On the Statistical Convergence of Metric-Valued Sequences
We study the conditions for the density of a subsequence of a statistically convergent sequence under which this subsequence is also statistically convergent. Some sufficient conditions of this type and almost converse necessary conditions are obtained in the setting of general metric spaces.
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508115583107072 |
|---|---|
| author | Değer, U. Dovgoshei, A. A. Küçükaslan, M. Дегер, У. Довгошей, О. А. Куцукаслан, М. |
| author_facet | Değer, U. Dovgoshei, A. A. Küçükaslan, M. Дегер, У. Довгошей, О. А. Куцукаслан, М. |
| author_sort | Değer, U. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
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| datestamp_date | 2019-12-05T10:25:31Z |
| description | We study the conditions for the density of a subsequence of a statistically convergent sequence under which this subsequence is also statistically convergent. Some sufficient conditions of this type and almost converse necessary conditions are obtained in the setting of general metric spaces. |
| first_indexed | 2026-03-24T02:20:05Z |
| format | Article |
| fulltext |
UDC 517.5
M. Küçükaslan, U. Değer (Mersin Univ., Turkey),
O. Dovgoshey (Inst. Appl. Math. and Mech. Nat. Acad. Sci. Ukraine, Donetsk)
ON STATISTICAL CONVERGENCE OF METRIC VALUED SEQUENCES
ПРО СТАТИСТИЧНУ ЗБIЖНIСТЬ
МЕТРИЧНОЗНАЧНИХ ПОСЛIДОВНОСТЕЙ
We study the conditions on the density of a subsequence of a statistical convergent sequence under which this subsequence
is also statistical convergent. Some sufficient conditions of this type and almost converse necessary conditions are obtained
in the setting of general metric spaces.
Вивчаються умови на щiльнiсть пiдпослiдовностi статистично збiжної послiдовностi, за яких ця пiдпослiдовнiсть
також є статистично збiжною. Деякi достатнi умови такого типу та майже оберненi необхiднi умови отримано в
постановцi загальних метричних просторiв.
1. Introduction and definitions. Analysis on metric spaces has rapidly developed in present time
(see [15, 18]). This development is usually based on some generalizations of the differentiability.
The generalizations of the differentiation involve linear structure by means of embeddings of metric
spaces in a suitable normed space or by use of geodesics.
A new intrinsic approach to the introduction of the smooth structure for general metric space was
proposed by O. Martio and O. Dovgoshey in [10] (see also [1, 3, 4, 7 – 9]). The approach in [10] is
completely based on the convergence of the metric valued sequences but it is not apriori clear that
the usual convergence is the best possible way to obtain the smooth structure for arbitrary metric
space.
The problem of convergence in different ways of a real (or complex) valued divergent sequence
goes back to the beginning of nineteenth century. A lot of different convergence methods were defined
(Cesaro, Nörlund, Weighted Mean, Abel et al.) and applied to many branches of mathematics. Almost
all convergence methods depend on the algebraic structure of the space. It is clear that metric space
does not have the algebraic structure in general. However, the notion of statistical convergence is
easy to extend for arbitrary metric spaces and this provides a general framework for summability
in such spaces [13, 21]. Thus, the studies of statistical convergence give a natural foundation for
upbuilding of different tangent spaces to general metric spaces.
The construction of tangent spaces in [3, 4, 7 – 10] is based on the following fundamental fact:
“If (xn) is a convergent sequence in a metric space, then each subsequence (xn(k)) of (xn) is
also convergent”. Thus the convergence of subsequence (xn(k)) does not depend on the choice of
(xn(k)). Unfortunately it is not the case for the statistical convergent sequences. The applications
of the statistical convergence to the infinitesimal geometry of metric spaces should be based on the
complete understanding of the structure of statistical convergent subsequences.
We study the conditions on the density of a subsequence of a statistical convergent sequence
under which this subsequence is also statistical convergent. Some sufficient conditions of such type
and “almost converse” them necessary conditions are obtained in the setting of general metric spaces.
Let us remember the main definitions. Let (X, d) be a metric space. For convenience denote by
X̃ the set of all sequences of points from X.
c© M. KÜÇÜKASLAN, U. DEĞER, O. DOVGOSHEY, 2014
712 ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 5
ON STATISTICAL CONVERGENCE OF METRIC VALUED SEQUENCES 713
Definition 1.1. A sequence (xn) ∈ X̃ is called convergent to a point a ∈ X, limn→∞ xn = a,
if for every ε > 0 there is an n0 = n0(ε) ∈ N such that n > n0 implies d(xn, a) < ε.
