigl(
2k
2+1 - 2k
2
\Bigr)
2n2 =
\sum n - 1
k=0
2k
2
2n2 \leq
\leq
\sum (n - 1)2
k=0
2k
2n2 =
2(n - 1)2+1 - 1
2n2 =
\Bigl(
22(1 - n) - 2 - n2
\Bigr)
,
we get
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\bigm| \bigm| \bigm| \Bigl\{ k \leq 2n
2
: k \in K
\Bigr\} \bigm| \bigm| \bigm|
2n2 = 0, (2.5)
i.e., \delta (K) = 0.
Consequently, K has no Cesàro density. However, by (2.5), we obtain \delta Sn (K) = 0, where
Sn = 2n
2
. K is 2n
2
-null set but is not Cesàro-null.
We now give some definitions.
Definition 2.2 [4]. Let (xn) \subset \BbbC and L \in \BbbC . If \delta (\{ k \in \BbbN : | xk - L| \geq \varepsilon \} ) = 0 for each
\varepsilon > 0, then (xn) is called statistically convergent to L.
Definition 2.3 [12]. Let (pk) be a sequence of nonnegative numbers such that p1 > 0, Pn =
=
\sum n
k=1
pk \rightarrow \infty as n \rightarrow \infty . A sequence x = (xk) is called weighted statistically convergent to
L if the set \{ k \in \BbbN : pk | xk - L| \geq \varepsilon \} is Pn-null set for every \varepsilon > 0.
Definition 2.4 [8]. Let A be an infinite matrix, (xn) \subset \BbbC and L \in \BbbR . Then (xn) is called
A-strongly convergent to L if \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty
\sum \infty
k=1
ank | xk - L| = 0. This convergence is denoted by
xn \rightarrow L [A] .
3. \bfitA -strongly statistical convergence.
Definition 3.1. Let
A = (ank) =
\left(
a11 a12 a13 \cdot \cdot \cdot
a21 a22 a23 \cdot \cdot \cdot
a31 a32 a33 \cdot \cdot \cdot
...
...
...
. . .
\right)
be a nonnegative matrix, rn =
\sum \infty
k=1
ank and Sn =
\sum n
i=1
ri. Assume that the matrix A satisfies
the following conditions:
(i) rn < +\infty for each n \in \BbbN ,
(ii) \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty Sn = +\infty .
We say that (xn) \subset \BbbC is A-strongly statistically convergent to L \in \BbbC if
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
1
Sn
\bigm| \bigm| \bigm| \bigm| \bigm|
\Biggl\{
k \leq Sn :
\infty \sum
i=1
aki | xi - L| \geq \varepsilon
\Biggr\} \bigm| \bigm| \bigm| \bigm| \bigm| = 0
for all \varepsilon > 0. We write xn
st\rightarrow L [A] when (xn) is A-strongly statistically convergent to L.
We will say that A matrix A satisfying (i) and (ii) in Definition 3.1 is an S -type matrix.
If
A = I = (ank) =
\Biggl\{
1, if n = k,
0, if n \not = k,
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 2
STRONGLY STATISTICAL CONVERGENCE 227
then A-strongly statistical convergence coincides with the statistical convergence which was intro-
duced in [4].
If
A = (ank) =
\Biggl\{
pn, if n = k,
0, if n \not = k,
then A-strongly statistical convergence coincides with the weighted statistical convergence which
was introduced in [12], where pn \geq 0, p1 > 0 and
\sum \infty
n=1
pn = +\infty .
Theorem 3.1. Let A be an S -type matrix, (xn) be a complex sequence, L \in \BbbC and Ak =
=
\sum \infty
i=1
aki | xi - L| . Hence, xn
st\rightarrow L [A] if and only if there exist two nonnegative sequences
(Bk) , (Ck) such that Ak = Bk + Ck, \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty Bn = 0 and \delta Sn (\{ k \in \BbbN : Ck \not = 0\} ) = 0.
