Some approximation properties of Szasz–Mirakyan–Bernstein operators of the Chlodovsky type
We motivate a new sequence of positive linear operators by means of the Chlodovsky-type Szasz–Mirakyan–Bernstein operators and investigate some approximation properties of these operators in the space of continuous functions defined on the right semiaxis. We also find the order of this approximation...
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irk-123456789-1660812020-02-19T01:26:17Z Some approximation properties of Szasz–Mirakyan–Bernstein operators of the Chlodovsky type Tunc, T. Simsek, E. Статті We motivate a new sequence of positive linear operators by means of the Chlodovsky-type Szasz–Mirakyan–Bernstein operators and investigate some approximation properties of these operators in the space of continuous functions defined on the right semiaxis. We also find the order of this approximation by using the modulus of continuity and present the Voronovskaya-type theorem. Метою даної статті є обґрунтування нової послідовності додатних лінійних onepaTopiB за допомогою onepaTopiB Сaсa-Мiрaкянa-Бернштейнa типу Хлодовського та дослідження деяких апроксимаційних властивостей цих операторів у просторі неперервних функцій, заданих на правій півосі. Крім того, встановлено порядок таких наближень за допомогою модуля неперервності та наведено теорему типу Вороновської. 2014 Article Some approximation properties of Szasz–Mirakyan–Bernstein operators of the Chlodovsky type / T. Tunc, E. Simsek // Український математичний журнал. — 2014. — Т. 66, № 6. — С. 826–834. — Бібліогр.: 12 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/166081 517.5 en Український математичний журнал Інститут математики НАН України |
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Статті Статті Tunc, T. Simsek, E. Some approximation properties of Szasz–Mirakyan–Bernstein operators of the Chlodovsky type Український математичний журнал |
| description |
We motivate a new sequence of positive linear operators by means of the Chlodovsky-type Szasz–Mirakyan–Bernstein operators and investigate some approximation properties of these operators in the space of continuous functions defined on the right semiaxis. We also find the order of this approximation by using the modulus of continuity and present the Voronovskaya-type theorem. |
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Article |
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Tunc, T. Simsek, E. |
| author_facet |
Tunc, T. Simsek, E. |
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Tunc, T. |
| title |
Some approximation properties of Szasz–Mirakyan–Bernstein operators of the Chlodovsky type |
| title_short |
Some approximation properties of Szasz–Mirakyan–Bernstein operators of the Chlodovsky type |
| title_full |
Some approximation properties of Szasz–Mirakyan–Bernstein operators of the Chlodovsky type |
| title_fullStr |
Some approximation properties of Szasz–Mirakyan–Bernstein operators of the Chlodovsky type |
| title_full_unstemmed |
Some approximation properties of Szasz–Mirakyan–Bernstein operators of the Chlodovsky type |
| title_sort |
some approximation properties of szasz–mirakyan–bernstein operators of the chlodovsky type |
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Інститут математики НАН України |
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2014 |
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Статті |
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http://dspace.nbuv.gov.ua/handle/123456789/166081 |
| citation_txt |
Some approximation properties of Szasz–Mirakyan–Bernstein operators of the Chlodovsky type / T. Tunc, E. Simsek // Український математичний журнал. — 2014. — Т. 66, № 6. — С. 826–834. — Бібліогр.: 12 назв. — англ. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT tunct someapproximationpropertiesofszaszmirakyanbernsteinoperatorsofthechlodovskytype AT simseke someapproximationpropertiesofszaszmirakyanbernsteinoperatorsofthechlodovskytype |
| first_indexed |
2025-07-14T20:43:29Z |
| last_indexed |
2025-07-14T20:43:29Z |
| _version_ |
1837656510155456512 |
| fulltext |
UDC 517.5
T. Tunc, E. Simsek (Mersin Univ., Turkey)
SOME APPROXIMATION PROPERTIES
OF SZASZ – MIRAKYAN – BERNSTEIN OPERATORS
OF CHLODOVSKY-TYPE
ДЕЯКI АПРОКСИМАЦIЙНI ВЛАСТИВОСТI ОПЕРАТОРIВ
САСА – МIРАКЯНА – БЕРНШТЕЙНА ТИПУ ХЛОДОВСЬКОГО
The aim of this paper is to motivate a new sequence of positive linear operators by means of Chlodovsky-type Szasz –
Mirakyan – Bernstein operators and to investigate some approximation properties of these operators in the space of conti-
nuous functions defined on the right semiaxis. We also find the order of this approximation by using the modulus of
continuity and give the Voronovskаya-type theorem.