Definition 1.2. A metric valued sequence x̃ = (xn) ∈ X̃ is d-statistical convergent to a ∈ X if
lim
n→∞
1
n
∣∣ {k : k ≤ n, d(xk, a) ≥ ε} ∣∣ = 0
holds for every ε > 0.
Here and later |B| denotes the number of elements of a set B.
The idea of statistical convergence goes back to Zygmund [22]. It was formally introduced by
Steinhous [20] and Fast [11]. In recent years, it has become an active research for mathematicians
(see, for example, [5, 6, 12 – 14, 17]).
Definition 1.3 [11] (Dense subset of N). A set K ⊆ N is called a statistical dense subset of N if
lim
n→∞
1
n
∣∣K(n)
∣∣ = 1,
where K(n) = {k ∈ K : k ≤ n}.
It may be proved that the intersection of two dense subsets is dense. Moreover it is clear that the
supersets of dense sets are also dense. Hence the family of all dense sets forms a filter on N. The d
-statistical convergence is simply the convergence in (X, d) with respect to this filter.
Definition 1.4 (Dense subsequence). If
(
n(k)
)
is an infinite, strictly increasing sequence of
natural numbers and x̃ = (xn) ∈ X̃, write x̃′ = (xn(k)) and Kx̃′ =
{
n(k) : k ∈ N
}
. The sub-
sequence x̃′ is a dense subsequence of x̃ if Kx̃′ is a dense subset of N.
In the next definition we introduce an equivalence relation on the set X̃.
Definition 1.5. Sequences x̃ = (xn) ∈ X̃ and ỹ = (yn) ∈ X̃ are statistical equivalent, x̃ � ỹ, if
there is a statistical dense M ⊆ N such that xn = yn for every n ∈M.
2. Convergent sequences and statistical convergent ones. In this section, some basic results
on d-statistical convergence will be given for an arbitrary metric space. In particular, it is shown that
there is some one-to-one correspondence between metrizable topologies on X and the subsets of X̃
consisting of all statistical convergent sequences.
Let (X, d) be a nonvoid metric space. It is clear that every convergent sequence (xn) ∈ X̃ is also
d-statistical convergent. Moreover, all statistical convergent sequences are convergent if and only if
|X| = 1. Nevertheless, we have the following result.
Theorem 2.1. Let (X, d1) and (X, d2) be two metric spaces with the same underlining set X.
Then the following statements are equivalent:
(i) The set of all d1-statistical convergent sequences coincides the set of all d2-statistical con-
vergent sequences.
(ii) The set of all sequences which are convergent in the space (X, d1) coincides the set of all
sequences which are convergent in the space (X, d2).
(iii) The metrics d1 and d2 induce one and the same topology on X.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 5
714 M. KÜÇÜKASLAN, U. DEĞER, O. DOVGOSHEY
Proof. The equivalence (ii) ⇔ (iii) is well known. Since every statistical convergent sequence
can be obtained by a variation of values of a suitable convergent sequence outside of a statistical
dense set, the implication (ii)⇒ (i) follows.
Suppose now that topologies induced by the metrics d1 and d2 are distinct. Then there exist a
point a ∈ X and ε0 > 0 such that either{
x ∈ X : d1(x, a) < ε0} + {x ∈ X : d2(x, a) < δ
}
(2.1)
for all δ > 0 or {
x ∈ X : d2(x, a) < ε0} + {x ∈ X : d1(x, a) < δ
}
for all δ > 0. We assume, without loss of generality, that (2.1) holds. Then there is a sequence
x̃ = (xn) such that
d2(xn, a) <
1
n
and d1(xn, a) ≥ ε0 (2.2)
for each n ∈ N. Let us define a new sequence ỹ = (yn) ∈ X̃ by the rule
yn =
xn if n is odd,
a if n is even.
This definition and (2.2) imply the equality
lim
n→∞
∣∣{k ∈ N : d1(yk, a) ≥ ε0, k ≤ n}
∣∣
n
=
1
2
. (2.3)
It is clear that the sequence ỹ is d2-statistical convergent to a. If statement (i) holds, then ỹ is
also d1-statistical convergent. Using Theorem 3.1 (the proof of Theorem 3.1 does not depend on
Theorem 2.1, see Section 3 of the paper) we obtain that ỹ is d1-statistically convergent to the same
a. Consequently we have
lim
n→∞
∣∣{k ∈ N : d1(yk, a) ≥ ε0, k ≤ n}
∣∣
n
= 0,
contrary to (2.3) The implication (i)⇒ (iii) follows.