Proof. Necessity. Assume that xn
st\rightarrow L [A] . Then we get
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
1
Sn
\bigm| \bigm| \bigm| \bigm| \bigm|
\Biggl\{
k \leq Sn :
\infty \sum
i=1
aki | xi - L| \geq \varepsilon
\Biggr\} \bigm| \bigm| \bigm| \bigm| \bigm| = 0
for each \varepsilon > 0. Therefore, if \varepsilon > 0 and r > 0, then there exists nr,\varepsilon \in \BbbN such that the inequality
1
Sn
\bigm| \bigm| \bigm| \bigm| \bigm|
\Biggl\{
k \leq Sn :
\infty \sum
i=1
aki | xi - L| \geq \varepsilon
\Biggr\} \bigm| \bigm| \bigm| \bigm| \bigm| < r (3.1)
holds for every n > nr,\varepsilon . We choose \varepsilon = r =
1
j
and Nj = nr,\varepsilon for j \in \BbbN . Then, the relation (3.1)
turns into the inequality
1
Sn
\bigm| \bigm| \bigm| \bigm| \bigm|
\Biggl\{
k \leq Sn :
\infty \sum
i=1
aki | xi - L| \geq 1
j
\Biggr\} \bigm| \bigm| \bigm| \bigm| \bigm| < 1
j
, (3.2)
where n > Nj . Note that the sequence (Nj) of naturals can be constructed as strictly increasing. We
define the sequences (Bk) and (Ck) as follows:
Bk :=
\left\{ Ak, if 1 \leq k \leq N1 or if Nj < k \leq Nj+1 and Ak <
1
j
,
0, otherwise,
and
Ck := Ak - Bk,
where k, j \in \BbbN . It is obvious that Ak = Bk + Ck and Bk, Ck \geq 0 for each k \in \BbbN .
Given \varepsilon > 0, there exists j \in \BbbN such that
1
j
< \varepsilon . Let k > Nj . Since (Nj) is a strictly
increasing sequence, there exists M \geq j such that NM < k \leq NM+1. There are two cases for Ak :
Ak <
1
M
or Ak \geq 1
M
. In the former case, Bk = Ak <
1
M
\leq 1
j
< \varepsilon . In the latter case, Bk = 0 < \varepsilon .
Therefore, \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty Bn = 0.
Now, we will show that \delta Sn (\{ k \in \BbbN : Ck \not = 0\} ) = 0. By the definition of (Ck) , the inclusion
\{ k \leq Sn : Nj < k \leq Nj+1, Ck \not = 0\} \subset
\biggl\{
k \leq Sn : Nj < k \leq Nj+1, Ak \geq 1
j
\biggr\}
(3.3)
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 2
228 U. KAYA, N. D. ARAL
holds, where n, j \in \BbbN . For n \in \BbbN , there exists j \in \BbbN \cup \{ 0\} such that
Nj < Sn \leq Nj+1, (3.4)
where N0 = 0. Suppose that Ck \not = 0 and k \leq Sn. Since (Nj) is a strictly increasing sequence, (3.4)
implies that there exists M1 \in \BbbN such that M1 \leq j and NM1 < k \leq NM1+1. By using (3.3), we
have Ak \geq 1
M1
and, so, Ak \geq 1
j
. Thus, we get
\{ k \leq Sn : Ck \not = 0\} \subset
\biggl\{
k \leq Sn : Ak \geq 1
j
\biggr\}
. (3.5)
By (3.2) and (3.5), we have
1
Sn
| \{ k \leq Sn : Ck \not = 0\} | \leq 1
Sn
\bigm| \bigm| \bigm| \bigm| \biggl\{ k \leq Sn : Ak \geq 1
j
\biggr\} \bigm| \bigm| \bigm| \bigm| < 1
j
.
(3.4) implies that j \rightarrow \infty as n \rightarrow \infty . Therefore,
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
1
Sn
| \{ k \leq Sn : Ck \not = 0\} | = 0.
This implies that \delta Sn (\{ k \in \BbbN : Ck \not = 0\} ) = 0.
Sufficiency. Assume that there exist (Bk) and (Ck) sequences which satisfy the following con-
ditions:
(i) Bk, Ck \geq 0 for all k \in \BbbN ,
(ii) Ak = Bk + Ck for all k \in \BbbN ,
(iii) \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty Bn = 0,
(iv) \delta Sn (\{ k \in \BbbN : Ck \not = 0\} ) = 0.
Let \varepsilon > 0. Condition (iii) implies that the set \{ k \in \BbbN : Bk \geq \varepsilon \} is finite. So, the equality
\delta Sn (\{ k \in \BbbN : Bk \geq \varepsilon \} ) = \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
1
Sn
| \{ k \leq Sn : Bk \geq \varepsilon \} | = 0
holds. By (i) and (ii), if Ak \geq \varepsilon , then either Ck = 0 and Bk \geq \varepsilon or Ck \not = 0. By using (iv), we have
the following:
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
1
Sn
| \{ k \leq Sn : Ak \geq \varepsilon \} | \leq
\leq \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
1
Sn
| \{ k \leq Sn : Bk \geq \varepsilon \} | + \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
1
Sn
| \{ k \leq Sn : Ck \not = 0\} | = 0 + 0 = 0.