Метою даної статтi є обґрунтування нової послiдовностi додатних лiнiйних операторiв за допомогою операторiв
Саса – Мiракяна – Бернштейна типу Хлодовського та дослiдження деяких апроксимацiйних властивостей цих опера-
торiв у просторi неперервних функцiй, заданих на правiй пiвосi. Крiм того, встановлено порядок таких наближень
за допомогою модуля неперервностi та наведено теорему типу Вороновської.
1. Introduction. Let N denotes the set of natural numbers and let N0 = N∪{0}. Let f be real-valued
function defined on the closed interval [0, 1]. The nth Bernstein operator of f, Bn(f) is defined as
Bn (f ;x) =
n∑
k=0
pn,k(x)f
(
k
n
)
, x ∈ [0, 1], n ∈ N,
where
pn,k(x) =
(
n
k
)
xk(1− x)n−k, 0 ≤ k ≤ n. (1)
The Bernstein polynomials Bn(f) was introduced to prove the Weierstrass approximation theorem by
S. N. Bernstein [1] in 1912. They have been studied intensively and their connection with different
branches of analysis, such as convex and numerical analysis, total positivity and the theory of mono-
tone operators have been investigated. Basic facts on Bernstein polynomials and their generalizations
can be found in [2, 7 – 9] and references therein.
In 1937, I. Chlodovsky [3] introduced a generalization of the Bernstein polynomials for un-
bounded intervals. This generalization is named as the Bernstein – Chlodovsky polynomials in the
literature and have the following form:
Cn (f ;x) =
n∑
k=0
pn,k
(
x
bn
)
f
(
k
n
bn
)
, x ∈ [0, bn], n ∈ N0, (2)
where pn,k is defined in (1) and (bn) is a increasing sequence of positive real numbers such that
limn→∞ bn =∞, limn→∞ bn/n = 0. If we take the case bn = 1, n ∈ N0, these polynomials become
the classical Bernstein polynomials. The approximation properties of the Bernstein – Chlodovsky
polynomials can be found in [2, 3, 5, 6].
c© T. TUNC, E. SIMSEK, 2014
826 ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 6
SOME APPROXIMATION PROPERTIES OF SZASZ – MIRAKYAN – BERNSTEIN OPERATORS . . . 827
For the function f which is continuous on [0,∞), the Szasz – Mirakyan operators which are
introduced by G. M. Mirakyan [10] in 1941 and then, are investigated by J. Favard [11] and O. Szasz
[4], are defined as
Sn(f ;x) =
∞∑
m=0
qn,m(x)f
(m
n
)
, x ∈ [0,∞), n ∈ N,
where
qn,m(x) = e−nx
(nx)m
m!
, m ∈ N0. (3)
Let I is a fixed interval (bounded or not) in R and µm be a sequence of density functions on the
interval I, that is, the functions µm have the following properties:
(i) µm nonnegative for all x ∈ I and m ∈ N0,
(ii)
∑∞
m=0
µm(x) = 1 for all x ∈ I.
Let (Ln) be a sequence of positive linear operators defined on the set of the continuous functions
on the interval I, say C(I). Now we defined the new operators Ln on C(I) by
Ln(f ;x) =
∞∑
m=0
µm(nx)Lϕ(f ;x), x ∈ I, m ∈ N0, n ∈ N, (4)
where µn are density functions on I and ϕ := ϕ(n,m) = αnβm where (αn) is a nondecreasing and
(βm) is a strictly increasing natural sequence. It is easy to check that the operators (Ln) are positive
and linear on C(I).