Theorem 2.1 is proved.
The next simple lemma gives us a tool for a reduction of some questions related to the d-statistical
convergence to the case of the statistical convergence in R.
Lemma 2.1. Let (X, d) be a metric space, a ∈ X and x̃ = (xn) ∈ X̃. Then x̃ is d-statistical
convergent to a in X if and only if the sequence
(
d(xn, a)
)
is statistical convergent to 0 in R.
The proof follows directly from the definitions.
Theorem 2.2. Let (X, d) be a metric space, a ∈ X and let x̃ = (xn) ∈ X̃ be a d-statistically
convergent to a sequence. There is ỹ = (yn) ∈ X̃ such that ỹ � x̃ and ỹ is convergent to a.
Proof. If X = R and d(x, y) = |x − y| for all x, y ∈ X, then the theorem is known (see
Theorem A in [14] or Lemma 1.1 in [17]). Now let (X, d) be an arbitrary metric space. By Lemma 2.1(
d(xn, a)
)
is statistically convergent to 0. Hence there is a subsequence
(
d(xn(k), a)
)
of the sequence(
d(xn, a)
)
such that limk→∞ d(xn(k), a) = 0 and the set K =
{
n(k) : k ∈ N
}
is a dense subset of
N. Define the sequence ỹ = (yn) ∈ X̃ as
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 5
ON STATISTICAL CONVERGENCE OF METRIC VALUED SEQUENCES 715
yn =
xn if n ∈ K,
a if n ∈ N \K.
It is easy to see that ỹ � x̃ and limn→∞ yn = a.
Theorem 2.2 is proved.
3. Statistical convergence of sequences and their subsequences. If a given sequence is d-
statistical convergent it is natural to ask how we can check that its subsequence is d-statistical
convergent to the same limit.
Theorem 3.1. Let (X, d) be a metric space, x̃ = (xn) ∈ X̃ and let x̃′ = (xn(k)) be a subse-
quence of x̃ such that
lim inf
n→∞
|Kx̃′(n)|
n
> 0.
If x̃ is d-statistical convergent to a ∈ X, then x̃′ is also d-statistical convergent to this a.
Proof. Suppose that (xn) is d-statistical convergent to a. It is clear that{
n(k) : n(k) ≤ n, d(xn(k), a) ≥ ε
}
⊆
{
m : m ≤ n, d(xm, a) ≥ ε
}
for all n. Consequently we have
1∣∣Kx̃′(n)
∣∣ ∣∣{n(k) : n(k) ≤ n, d(xn(k), a) ≥ ε
}∣∣ ≤ 1∣∣Kx̃′(n)
∣∣ ∣∣{m : m ≤ n, d(xm, a) ≥ ε
}∣∣. (3.1)
The sequence x̃ = (xn(k)) is d-statistical convergent if we obtain
lim sup
n→∞
∣∣n(k) : n(k) ≤ n, d(xn(k), a) ≥ ε∣∣∣∣Kx̃′(n)
∣∣ = 0
for every ε > 0. The last limit relation holds if
lim sup
n→∞
∣∣{m : m ≤ n, d(xm, a) ≥ ε}
∣∣∣∣Kx̃′(n)
∣∣ = 0. (3.2)
To prove (3.2) we can use the inequality
lim inf
n→∞
yn lim sup
n→∞
zn ≤ lim sup
n→∞
ynzn (3.3)
which holds for all sequences of nonnegative real numbers with 0 6= lim infn→∞ yn 6= ∞ (see, for
example, [2]). Putting in (3.3)
yn =
∣∣Kx̃′(n)
∣∣
n
and zn =
∣∣{m : m ≤ n, d(xm, a) ≥ ε}
∣∣∣∣Kx̃′(n)
∣∣
we see that
ynzn =
∣∣{m : m ≤ n, d(xm, a) ≥ ε}
∣∣
n
.
Hence we get
lim inf
n→∞
∣∣Kx̃′(n)
∣∣
n
lim sup
n→∞
∣∣{m : m ≤ n, d(xm, a) ≥ ε}
∣∣∣∣Kx̃′(n)
∣∣ ≤ lim sup
n→∞
∣∣{m : m ≤ n, d(xm, a) ≥ ε}
∣∣
n
.