Theorem 3.1 is proved.
By choosing Bk = Ak and Ck = 0, for k \in \BbbN , we have the following corollary.
Corollary 3.1. Let A be an S -type matrix, (xn) \subset \BbbC and L \in \BbbC . Then xn\rightarrow L [A] implies
xn
st\rightarrow L [A] .
The converse of Corollary 3.1 is not true. There exists a statistically convergent but not conver-
gent sequence when A = I (see [6]).
We will use \delta Sn (mn) instead of \delta Sn (\{ mn \in \BbbN : n \in \BbbN \} ) for short notation.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 2
STRONGLY STATISTICAL CONVERGENCE 229
Corollary 3.2. Let A be an S -type matrix, (xn) \subset \BbbC and L \in \BbbC . Hence, xn
st\rightarrow L [A] if and
only if there exists a strictly increasing sequence (mn) of naturals such that \delta Sn (mn) = 1, B =
= (bnk) = (amnk) and xn \rightarrow L [B] .
Remark 3.1. Corollary 3.2 asserts that A-strongly statistical convergence is strong convergence
on the matrix obtained by eliminating some rows from the matrix A, where the density of the indices
of these rows is 0.
4. Main results.
Definition 4.1. Let (Sn) be a nondecreasing, nonnegative and unbounded sequence. We say
that a nonnegative sequence (rn) is Sn-dense positive provided the condition
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n\rightarrow \infty
rmn > 0
holds for every indices of (mn) satisfying \delta Sn (mn) = 1. When Sn = n, we write dense positive
instead of n-dense positive for a sequence.
Remark 4.1. Sn-dense positivity of a sequence (rn) is weaker than the condition \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
n\rightarrow \infty
(rn) >
> 0 and stronger than the condition \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n\rightarrow \infty
(rn) > 0. That is, if the limit inferior of a nonnegative
sequence is positive, then it is Sn-dense positive. Similarly, if a nonnegative sequence is Sn-dense
positive, then its limit superior is positive. However, the converses of these assertions are not true in
general, as we show in next two examples.
Example 4.1. Let
an =
\Biggl\{
1, if k = n2,
0, if k \not = n2,
and Sn = n. Obviously, \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n\rightarrow \infty
an = 1. However, if we choose (mn) as all nonsquare numbers,
the conditions \delta (mn) = 1 and \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n\rightarrow \infty
amn = 0 hold. So, \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n\rightarrow \infty
an > 0 but (an) is not dense
positive.
Example 4.2. Let
bn =
\Biggl\{
1, if k \not = n2,
0, if k = n2,
and Sn = n. Obviously, \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
n\rightarrow \infty
bn = 0. However, for each (mn) \subset \BbbN satisfying \delta (mn) = 1, the
equality \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n\rightarrow \infty
bmn = 1 holds. Hence, (bn) is dense positive but \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
n\rightarrow \infty
bn = 0.
Theorem 4.1. Let A = (ank) be an S -type matrix. A-strongly statistical limit is unique if and
only if (rn) is Sn-dense positive, where rn =
\sum \infty
k=1
ank.
Proof. Sufficiency. Assume that (rn) is Sn-dense positive and a sequence (xn) of complex num-
bers strongly statistically tends to both L and R. By Corollary 3.2, since xn
st\rightarrow L [A] and xn
st\rightarrow R [A] ,
there exist indices (mn) and (jn) such that B = (amnk) , D = (ajnk) , xn \rightarrow L [B] , xn \rightarrow R [D]
and \delta Sn (mn) = \delta Sn (jn) = 1. By Proposition 2.2, we get \delta Sn ((mn) \cap (jn)) = 1. We now define
(in) by (in) := (mn) \cap (jn) . Obviously, the inclusions (in) \subset (mn) and (in) \subset (jn) hold. From
here, we have xn \rightarrow L [E] and xn \rightarrow R [E] , where E = (aink) . Since (rn) is dense positive, we
obtain \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n\rightarrow \infty
rin > 0. Then, by [8] (Theorem 3), we have L = R.