Taking βm = m + 1, µm = qn,m and Lϕ = Cϕ, where Cϕ and qn,m defined in (2) and (3)
respectively, we can rewrite (4) as
En(f ;x) =
∞∑
m=1
e−nx(nx)m−1
(m− 1)!
αnm∑
k=0
(
αnm
k
)(
x
bn
)k (
1− x
bn
)αnm−k
f
(
kbn
mαn
)
. (5)
The operators En defined in (5) is called the Szasz – Mirakyan – Bernstein operators of Chlodovsky-
type (SMBC). In this study, we investigate some approximation properties of these operators and find
Voronovskya-type theorem and the order of this approximation by using modulus of continuity.
2. Notations and auxiliary facts. Let I = [0,∞), and let C(I) be the space of real-valued
continuous function on I equipped with the uniform norm:
‖f‖I := sup{|f(x)| : x ∈ I}
and Cr(I), r ∈ N0, be the set all r-times continuously differentiable functions f ∈ C(I).
For the real-valued function f defined on I and δ ≥ 0, the modulus of continuity ω(f, δ) of f
with argument δ is defined by
ω(f, δ) := sup{|f(x+ h)− f(x)| : x, x+ h ∈ I, |h| < δ}.
For M > 0 and 0 < µ ≤ 1, the class of the function C(I) satisfying the relation ω(f, δ) ≤Mδµ
for all δ ≥ 0, is called Lipschitz class and denoted by LipM µ.
Let er, r ∈ N0, denote the test functions defined by er(x) = xr and xr denote the functions
defined by xr(t) = (t − x)r for the fixed real numbers x. By the simple calculations we have the
following lemma.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 6
828 T. TUNC, E. SIMSEK
Lemma 2.1. For all x ∈ [0, bn], n ∈ N, we have
En(e0;x) = 1, En(e1;x) = x, En(e2;x) = x2 +
(bn − x)(1− e−nx)
nαn
and for the central moments, we have
En(x0;x) = 1, En(x1;x) = 0, En(x2;x) =
(bn − x)(1− e−nx)
nαn
,
En(x3;x) =
(bn − x)(bn − 2x)
nα2
n
∞∑
m=1
e−nx(nx)m
m.m!
,
En(x4;x) =
(bn − x)(6x2 − 6bnx+ b2n)
nα3
n
∞∑
m=1
e−nx(nx)m
m2m!
+
3x(bn − x)2
nα2
n
∞∑
m=1
e−nx(nx)m
m.m!
.
By using Lemma 2.1, we have the following estimation for En(x4) at the point x ∈ (0, bn]:
En(x4;x) ≤ cn(x)
(
bn
nαn
)2
, (6)
where limn→∞ cn(x) = 6.
3. Convergence of sequence of the operators En. In this section, we assume that f is a function
defined on the semiaxis [0,∞). The aim of this section is to preserve the relation
lim
n→∞
En(f ;x) = f(x), x ∈ [0,∞),
for the reasonably general classes of functions.
Theorem 3.1. If bn = o(n), and the function f is bounded on the semiaxis [0,∞), then
lim
n→∞
En(f ;x) = f(x) (7)
holds at any continuity point x of the function f.
Proof. Let ε > 0 and let x ∈ [0,∞) be a continuity point of the function f, then there exist a
δ > 0 such that |f(t)− f(x)| < ε holds for all t ∈ [0,∞) satisfying the inequality |t− x| < δ. Since
the relation (7) is clear for x = 0, we assume that x > 0. Let N ∈ N such that bN ≥ x so that for all
n ≥ N, bn ≥ x. For n ≥ N, by using Lemma 2.1 we have
|En(f ;x)− f(x)| ≤
∞∑
m=1
qn,m−1(x)
αnm∑
k=0
pmαn,k
(
x
bn
) ∣∣∣∣f ( kbn
mαn
)
− f(x)
∣∣∣∣ ≤
≤
∞∑
m=1
qn,m−1(x)
∑
∣∣∣ kbnmαn
−x
∣∣∣<δ
pmαn,k
(
x
bn
) ∣∣∣∣f ( kbn
mαn
)
− f(x)
∣∣∣∣+
+
∞∑
m=1
qn,m−1(x)
∑
∣∣∣ kbnmαn
−x
∣∣∣≥δ
pmαn,k
(
x
bn
) ∣∣∣∣f ( kbn
mαn
)
− f(x)
∣∣∣∣ .