The last inequality implies (3.2) because (xn) is d-statistical convergent.
Theorem 3.1 is proved.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 5
716 M. KÜÇÜKASLAN, U. DEĞER, O. DOVGOSHEY
Theorem 3.2. Let (X, d) be a metric space and let x̃ ∈ X̃. The following statements are equi-
valent:
(i) The sequence x̃ is d-statistical convergent.
(ii) Every subsequence x̃′ of x̃ with
lim inf
n→∞
∣∣Kx̃′(n)
∣∣
n
> 0
is d-statistical convergent.
(iii) Every dense subsequence x̃′ of x̃ is d-statistical convergent.
Proof. The implication (i)⇒ (ii) was proved in Theorem 3.1. Since every dense subsequence x̃′
of x̃ satisfies the inequality
lim inf
n→∞
∣∣Kx̃′(n)
∣∣
n
> 0,
we have (ii)⇒ (iii). The implication (iii)⇒ (i) holds because x̃ is a dense subsequence of it-self.
Theorem 3.2 is proved.
Lemma 3.1. Let (X, d) be a metric space with |X| ≥ 2, let x̃ = (xn) ∈ X̃ and let x̃′ =
(
xn(k)
)
be an infinite subsequence of x̃ such that
lim sup
n→∞
∣∣Kx̃′(n)
∣∣
n
= 0. (3.4)
There are a sequence ỹ ∈ X̃ and a subsequence ỹ′ of ỹ such that: x̃ � ỹ and Kỹ′ = Kx̃′ and ỹ′ is
not d-statistical convergent.
Proof. Let a and b be two distinct points of X. Define the sequence ỹ = (yn) ∈ X̃ by the rule
yn =
xn if n ∈ N \Kx̃′ ,
a if n = n(k) ∈ Kx̃′ and k is odd,
b if n = n(k) ∈ Kx̃′ and k is even.
(3.5)
The set N \Kx̃′ is a statistical dense subset of N. Indeed, the equality
n =
∣∣{m ∈ Kx̃′ : m ≤ n}
∣∣+ ∣∣{m ∈ N \Kx̃′ : m ≤ n}
∣∣
holds for each n ∈ N. It implies the inequality
lim inf
n→∞
∣∣{m ∈ N \Kx̃′ : m ≤ n}
∣∣
n
≥ 1− lim sup
n→∞
∣∣{m ∈ Kx̃′ : m ≤ n}
∣∣
n
. (3.6)
Using (3.4) we obtain
1 ≤ lim inf
n→∞
∣∣{m ∈ N \Kx̃′ : m ≤ n}
∣∣
n
≤ lim sup
n→∞
∣∣{m ∈ N \Kx̃′ : m ≤ n}
∣∣
n
≤ 1.
Consequently,
lim
n→∞
∣∣{m ∈ N \Kx̃′ : m ≤ n}
∣∣
n
= 1. (3.7)
The equivalence x̃ � ỹ follows.
Define the desired subsequence ỹ′ of ỹ as ỹ′ = (yn(k)). It is easy to see that ỹ′ is not d-statistical
convergent.
Lemma 3.1 is proved.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 5
ON STATISTICAL CONVERGENCE OF METRIC VALUED SEQUENCES 717
Lemma 3.2. Let (X, d) be a metric space, a ∈ X, x̃ and ỹ belong to X̃ and let x̃ be d-statistical
convergent to a. If x̃ � ỹ, then ỹ is also d-statistical convergent to a.
Proof. Suppose that ỹ � x̃. Define a subset M of the set N as
(n ∈M)⇔ (xn 6= yn).
Then, by Definition 1.5, N \M is statistical dense. It implies the equality
lim
n→∞
∣∣{m ∈M : m ≤ n}
∣∣
n
= 0. (3.8)
Let ε be a strictly positive number. It follows directly from the definition of the set M that the
inclusion{
m ∈ N : m ≤ n, d(ym, a) ≥ ε
}
⊆
{
m ∈M : m ≤ n
}
∪
{
m ∈ N : m ≤ n, d(xm, a) ≥ ε
}
(3.9)
holds for each n ∈ N. Using this inclusion and equality (3.8) we obtain
lim sup
n→∞
∣∣{m ∈ N : m ≤ n, d(ym, a) ≥ ε}
∣∣
n
≤
≤ lim sup
n→∞
∣∣{m ∈M : m ≤ n}
∣∣
n
+ lim sup
n→∞
∣∣{m ∈ N : m ≤ n, d(xm, a) ≥ ε}
∣∣
n
=
= lim sup
n→∞
∣∣{m ∈ N : m ≤ n, d(xm, a) ≥ ε}
∣∣
n
.