Necessity. Assume that A-strongly statistical limit is unique. We claim that the sequence (rn) is
Sn-dense positive. Consider a set of indices (mn) \subset \BbbN satisfying \delta Sn (mn) = 1. Let B = (amnk) .
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 2
230 U. KAYA, N. D. ARAL
We now define two classes of sequences as follows:
Cst
[A] :=
\Bigl\{
(xn) \subset \BbbC : \exists L \in \BbbC : xn
st\rightarrow L [A]
\Bigr\}
,
C[B] := \{ (xn) \subset \BbbC : \exists L \in \BbbC : xn \rightarrow L [B]\} .
By Corollary 3.2, we have C[B] \subset Cst
[A]. From here, since A-statistically strong limit is unique,
B-strong limit is also unique. So, \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n\rightarrow \infty
rmn > 0 by [8] (Theorem 3).
Theorem 4.1 is proved.
Corollary 4.1. If A is nonnegative Toeplitz matrix, then A-strongly statistical limit is unique (see
[11], Chapter 7.1, Theorem 3).
Remark 4.2. Recall the Definition 2. By Theorem 4.1, the uniqueness of weighted statistical
limit requires that (pn) is Pn-dense positive. Indeed, this condition is necessary and sufficient
for uniqueness. A sequence may have infinitely many weighted statistical limits when (pn) is not
Pn-dense positive, as we show in next example.
Example 4.3. Consider the matrix
A = (ank) =
\left\{
1
n
, if n = k,
0, if n \not = k,
and the constant sequence (xn) = (0, 0, . . . ) . By Definition 3.1, we get rn =
1
n
and Sn =
\sum n
i=1
1
i
.
Obviously, the equality \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty Sn = +\infty holds. So, A is an S -type matrix. Let L \in \BbbC and \varepsilon > 0.
Since the sequences (Sn) and
\biggl(
| L|
n
\biggr)
tend to infinity and zero, respectively, then the cardinality of
the set \biggl\{
k \leq Sn :
| L|
k
\geq \varepsilon
\biggr\}
is a constant for sufficiently large n. Thus, we have the following:
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
1
Sn
\bigm| \bigm| \bigm| \bigm| \biggl\{ k \leq Sn :
1
k
| xk - L| \geq \varepsilon
\biggr\} \bigm| \bigm| \bigm| \bigm| = \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
1
Sn
\bigm| \bigm| \bigm| \bigm| \biggl\{ k \leq Sn :
| L|
k
\geq \varepsilon
\biggr\} \bigm| \bigm| \bigm| \bigm| = 0.
This shows that each complex number L is A-strongly statistical limit or
\biggl(
1
n
\biggr)
-weighted statistical
limit of (xn) .
Theorem 4.2. Let A = (ank) be an S -type matrix. If there exist some indices mn \in \BbbN
such that \delta Sn (mn) = 1, \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty amnk = 0 for each k \in \BbbN and \mathrm{s}\mathrm{u}\mathrm{p}n\in \BbbN rmn < +\infty , then A-
strongly statistical convergence is a regular summability method, i.e., xn \rightarrow L implies xn
st\rightarrow L [A]\Bigl(
rmn =
\sum \infty
k=1
amnk
\Bigr)
.
Proof. Let xn \rightarrow L and B = (amnk) . By the hypothesis and [8] (Theorem 1), xn \rightarrow L [B] .
Finally, by Corollary 3.2, we obtain xn
st\rightarrow L [A] .
Proposition 4.1. If (xn) \subset \BbbC is statistically strongly summable to L by a matrix A = (ank) of
which (rn) is dense positive, then L must be a limit point of (xn) .
The proof of this assertion is directly obtained from Corollary 3.2 and [8] (Theorem 4).
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 2
STRONGLY STATISTICAL CONVERGENCE 231
Theorem 4.3. Let A = (ank) be an S -type matrix, (xn) , (yn) be two complex sequences and
L,M,\alpha , \beta \in \BbbC . Then the followings hold:
(i) xn
st\rightarrow L [A] , yn
st\rightarrow M [A] implies (\alpha xn + \beta yn)
st\rightarrow \alpha L+ \beta M [A] ,
(ii) xn
st\rightarrow L [A] , yn
st\rightarrow M [A] implies (xnyn)
st\rightarrow LM [A] if either (xn) or (yn) is bounded,
(iii) xn
st\rightarrow L [A] , yn
st\rightarrow M [A] implies
\biggl(
xn
yn
\biggr)
st\rightarrow L
M
[A] if there exists a positive number d such
that | yn| \geq d for sufficiently large n and one of the conditions that either M \not = 0 or (rn) of A is
dense positive is true.
