On statistical convergence of vector-valued sequences associated with multiplier sequences
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irk-123456789-1640502020-02-09T01:25:55Z On statistical convergence of vector-valued sequences associated with multiplier sequences Altinok, H. Et, M. Gökhan, A. Короткі повідомлення 2006 Article On statistical convergence of vector-valued sequences associated with multiplier sequences / H.Altinok, M. Et, A. Gökhan // Український математичний журнал. — 2006. — Т. 58, № 1. — С. 125–131. — Бібліогр.: 24 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/164050 517.5 en Український математичний журнал Інститут математики НАН України |
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Короткі повідомлення Короткі повідомлення Altinok, H. Et, M. Gökhan, A. On statistical convergence of vector-valued sequences associated with multiplier sequences Український математичний журнал |
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Altinok, H. Et, M. Gökhan, A. |
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Altinok, H. Et, M. Gökhan, A. |
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Altinok, H. |
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On statistical convergence of vector-valued sequences associated with multiplier sequences |
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On statistical convergence of vector-valued sequences associated with multiplier sequences |
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On statistical convergence of vector-valued sequences associated with multiplier sequences |
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On statistical convergence of vector-valued sequences associated with multiplier sequences |
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On statistical convergence of vector-valued sequences associated with multiplier sequences |
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on statistical convergence of vector-valued sequences associated with multiplier sequences |
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Інститут математики НАН України |
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On statistical convergence of vector-valued sequences associated with multiplier sequences / H.Altinok, M. Et, A. Gökhan // Український математичний журнал. — 2006. — Т. 58, № 1. — С. 125–131. — Бібліогр.: 24 назв. — англ. |
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Український математичний журнал |
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AT altinokh onstatisticalconvergenceofvectorvaluedsequencesassociatedwithmultipliersequences AT etm onstatisticalconvergenceofvectorvaluedsequencesassociatedwithmultipliersequences AT gokhana onstatisticalconvergenceofvectorvaluedsequencesassociatedwithmultipliersequences |
| first_indexed |
2025-07-14T16:36:21Z |
| last_indexed |
2025-07-14T16:36:21Z |
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1837640956154740736 |
| fulltext |
UDC 517.5
M. Et, A. Gökhan, H. Altinok (Fırat Univ., Turkey)
ON STATISTICAL CONVERGENCE OF VECTOR-VALUED
SEQUENCES ASSOCIATED WITH MULTIPLIER SEQUENCES
PRO STATYSTYÇNU ZBIÛNIST\ VEKTORNOZNAÇNYX
POSLIDOVNOSTEJ, WO POV’QZANI
Z KOEFICI{NTNYMY POSLIDOVNOSTQMY
In this paper we introduce the vector-valued sequence spaces w∞(F, Q, p, u), w1(F, Q, p, u), w0(F, Q, p, u),
Sq
u, and Sq
0u using a sequence of modulus functions and the multiplier sequence u = (uk) of nonzero complex
numbers. We give some relations related to these sequence spaces. It is also shown that if a sequence is strongly
uq-Cesàro summable with respect to the modulus function then it is uq-statistically convergent.
Vvedeno prostory vektornoznaçnyx poslidovnostej w∞(F, Q, p, u), w1(F, Q, p, u), w0(F, Q, p, u),
Sq
u ta Sq
0u z vykorystannqm poslidovnosti modul\-funkcij i koefici[ntno] poslidovnosti u = (uk)
nenul\ovyx kompleksnyx çysel. Navedeno deqki spivvidnoßennq, wo stosugt\sq cyx prostoriv posli-
dovnostej. TakoΩ pokazano, wo qkwo poslidovnist\ syl\no uq-Çezaro-sumovna po vidnoßenng do
modul\-funkci], to vona uq-statystyçno zbiΩna.
1. Introduction. Let w be the set of all sequences of real or complex numbers and �∞,
c, and c0 be, respectively, the Banach spaces of bounded, convergent, and null sequences
x = (xk) with the usual norm ‖x‖ = sup |xk|, where k ∈ N = {1, 2, . . .} is the set of
positive integers.