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 6
SOME APPROXIMATION PROPERTIES OF SZASZ – MIRAKYAN – BERNSTEIN OPERATORS . . . 829
Since f is bounded on [0,∞) there is a number M > 0 such that |f(t)| ≤M, for all t ∈ [0,∞).
Therefore, putting t = xb−1n and using Lemma 2.1 and the inequality (7) in [2, p. 6], we obtain
|En(f ;x)− f(x)| ≤ εEn(e0;x) + 2M
∞∑
m=1
qn,m−1(x)
∑
∣∣∣ kbnαnm
−x
∣∣∣≥δ
pαnm,k
(
x
bn
)
=
= ε+ 2M
∞∑
m=1
e−ntbn(ntbn)m−1
(m− 1)!
∑
∣∣∣ k
αnm
−t
∣∣∣≥ δ
bn
pαnm,k(t) ≤
≤ ε+ 2M
t(1− t)
αn(δb−1n )2
∞∑
m=1
e−ntbn(ntbn)m−1
(m− 1)!m
≤
≤ ε+ 2M
(bn − x)
nαnδ2
(1− e−nx) ≤
≤ ε+
2Mbn
δ2nαn
since bn = o(n), we get the desired assertion.
The Theorem 3.1 also remains true for unbounded functions which do not grow too rapidly for
x→∞. Let M(bn) = ‖f‖[0,bn]. We will use the following lemma due to Albrycht and Radecki [12]
for the proof of the following theorem.
Lemma 3.1 [12]. For 0 < δ ≤ x < bn and sufficiently large n, we have∑
| kbnn −x|≥δ
pn,k
(
x
bn
)
≤ 2 exp
(
− δ2n
4xbn
)
.
Theorem 3.2. Let x ∈ (0,∞) be a continuity point of the function f. If bn = o(n) and
lim
n→∞
M(bn)e
− δ
2αn
4xbn
−nx
(
1−exp
(
− δ
2αn
4xbn
))
= 0, (8)
then (7) holds.
Proof. Using the inequality in the proof of Theorem 3.1 with M(bn) instead of M and the
Lemma 3.1, we have
|En(f ;x)− f(x)| ≤ ε+ 2M(bn)
∞∑
m=1
qn,m−1(x)
∑
∣∣∣ kbnmαn
−x
∣∣∣≥δ
pαnm,k
(
x
bn
)
=
= ε+ 4M(bn)
∞∑
m=1
e−nx(nx)m−1
(m− 1)!
e
− δ
2mαn
4xbn =
= ε+ 4M(bn)e−nxenxe
−
δ2αn
4xbn e
− δ
2αn
4xbn =
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 6
830 T. TUNC, E. SIMSEK
= ε+ 4M(bn)e
− δ
2αn
4xbn
−nx
(
1−exp
(
− δ
2αn
4xbn
))
.
Remark 3.1. If we assume αn = O(1) in Theorem 3.2, then the condition (8) is equal to the
condition
lim
n→∞
M(bn)e
−γ nbn , γ =
δ2
4x
,
which is the same with the condition for Bernstein operators of Chlodovsky-type [3].
Remark 3.2. If we assume αn = o(bn), the condition
lim
n→∞
M(bn)e
−γ nbn , γ =
δ2
4x
,
and if αn = O(bn), the condition
lim
n→∞
M(bn)e−γn, γ =
δ2
4x
,
is equal to the condition (8).
In view of the remarks, we conclude the following: Under the condition αn = O(bλn), λ > 0, for
increasing values of λ, the relation (7) is satisfied by larger class of functions.