Proof. By Corollary 3.2, xn
st\rightarrow L [A] implies xn \rightarrow L [B] and yn
st\rightarrow M [A] implies yn \rightarrow M [D] ,
where B = (amnk) , D = (ajnk) and \delta Sn (mn) = \delta Sn (jn) = 1. Thus, we have xn \rightarrow L [E]
and yn \rightarrow M [E] , where E = (aink) and (in) = (mn) \cap (jn) . By Proposition 2.2, the equality
\delta Sn (in) = 1 is true. By [10] (Theorem 1), we have the followin:
(i) For each \alpha , \beta \in \BbbC , (\alpha xn + \beta yn) \rightarrow \alpha L + \beta M [E] . Hence, we get, by Corollary 3.2, that
(\alpha xn + \beta yn)
st\rightarrow \alpha L+ \beta M [A] .
(ii) If either (xn) or (yn) is bounded sequence, then (xkyk) \rightarrow LM [E] . Similarly, (xnyn)
st\rightarrow
st\rightarrow LM [A] .
(iii) If (rn) of A is dense positive, we conclude | M | \geq d by Proposition 4.1. So, we can assume
that (rn) of A is dense positive or M \not = 0 . Since (yn) satisfies the condition | yn| \geq d, by (ii), we
have
\biggl(
xn
yn
\biggr)
st\rightarrow L
M
[A].
References
1. C. Belen, S. A. Mohiuddine, Generalized weighted statistical convergence and application, Appl. Math. and Comput.,
219, № 18, 9821 – 9826 (2013).
2. J. Connor, On strong matrix summability with respect to a modulus and statistical convergence, Canad. Math. Bull.,
32, № 2, 194 – 198 (1989).
3. O. H. Edely, M. Mursaleen, A. Khan, Approximation for periodic functions via weighted statistical convergence,
Appl. Math. and Comput., 219, № 15, 8231 – 8236 (2013).
4. H. Fast, Sur la convergence statistique, Colloq. Math., 2, 241 – 244 (1951).
5. A. R. Freedman, J. J. Sember, Density and summability, Pacif. J. Math., 95, № 2, 293 – 305 (1981).
6. J. A. Fridy, On statistical convergence, Analysis, 5, № 4, 301 – 313 (1985).
7. S. Ghosal, Generalized weighted random convergence in probability, Appl. Math. and Comput., 249, 502 – 509
(2014).
8. H. J. Hamilton, J. D. Hill, On strong summability, Amer. J. Math., 60, № 3, 588 – 594 (1938).
9. V. Karakaya, T. A. Chishti, Weighted statistical convergence, Iran. J. Sci. and Technol. Trans. A Sci., 33, № 3,
219 – 223 (2009).
10. I. J. Maddox, Spaces of strongly summable sequences, Quart. J. Math., 18, № 1, 345 – 355 (1967).
11. I. J. Maddox, Elements of functional analysis, Cambridge Univ. Press, Cambridge, UK (1970).
12. M. Mursaleen, V. Karakaya, M. Erturk, F. Gursoy, Weighted statistical convergence and its application to Korovkin
type approximation theorem, Appl. Math. and Comput., 218, 9132 – 9137 (2012).