Studies on vector-valued sequence spaces were carried out by Rath and Srivastava [1],
Das and Choudhary [2], Leonard [3], Srivastava and Srivastava [4], Tripathy and Sen [5],
Tripathy and Mahanta [6], and many others.
Throughout the article, for all k ∈ N Ek are seminormed spaces seminormed by qk
and X is a seminormed space seminormed by q. If what follows, w(Ek), c(Ek), �∞(Ek),
and �p(Ek) denote the spaces of all, convergent, bounded, and p-absolutely summable
Ek-valued sequences, respectively. In the case where Ek = C (the field of complex
numbers) for all k ∈ N, one has the corresponding scalar-valued sequence spaces. The
zero elements of Ek are denoted by θk. The zero sequence is denoted by θ̄ = (θk).
Let u = (uk) be a sequence of nonzero scalar. Then for a sequence space E, the
multiplier sequence space E(u) associated with the multiplier sequence u is defined as
E(u) = {(xk) ∈ w : (ukxk) ∈ E} .
Studies on the multiplier sequence spaces were carried out by Çolak [7], Çolak et
al. [8], Srivastava and Srivastava [4], Tripathy and Mahanta [6], and many others.
The notion of a modulus was introduced by Nakano [9]. We recall that a modulus f
is a function from [0,∞) to [0,∞) such that:
i) f(x) = 0 if and only if x = 0,
ii) f(x+ y) ≤ f(x) + f(y) for x, y ≥ 0,
iii) f is increasing,
iv) f is continuous from the right at 0.
It follows that f must be continuous everwhere on [0,∞). A modulus may be un-
bounded or bounded. Ruckle [10], Maddox [11] used a modulus f to construct some
sequence spaces.
2. Main results. In this section, we prove some results involving the sequence spaces
w0(F,Q, p, u), w1(F,Q, p, u), and w∞(F,Q, p, u).
c© M. ET, A. GÖKHAN, H. ALTINOK, 2006
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 1 125
126 M. ET, A. GÖKHAN, H. ALTINOK
Definition 1. Let p = (pk) be a sequence of strictly positive real numbers, let F =
= (fk) be a sequence of modulus functions, and let u = (uk) be any fixed sequence of
nonzero complex numbers uk. We define the following sequence spaces:
w0(F,Q, p, u) =
{
xk ∈ Ek :
1
n
n∑
k=1
[fk (qk(ukxk))]pk → 0, as n → ∞
}
,
w1(F,Q, p, u) =
xk ∈ Ek :
1
n
n∑
k=1
[fk (qk (ukxk − �))]pk → 0,
as n → ∞ and � ∈ Ek
,
w∞(F,Q, p, u) =
{
xk ∈ Ek : sup
n
1
n
n∑
k=1
[fk (qk(ukxk))]pk < ∞
}
.
In the case where fk = f and qk = q for all k ∈ N, we shall write w0(f, q, p, u),
w1(f, q, p, u), and w∞(f, q, p, u) instead of w0(F,Q, p, u), w1(F,Q, p, u), and w∞(F,
Q, p, u), respectively.
Throughout the paper, Z will denote any one of the notation 0, 1, or ∞.
If x ∈ w1(f, q, p, u), we say that x is strongly uq-Cesàro summable with respect to
the modulus function f and we will write xk → �(w1(f, q, p, u)); � will be called uq-limit
of x with respect to the modulus f.
The proofs of the following theorems are obtained by using the known standard tech-
niques, therefore we give them without proofs.
Theorem 1. Let the sequence (pk) be bounded. Then the spaces wZ(F,Q, p, u) are
linear spaces.
Theorem 2. Let f be a modulus function and the sequence (pk) be bounded, then
w0(f, q, p, u) ⊂ w1(f, q, p, u) ⊂ w∞(f, q, p, u)
and the inclusions are strict.
Theorem 3. w0(F,Q, p, u) is a paranormed (need not total paranorm) space with
g (x) = sup
n
(
1
n
n∑
k=1
[fk (qk(ukxk))]pk
)1
M
, (1)
where M = max(1, sup pk).
Theorem 4. Let F = (fk) and G = (gk) be any two sequences of modulus functions.