4. Voronovskya-type theorem.
Theorem 4.1. Let f be defined on (0,∞) and satisfies the growth condition
lim
n→∞
M(bn)
nαn
bn
e
− δ
2αn
4xbn
−nx
(
1−exp
(
− δ
2αn
4xbn
))
= 0, δ > 0, x ∈ (0,∞). (9)
Then we have
lim
n→∞
nαn
bn
[En(f ;x)− f(x)] =
1
2
f ′′(x) (10)
at each point x > 0 for which f ′′(x) exists.
Proof. Let x ≤ bn and f has the second derivative at x. Then, by Taylor’s formula, we have
f(t) = f(x) + (t− x)f ′(x) + (t− x)2
[
f ′′(x)
2
+ h(t− x)
]
(11)
where h(ξ) tends to zero with ξ. Applying En to the formula (11), by Lemma 2.1, we obtain
En(f ;x) = f(x) +
1
2
f ′′(x)
(bn − x)(1− e−nx)
nαn
+Rn(x),
where
Rn(x) :=
∞∑
m=1
qn,m−1(x)
αnm∑
k=0
pmαn,k
(
x
bn
)(
kbn
mαn
− x
)2
h
(
kbn
mαn
− x
)
.
To complete the proof, we have to prove that
lim
n→∞
nαn
bn
Rn(x) = 0.
For any ε > 0 there exists a δ > 0 such that |h(ξ)| < ε for |ξ| < δ, and we choose δ so small that
δ ≤ x. Let us split the second sum in Rn(x) into two parts as follows:
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 6
SOME APPROXIMATION PROPERTIES OF SZASZ – MIRAKYAN – BERNSTEIN OPERATORS . . . 831
Rn(x) =
∞∑
m=1
qn,m−1(x)
∑∣∣ kbn
mαn
−x
∣∣<δ+
∑∣∣ kbn
mαn
−x
∣∣≥δ
pmαn,k×
×
(
x
bn
)(
kbn
mαn
− x
)2
h
(
kbn
mαn
− x
)
=:
=: Rn,1(x) +Rn,2(x).
For the sum Rn,1(x), we have by Lemma 2.1
nαn
bn
Rn,1(x) < ε
(bn − x)(1− e−nx)
bn
< ε.
Let us now estimate Rn,2(x). If we write t =
kbn
mαn
in (11), we get
(
kbn
mαn
− x
)2
h
(
kbn
mαn
− x
)
=
= f
(
kbn
mαn
)
− f(x)−
(
kbn
mαn
− x
)
f ′(x)−
(
kbn
mαn
− x
)2 f ′′(x)
2
and hence
|Rn,2(x)| ≤
∞∑
m=1
qn,m−1(x)
∑
∣∣∣ kbnmαn
−x
∣∣∣≥δ
pmαn,k
(
x
bn
) ∣∣∣∣f ( kbn
mαn
)
− f(x)
∣∣∣∣+
+
∣∣f ′(x)
∣∣ ∞∑
m=1
qn,m−1(x)
∑
∣∣∣ kbnmαn
−x
∣∣∣≥δ
pmαn,k
(
x
bn
) ∣∣∣∣ kbnmαn
− x
∣∣∣∣+
+
|f ′′(x)|
2
∞∑
m=1
qn,m−1(x)
∑
∣∣∣ kbnmαn
−x
∣∣∣≥δ
pmαn,k
(
x
bn
)(
kbn
mαn
− x
)2
=:
=: Σ1 + Σ2 + Σ3.