Since x̃ is d-statistical convergent to a we have
lim sup
n→∞
∣∣{m ∈ N : m ≤ n, d(xm, a) ≥ ε}
∣∣
n
= 0
for every ε > 0. Consequently the inequality
lim sup
n→∞
∣∣{m ∈ N : m ≤ n, d(ym, a) ≥ ε}
∣∣
n
≤ 0 (3.10)
holds for every ε > 0. Using (3.10) we get
0 ≤ lim inf
n→∞
∣∣{m ∈ N : m ≤ n, d(ym, a) ≥ ε}
∣∣
n
≤ lim sup
n→∞
∣∣{m ∈ N : m ≤ n, d(ym, a) ≥ ε}
∣∣
n
≤ 0.
Hence the limit relation
lim
n→∞
∣∣{m ∈ N : m ≤ n, d(ym, a) ≥ ε}
∣∣
n
= 0
holds. The last limit relation holds for every ε > 0 if and only if ỹ is d-statistical convergent to a.
Lemma 3.2 is proved.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 5
718 M. KÜÇÜKASLAN, U. DEĞER, O. DOVGOSHEY
Theorem 3.3. Let (X, d) be a metric space with |X| ≥ 2, a ∈ X, and let x̃ ∈ X̃ be a d-
statistical convergent to a. Then for every infinite subsequence x̃′ of x̃ with
lim sup
n→∞
∣∣Kx̃′(n)
∣∣
n
= 0
there are a sequence ỹ ∈ X̃ and a subsequence ỹ′ of ỹ such that:
(i) ỹ � x̃ and Kx̃′ = Kỹ′ ;
(ii) ỹ is d-statistical convergent to a;
(iii) ỹ′ is not d-statistical convergent.
Proof. By Lemma 3.1 there are ỹ and ỹ′ such that (i) and (iii) holds. To prove (ii) note that
ỹ � x̃ by (i) and x̃ is d-statistical convergent to a. Consequently, by Lemma 3.2, ỹ is also d-statistical
convergent to a.
Theorem 3.3 is proved.
Using this theorem we obtain the following “weak” converse of Theorem 3.1.
Theorem 3.4. Let (X, d) be a metric space with |X| ≥ 2 and let x̃ ∈ X̃ be a d-statistical
convergent sequence. Assume x̃′ is a subsequence of x̃ having the following property: if ỹ � x̃ and
ỹ′ is a subsequence of ỹ such that Kx̃′ = Kỹ′ , then ỹ′ is d-statistical convergent. Then the inequality
lim sup
n→∞
∣∣Kx̃′(n)
∣∣
n
> 0 (3.11)
holds.
Proof. We have either (3.11) or
lim sup
n→∞
∣∣Kx̃′(n)
∣∣
n
= 0.
If the last equality holds then, by Theorem 3.3, there are ỹ and ỹ′ such that ỹ � x̃ Kx̃′ = Kỹ′ and ỹ′
is not d-statistical convergent, contrary to the assumption.
Theorem 3.4 is proved.
Similarly we have a “weak” converse of Theorem 3.3.
Theorem 3.5. Let (X, d) be a metric space, a ∈ X, and let x̃ ∈ X̃ be a d-statistical convergent
to a sequence. Suppose x̃′ =
(
xn(k)
)
is a subsequence of x̃ for which there are ỹ ∈ X̃ and ỹ′ such
that conditions (i) and (iii) of Theorem 3.3 hold. Then we have the equality
lim inf
n→∞
∣∣Kx̃′(n)
∣∣
n
= 0. (3.12)
To prove this result we shall use the next lemma.