For any bounded sequences p = (pk) and t = (tk) of strictly positive real numbers and
for any two sequences of seminorms q = (qk) and r = (rk), we have:
i) wZ(f,Q, u) ⊂ wZ(f ◦ g,Q, u),
ii) wZ(F,Q, p, u) ∩ wZ(F,R, p, u) ⊂ wZ(F,Q+R, p, u),
iii) wZ(F,Q, p, u) ∩ wZ(G,Q, p, u) ⊂ wZ(F +G,Q, p, u),
iv) if q is stronger than r, then wZ(F,Q, p, u) ⊂ wZ(F,R, p, u),
v) if q is equivalent to r, then wZ(F,Q, p, u) = wZ(F,R, p, u),
vi) wZ(F,Q, p, u) ∩ wZ(F,R, p, u) �= ∅.
Proof. i) We shall only prove i) for Z = 0, and the other cases can be proved by using
similar arguments. Let ε > 0. We choose δ, 0 < δ < 1, such that f(t) < ε for 0 ≤ t ≤ δ
and all k ∈ N. Write yk = g (qk(ukxk)) and consider
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ON STATISTICAL CONVERGENCE OF VECTOR-VALUED SEQUENCES ASSOCIATED ... 127
n∑
k=1
[f(yk)] =
∑
1
[f(yk)] +
∑
2
[f(yk)] ,
where the first summation is over yk ≤ δ and second summation is over yk > δ. Since f
is continuous, we have ∑
1
[f(yk)] < nε. (2)
By the definition of f, we have the following relation for yk > δ :
f(yk) < 2f(1)
yk
δ
.
Hence,
1
n
∑
2
[f(yk)] ≤ 2δ−1f(1)
1
n
n∑
k=1
yk. (3)
It follows from (2) and (3) that w0(f,Q, u) ⊂ w0(f ◦ g,Q, u).
The following result is a consequence of Theorem 4 (i).
Proposition 1. Let fbe a modulus function. Then wZ(Q, u) ⊂ wZ(f,Q, u).
Theorem 5. Let Ek be a complete seminormed space for each k ∈ N. Then the
sequence space w0(F,Q, p, u) is complete and seminormed by (1).
Proof. Let (xit) be a Cauchy sequence in w0(F,Q, p, u), where xi = (xi
k)∞k=1. Then
g(xi − xj) → 0, as i, j → ∞. (4)
Hence, for each fixed k, we have[
fk
(
qk
(
uk
(
xi
k − xj
k
)))]pk
→ 0 as i, j → ∞.
By continuity of fk for all k ∈ N, we have
lim
i,j→∞
[
fk
(
qk
(
uk
(
xi
k − xj
k
)))]pk
=
[
fk
(
lim
i,j→∞
qk
(
ukx
i
k − ukx
j
k
))]pk
= 0.
Since fk is a modulus for all k ∈ N,
lim
i,j→∞
qk
(
ukx
i
k − ukx
j
k
)
= 0.
Let yi
k = ukx
i
k for all k ∈ N. Then
(
yi
k
)∞
i=1
is a Cauchy sequence in Ek for each
k ∈ N. Since Ek are complete, there exists yk ∈ Ek such that yi
k → yk as i → ∞ for all
k ∈ N. Since Ek are linear, we can express yk as yk = ukxk, where k ∈ N.
Since g is continuous, taking j → ∞ in ( 4), we have g(xi − x) < ε for all i ≥ n0.
Hence,
g(xi − x) ∈ w0(F,Q, p, u) for all i ≥ n0.
Since (xi−x), (xit) ∈ w0(F,Q, p, u), and the space w0(F,Q, p, u) is linear, we have
x = xi − (xi − x) ∈ w0(F,Q, p, u). Hence w0(F,Q, p, u) is complete.
Theorem 6. Let 0 < pk ≤ tk and let
(
tk
pk
)
be bounded. Then wZ(F,Q, t, u) ⊂
⊂ wZ(F,Q, p, u).