By the Lemma 3.1, we obtain
Σ1 ≤ 4M(bn)e
− δ
2αn
4xbn
−nx
(
1−exp
(
− δ
2αn
4xbn
))
,
thus
nαn
bn
Σ1 ≤ 4M(bn)
nαn
bn
e
− δ
2αn
4xbn
−nx
(
1−exp
(
− δ
2αn
4xbn
))
.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 6
832 T. TUNC, E. SIMSEK
If we consider the Cauchy – Schwartz inequality, the inequality (6) and Lemma 3.1 for the second
sum Σ2, we have
Σ2 ≤ |f ′(x)|
∞∑
m=1
qn,m−1(x)
∑
∣∣∣ kbnmαn
−x
∣∣∣≥δ
pmαn,k
(
x
bn
)
1/2
×
×
∞∑
m=1
qn,m−1(x)
∑
∣∣∣ kbnmαn
−x
∣∣∣≥δ
pmαn,k
(
x
bn
)(
kbn
mαn
− x
)2
1/2
≤
≤
√
2|f ′(x)|e−
δ2αn
8xbn
−nx2
(
1−exp
(
− δ
2αn
4xbn
))
×
×
δ−2
∞∑
m=1
qn,m−1(x)
∑
∣∣∣ kbnmαn
−x
∣∣∣≥δ
pmαn,k
(
x
bn
)(
kbn
mαn
− x
)4
1/2
≤
≤
√
2|f ′(x)|δ−1
√
cn(x)
bn
nαn
e
− δ
2αn
8xbn
−nx2
(
1−exp
(
− δ
2αn
4xbn
))
,
thus
nαn
bn
Σ2 ≤
√
2|f ′(x)|δ−1
√
cn(x)e
− δ
2αn
8xbn
−nx2
(
1−exp
(
− δ
2αn
4xbn
))
.
By the similar arguments, we obtain the estimate
nαn
bn
Σ3 ≤
|f ′′(x)|
√
2
2
√
cn(x)e
− δ
2αn
8xbn
−nx2
(
1−exp
(
− δ
2αn
4xbn
))
.
Therefore, we have
lim
n→∞
nαn
bn
Σi = 0, i = 1, 2, 3,
under the condition (9).
Corollary 4.1. If the function f is bounded on the semiaxis [0,∞), then (10) holds at each point
x > 0 for which f ′′(x) exists.
5. Rates of convergence.
Theorem 5.1. If f is uniformly continuous on the semiaxis (0,∞), then
‖En(f)− f‖[0,bn] ≤ 2ω
(
f ;
√
bn
nαn
)
.
Proof. Let x ∈ (0, bn]. Since
En(f ;x)− f(x) =
∞∑
m=1
e−nx(nx)m−1
(m− 1)!
αnm∑
k=0
pmαn,k
(
x
bn
)[
f
(
kbn
mαn
)
− f(x)
]
,
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SOME APPROXIMATION PROPERTIES OF SZASZ – MIRAKYAN – BERNSTEIN OPERATORS . . . 833
then we have
|En(f ;x)− f(x)| ≤
∞∑
m=1
e−nx(nx)m−1
(m− 1)!
αnm∑
k=0
pmαn,k
(
x
bn
)
ω
(
f ;
∣∣∣∣ kbnmαn
− x
∣∣∣∣) .
Taking into account that ω(f ;λδ) ≤ (λ+ 1)ω(f ; δ) and using Cauchy – Schwartz inequality, for
δ > 0 we obtain
|En(f ;x)− f(x)| ≤ ω(f ; δ)
{
1
δ
∞∑
m=1
e−nx(nx)m−1
(m− 1)!
αnm∑
k=0
pmαn,k
(
x
bn
) ∣∣∣∣ kbnmαn
− x
∣∣∣∣+ 1
}
≤
≤ ω(f ; δ)
{
1
δ
√
En(x2;x) + 1
}
=
= ω(f ; δ)
1
δ
√
(bn − x)(1− e−nx)
nαn
+ 1
≤
≤ ω(f ; δ)
{
1
δ
√
bn
nαn
+ 1
}
.
Choosing δ =
√
bn
nαn
, we have the inequality
|En(f ;x)− f(x)| ≤ 2ω
(
f ;
√
bn
nαn
)
which is also trivial for x = 0.
Theorem 5.1 is proved.
Corollary 5.1. Let f ∈ C(0,∞). If f ∈ LipM µ, then
‖En(f)− f‖[0,bn] ≤ 2M
(
bn
nαn
)µ/2
.
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834 T. TUNC, E. SIMSEK
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Received 08.01.13
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