13. T. Salat, On statistically convergent sequences of real numbers, Math. Slovaca, 30, № 2, 139 – 150 (1980).
14. H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math., 2, № 1, 73 – 74 (1951).
15. L. Wlodarski, On some strong continuous summability methods, Proc. London Math. Soc., 3, № 1, 273 – 289 (1963).
Received 25.12.16,
after revision — 03.08.17
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 2
|
| id | umjimathkievua-article-2368 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:22:29Z |
| publishDate | 2020 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/a9/3fee302f1a626dd316ecc5b180c700a9.pdf |
| spelling | umjimathkievua-article-23682020-04-07T12:13:59Z Strongly statistical convergence Сильно статистична збіжність Kaya, U. Aral , N. D. Kaya, U. Kaya, U. Aral , N. D. збіжність convergence UDC 519.21 We introduce $A$-strongly statistical convergence for sequences of complex numbers, where $A=\left(a_{nk}\right)_{n,k\in \mathbb{N}}$ is an infinite matrix with nonnegative entries. A sequence $\left(x_{n}\right)$ is called strongly convergent to $L$ if $\displaystyle{\lim\nolimits_{n\to\infty} \sum\nolimits_{k=1}^{\infty}a_{nk}\left|x_{k}-L\right|=0}$ in the ordinary sense. In the definition of $A$-strongly statistical limit, we use the statistical limit instead of the ordinary limit via a convenient density. We study some densities and show that the $\left(a_{nk}\right)$-strongly statistical limit is a $\left(a_{m_{n}k}\right)$-strong limit, where the density of the set $\left\{m_{n}\in\mathbb{N}\colon n\in\mathbb{N}\right\}$ is equal to 1. We introduce the notion of dense positivity for nonnegative sequences. A nonnegative sequence $\left(r_{n}\right)$ is dense positive provided the limit superior of a subsequence $\left(r_{m_{n}}\right)$ is positive for all $\left(m_{n}\right)$ with density equal to 1. We show that the dense positivity of $\left(r_{n}\right)$ is a necessary and sufficient condition for the uniqueness of $A$-strongly statistical limit, where $A=\left(a_{nk}\right)$ and $\displaystyle{r_{n}=\sum\nolimits_{k=1}^{\infty}a_{nk}}.$ Furthermore, necessary conditions for the regularity, linearity and multiplicativity of $A$-strongly statistical limit are established. УДК 519.21 Введено поняття $A$-сильно статистичної збіжності для послідовностей комплексних чисел, де $A=\left(a_{nk}\right)_{n,k\in \mathbb{N}}$ --- нескінченна матриця з невід’ємними елементами.Послідовність $\left(x_{n}\right)$ називається сильно збіжною до $L,$ якщо $\displaystyle{\lim\nolimits_{n\to\infty} \sum\nolimits_{k=1}^{\infty}a_{nk}\left|x_{k}-L\right|=0}$ у звичайному сенсі.У визначенні $A$-сильно статистичної границі застосовується поняття статистичної границі замість звичайної границі з відповідною щільністю.Вивчено деякі щільності і показано, що $\left(a_{nk}\right)$-сильно статистична границя --- це $\left(a_{m_{n}k}\right)$-сильна границя, де щільність множини $\left\{m_{n}\in\mathbb{N}\colon n\in\mathbb{N}\right\}$ дорівнює 1. Введено поняття щільної позитивності для невід'ємних послідовностей.Невід'ємна послідовність $\left(r_{n}\right)$ є щільно позитивною за умови, що верхня границя підпослідовності $\left(r_{m_{n}}\right)$ є додатною для всіх $\left(m_{n}\right)$ з щільністю, що дорівнює 1.Показано, що щільна позитивність $\left(r_{n}\right)$ є необхідною та достатньою умовою для єдиності $A$-сильно статистичної границі, де $A=\left(a_{nk}\right)$ та $\displaystyle{r_{n}=\sum\nolimits_{k=1}^{\infty}a_{nk}}.$Крім того, встановлено необхідні умови регулярності, лінійності та мультиплікативності $A$-сильно статистичної границі. Institute of Mathematics, NAS of Ukraine 2020-02-15 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2368 Ukrains’kyi Matematychnyi Zhurnal; Vol. 72 No. 2 (2020); 221-231 Український математичний журнал; Том 72 № 2 (2020); 221-231 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2368/1562 Copyright (c) 2020 U. Kaya,N. D. Aral |
| spellingShingle | Kaya, U. Aral , N. D. Kaya, U. Kaya, U. Aral , N. D. Strongly statistical convergence |
| title | Strongly statistical convergence |
| title_alt | Сильно статистична збіжність |
| title_full | Strongly statistical convergence |
| title_fullStr | Strongly statistical convergence |
| title_full_unstemmed | Strongly statistical convergence |
| title_short | Strongly statistical convergence |
| title_sort | strongly statistical convergence |
| topic_facet | збіжність convergence |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2368 |
| work_keys_str_mv | AT kayau stronglystatisticalconvergence AT aralnd stronglystatisticalconvergence AT kayau stronglystatisticalconvergence AT kayau stronglystatisticalconvergence AT aralnd stronglystatisticalconvergence AT kayau silʹnostatističnazbížnístʹ AT aralnd silʹnostatističnazbížnístʹ AT kayau silʹnostatističnazbížnístʹ AT kayau silʹnostatističnazbížnístʹ AT aralnd silʹnostatističnazbížnístʹ |