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 1
128 M. ET, A. GÖKHAN, H. ALTINOK
Proof. By taking wk = [fk (qk(ukxk))]tk for all k and using the same technique as
in Theorem 5 of Maddox [12], one can easily prove the theorem.
Theorem 7. Let f be a modulus function. If lim
t→∞
f(t)
t
= β > 0, then w1(Q, p, u) =
= w1(f, q, p, u).
Proof. Omitted.
3. uq-Statistical convergence. The notion of statistical convergence was introduced
by Fast [13] and Schoenberg [14] independently. Over the years and under different
names, statistical convergence has been discussed in the theory of Fourier analysis, er-
godic theory, number theory. Later on, it was further investigated from sequence space
point of view and linked with summability theory by Fridy [15], Connor [16], Šalát [17],
Mursaleen [18], Işık [19], Savaş [20], Malkowsky and Savaş [21], Kolk [22], Maddox
[23], Tripathy and Sen [24], and many others. In recent years, generalizations of sta-
tistical convergence have appeared in the study of strong integral summability and the
structure of ideals of bounded continuous functions on locally compact spaces. Statistical
convergence and its generalizations are also connected with subsets of the Stone – Čech
compactification of the natural numbers. Moreover, statistical convergence is closely re-
lated to the concept of convergence in probability. The notion depends on the density of
subsets of the set N of natural numbers.
A subset E of N is said to have density positive integers is defined by δ (E) if
δ (E) = lim
n→∞
1
n
n∑
k=1
χE(k)
exists, where χE is the characteristic function of E. It is clear that any finite subset of N
have zero natural density and δ (Ec) = 1 − δ (E) .
In this section, we introduce uq-statistically convergent sequences and give some in-
clusion relations between uq-statistically convergent sequences and w1(f, q, p, u)-sum-
mable sequences.
Definition 2. A sequence x = (xk) is said to be uq-statistically convergent to � if, for
every ε > 0,
δ
(
{k ∈ N : q(ukxk − �) ≥ ε}
)
= 0.
In this case, we write xk → � (Sq
u) . The set of all uq-statistically convergent sequences is
denoted by Sq
u.
By S, we denote the set of all statistically convergent sequences. If q (x) = |x| and
uk = 1 for all k ∈ N, then Sq
u is the same as S. In the case � = 0, we shall write Sq
0u
instead of Sq
u.
Theorem 8. Let fbe a modulus function. Then:
i) if xk → � (w1(Q, u)) , then xk → � (Sq
u),
ii) if x ∈ �∞ (uq) and xk → � (Sq
u) , then xk → � (w1(Q, u)),
iii) Sq
u ∩ �∞ (uq) = w1(Q, u) ∩ �∞ (uq) ,
where �∞ (uq) = {x ∈ w(X) : supk q(ukxk) < ∞} .
Proof. Omitted.
In the following theorems, we shall assume that the sequence p = (pk) is bounded
and 0 < h = infk pk ≤ pk ≤ supk pk = H < ∞.
Theorem 9. Let fbe a modulus function. Then w1(f, q, p, u) ⊂ Sq
u.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 1
ON STATISTICAL CONVERGENCE OF VECTOR-VALUED SEQUENCES ASSOCIATED ... 129
Proof. Let x ∈ w1(f, q, p, u) and let ε > 0 be given. Let
∑
1
and
∑
2
denote the
sums over k ≤ n with q (ukxk − �) ≥ ε and q (ukxk − �) < ε, respectively. Then
1
n
n∑
k=1
[f (q (ukxk − �))]pk ≥
≥ 1
n
∑
1
[f (q (ukxk − �))]pk ≥ 1
n
∑
1
[f (ε)]pk ≥
≥ 1
n
∑
1
min
(
[f (ε)]h , [f (ε)]H
)
≥
≥ 1
n
∣∣∣{k ≤ n : q (ukxk − �) ≥ ε
}∣∣∣ min
(
[f (ε)]h , [f (ε)]H
)
.
Hence, x ∈ Sq
u.
Theorem 10. Let fbe bounded. Then Sq
u ⊂ w1(f, q, p, u).