Lemma 3.3. Let (X, d) be a metric space, x̃ and ỹ belong to X̃ and let x̃ � ỹ. If K is a subset
of N such that
lim inf
n→∞
∣∣K(n)
∣∣
n
> 0 (3.13)
and if x̃′ =
(
xn(k)
)
and ỹ′ = (yn(k)) are subsequences of x̃ and, respectively, of ỹ such that
Kx̃′ = Kỹ′ = K, then the relation ỹ′ � x̃′ holds.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 5
ON STATISTICAL CONVERGENCE OF METRIC VALUED SEQUENCES 719
Proof. It is sufficient to show that
lim sup
m→∞
∣∣{n(k) ∈ K : xn(k) 6= yn(k), n(k) ≤ m}
∣∣∣∣K(m)
∣∣ = 0. (3.14)
Since the inclusion{
n(k) ∈ K : xn(k) 6= yn(k), n(k) ≤ m} ⊆ {n ∈ N : xn 6= yn, n ≤ m}
holds for each m ∈ N, we have
lim sup
m→∞
∣∣{n(k) ∈ K : xn(k) 6= yn(k), n(k) ≤ m}
∣∣
|K(m)|
≤
≤ lim sup
m→∞
∣∣{n ∈ N : xn 6= yn, n ≤ m}
∣∣∣∣K(m)
∣∣ ≤ lim sup
m→∞
m∣∣K(m)
∣∣ lim sup
m→∞
∣∣{n ∈ N : xn 6= yn, n ≤ m}
∣∣
m
=
= lim sup
m→∞
∣∣{n ∈ N : xn 6= yn, n ≤ m}
∣∣
m
(
lim inf
m→∞
∣∣K(m)
∣∣
m
)−1
. (3.15)
Inequality (3.13) implies that
0 ≤
(
lim inf
m→∞
∣∣K(m)
∣∣
m
)−1
< +∞. (3.16)
Moreover we have
lim sup
m→∞
∣∣{n ∈ N : xn 6= yn, n ≤ m}
∣∣
m
= 0
because x̃ � ỹ. Now (3.14) follows from the last equality, (3.15) and (3.16).
Lemma 3.3 is proved.
Proof of Theorem 3.5. Suppose that
lim inf
n→∞
∣∣Kx̃′(n)
∣∣
n
> 0. (3.17)
Let ỹ ∈ X̃ and let ỹ′ be a subsequence of ỹ such that conditions (i) and (iii) of Theorem 3.3 hold.
Then we have Kx̃′ = Kỹ′ and x̃ � ỹ. It follows from (3.17) and Lemma 3.3 that x̃′ � ỹ′. Moreover,
applying Theorem 3.1, we see that x̃′ is d-statistical convergent to a. Since x̃′ � ỹ′, Lemma 3.2
shows that ỹ′ is also d-statistical convergent to a, contrary to condition (iii) of Theorem 3.3. Hence
equality (3.12) holds.
Theorem 3.5 is proved.
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| id | umjimathkievua-article-2172 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:20:05Z |
| publishDate | 2014 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| resource_txt_mv | umjimathkievua/b1/1e245e7b9be583a836f07dd81203f9b1.pdf |
| spelling | umjimathkievua-article-21722019-12-05T10:25:31Z On the Statistical Convergence of Metric-Valued Sequences Про статистичну збіжність метричнозначних послідовностей Değer, U. Dovgoshei, A. A. Küçükaslan, M. Дегер, У. Довгошей, О. А. Куцукаслан, М. We study the conditions for the density of a subsequence of a statistically convergent sequence under which this subsequence is also statistically convergent. Some sufficient conditions of this type and almost converse necessary conditions are obtained in the setting of general metric spaces. Вивчаються умови на щільність підпослідовності статистично з6іжної послідовності, за яких ця підпослідовність також є статистично збіжною. Деякі достатні умови такого типу та майже обернені необхідні умови отримано в постановці загальних метричних просторів. Institute of Mathematics, NAS of Ukraine 2014-05-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2172 Ukrains’kyi Matematychnyi Zhurnal; Vol. 66 No. 5 (2014); 712–720 Український математичний журнал; Том 66 № 5 (2014); 712–720 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2172/1345 https://umj.imath.kiev.ua/index.php/umj/article/view/2172/1346 Copyright (c) 2014 Değer U.; Dovgoshei A. A.; Küçükaslan M. |
| spellingShingle | Değer, U. Dovgoshei, A. A. Küçükaslan, M. Дегер, У. Довгошей, О. А. Куцукаслан, М. On the Statistical Convergence of Metric-Valued Sequences |
| title | On the Statistical Convergence of Metric-Valued Sequences |
| title_alt | Про статистичну збіжність метричнозначних послідовностей |
| title_full | On the Statistical Convergence of Metric-Valued Sequences |
| title_fullStr | On the Statistical Convergence of Metric-Valued Sequences |
| title_full_unstemmed | On the Statistical Convergence of Metric-Valued Sequences |
| title_short | On the Statistical Convergence of Metric-Valued Sequences |
| title_sort | on the statistical convergence of metric-valued sequences |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2172 |
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