Proof. Suppose that f is bounded. Let ε > 0 and let
∑
1
and
∑
2
be the sums
introduced in previous theorem. Since f is bounded, there exists an integer K such that
f (x) < K for all x ≥ 0. Then
1
n
n∑
k=1
[
f (q (ukxk − �))
]pk
≤
≤ 1
n
(∑
1
[
f (q (ukxk − �))
]pk
+
∑
2
[
f (q (ukxk − �))
]pk
)
≤
≤ 1
n
∑
1
max(Kh,KH) +
1
n
∑
2
[
f(ε)
]pk ≤
≤ max(Kh,KH)
1
n
∣∣∣{k ≤ n : q (ukxk − �) ≥ ε
}∣∣∣ +
+ max
(
f(ε)h, f(ε)H
)
.
Hence, x ∈ w1(f, q, p, u).
Theorem 11. Sq
u = w1(f, q, p, u) if and only if f is bounded.
Proof. Let f be bounded. By Theorems 9 and 10, we have Sq
u = w1(f, q, p, u).
Conversely, suppose that f is unbounded. Then there exists a sequence (tk) of positive
numbers with f(tk) = k2 for k = 1, 2, . . . . If we choose
uixi =
{
tk, i = k2, k = 1, 2, . . . ,
0, otherwise,
then we have
1
n
∣∣∣ {
k ≤ n :
∣∣∣ukxk
∣∣∣ ≥ ε
}∣∣∣ ≤ √
n
n
for all n and so x ∈ Sq
u, but x /∈ w1(f, q, p, u) for X = C , q (x) = |x| and pk = 1 for all
k ∈ N. This contradicts to Sq
u = w1(f, q, p, u).
4. Special cases. Firstly, we note that w∞(F,Q, p, u) and w∞(F,Q, p) overlap but
neither one contains the other. For example, pk = 1, fk(x) = x, and qk(x) = |x|
for all k ∈ N. If we choose x = (1) and u = (k), then x ∈ w∞(F,Q, p), but
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 1
130 M. ET, A. GÖKHAN, H. ALTINOK
x /∈ w∞(F,Q, p, u), conversely, if we choose x = (k) and u =
(
1
k
)
, then x /∈
/∈ w∞(F,Q, p), but x ∈ w∞(F,Q, p, u). Similarly:
i) w0(F,Q, p, u) and w0(F,Q, p),
ii) w1(F,Q, p, u) and w1(F,Q, p),
iii) Sq
u and Sq,
iv) Sq
0u and Sq
0
overlap but neither one contains the other.
The definition of v-invariance of a sequence spaces E was given by Çolak [7] and the
v-invariantness of the sequence spaces �∞, c, c0, and �p was examined.
Definition 3. Let X be any sequence space and u = (uk) be any sequence of nonzero
complex numbers. We say that the sequence space X is uq-invariant if Xq
u = Xq.
By E[u], we denote one of the sequence spaces w∞(F,Q, p, u), w1(F,Q, p, u),
w0(F,Q, p, u), Sq
u, S
q
0u, and also, by E, we denote one of the sequence spaces
w∞(F,Q, p), w1(F,Q, p), w0(F,Q, p), Sq, Sq
0 . What conditions should satisfy u = (uk)
in order that E[u] = E?
If one considers the sequnce spaces:
1) wZ(f, q, p, u) instead of wZ(F,Q, p, u),
2) wZ(f,Q, p, u) instead of wZ(F,Q, p, u),
3) wZ(F, q, p, u) instead of wZ(F,Q, p, u),
4) wZ(F,Q, p) instead of wZ(F,Q, p, u),
5) wZ(F,Q, u) instead of wZ(F,Q, p, u),
6) wZ (F,Q) instead of wZ(F,Q, p, u),
7) wZ (F, p, u) instead of wZ(F,Q, p, u),
8) wZ (Q, p, u) instead of wZ(F,Q, p, u),
9) wZ (p, u) instead of wZ(F,Q, p, u),
10) Sq and Sq
0 instead of Sq
u and Sq
0u,
11) Su and S0u instead of Sq
u and Sq
0u,
one will get that most of the results proved in the previous sections will be true for these
spaces too.
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Received 16.05.2005
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