APPROXIMATION OF B-CONTINUOUS AND
B-DIFFERENTIABLE FUNCTIONS BY GBS
OPERATORS DEFINED BY INFINITE SUM
Approximation of B-Continuous
and B-Differentiable Functions
Ovidiu T. Pop
OVIDIU T. POP
vol. 10, iss. 1, art. 7, 2009
National College "Mihai Eminescu"
5 Mihai Eminescu Street
Satu Mare 440014, Romania
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EMail: ovidiutiberiu@yahoo.com
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18 March, 2009
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Communicated by:
S.S Dragomir
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2000 AMS Sub. Class.:
41A10, 41A25, 41A35, 41A36, 41A63
Key words:
Linear positive operators, GBS operators, B-continuous and B-differentiable
functions, approximation of B-continuous and B-differentiable functions by
GBS operators.
Received:
27 June, 2008
Accepted:
Abstract:
In this paper we start from a class of linear and positive operators defined by
infinite sum. We consider the associated GBS operators and we give an approximation of B-continuous and B-differentiable functions with these operators.
Through particular cases, we obtain statements verified by the GBS operators of
Mirakjan-Favard-Szász, Baskakov and Meyer-König and Zeller.
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Contents
1
Introduction
3
2
Preliminaries
9
3
Main Results
11
Approximation of B-Continuous
and B-Differentiable Functions
Ovidiu T. Pop
vol. 10, iss. 1, art. 7, 2009
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1.
Introduction
In this section, we recall some notions and results which we will use in this article.
Let N be the set of positive integers and N0 = N ∪ {0}.
In the following, let X and Y be real intervals.
A function f : X ×Y → R is called a B-continuous function in (x0 , y0 ) ∈ X ×Y
if and only if
lim
∆f [(x, y), (x0 , y0 )] = 0,
(x,y)→(x0 ,y0 )
where
Approximation of B-Continuous
and B-Differentiable Functions
Ovidiu T. Pop
vol. 10, iss. 1, art. 7, 2009
∆f [(x, y), (x0 , y0 )] = f (x, y) − f (x0 , y) − f (x, y0 ) + f (x0 , y0 )
denotes a so-called mixed difference of f .
A function f : X × Y → R is called a B-continuous function on X × Y if and
only if it is B-continuous in any point of X × Y .
A function f : X × Y → R is called a B-differentiable function in (x0 , y0 ) ∈
X × Y if and only if it exists and if the limit is finite
∆f [(x, y), (x, y0 )]
.
(x,y)→(x0 ,y0 ) (x − x0 )(y − y0 )
lim
This limit is called the B-differential of f in the point (x0 , y0 ) and is noted by
DB f (x0 , y0 ).
A function f : X × Y → R is called a B-differentiable function on X × Y if and
only if it is B-differentiable in any point of X × Y .
The definition of B-continuity and B-differentiability was introduced by K. Bögel
in the papers [8] and [9].
The function f : X × Y → R is B-bounded on X × Y if and only if there exists
k > 0 so that |∆f [(x, y), (s, t)]| ≤ k for any (x, y), (s, t) ∈ X × Y .
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We shall use the function sets B(X × Y ) = {f |f : X × Y → R, f bounded on
X × Y } with the usual sup-norm k·k∞ , Bb (X × Y ) = {f |f : X × Y → R, f is
B-bounded on X × Y }, Cb (X × Y ) = {f |f : X × Y → R, f is B-continuous on
X × Y } and Db (X × Y ) = {f |f : X × Y → R, f is B-differentiable on X × Y }.
Let f ∈ Bb (X × Y ). The function ωmixed (f ; · , ·) : [0, ∞) × [0, ∞) → R defined
by
ωmixed (f ; δ1 , δ2 ) = sup{∆f [(x, y), (s, t)]| : |x − s| ≤ δ1 , |y − t| ≤ δ2 }
for any (δ1 , δ2 ) ∈ [0, ∞) × [0, ∞) is called the mixed modulus of smoothness.
Theorem 1.1. Let X and Y be compact real intervals and f ∈ Bb (X × Y ).
Then lim ωmixed (f ; δ1 , δ2 ) = 0 if and only if f ∈ Cb (X × Y ).
δ1 ,δ2 →0
For any x ∈ X consider the function ϕx : X → R, defined by ϕx (t) = |t − x|,
for any t ∈ X. For additional information, see the following papers: [1], [3], [15]
and [19].
Let m ∈ N and the operator Sm : C2 ([0, ∞)) → C([0, ∞)) defined for any
function f ∈ C2 ([0, ∞)) by
∞
X
(mx)k
k
−mx
(1.1)
(Sm f )(x) = e
f
,
k!
m
k=0
lim f (x)2
x→∞ 1+x
for any x ∈ [0, ∞), where C2 ([0, ∞)) = f ∈ C([0, ∞)) :
exists and
is finite . The operators (Sm )m≥1 are called the Mirakjan-Favard-Szász operators,
introduced in 1941 by G. M. Mirakjan in the paper [13].
These operators were intensively studied by J. Favard in 1944 in the paper [11]
and O. Szász in the paper [20].
From [18], the following three lemmas result.
Approximation of B-Continuous
and B-Differentiable Functions
Ovidiu T. Pop
vol. 10, iss. 1, art. 7, 2009
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Lemma 1.2. For any m ∈ N, we have that
x
(1.2)
Sm ϕ2x (x) =
,
m
(1.3)
3mx2 + x
Sm ϕ4x (x) =
m3
for any x ∈ [0, ∞) and
Ovidiu T. Pop
2
Sm ϕ x
(1.4)
(1.5)
Approximation of B-Continuous
and B-Differentiable Functions
Sm ϕ4x
a
(x) ≤
,
m
a(3a + 1)
(x) ≤
m2
vol. 10, iss. 1, art. 7, 2009
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for any x ∈ [0, a], where a > 0.
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Let m ∈ N and the operator Vm : C2 ([0, ∞)) → C([0, ∞)), defined for any
function f ∈ C2 ([0, ∞)) by
k ∞ X
m+k−1
x
k
−m
(1.6)
(Vm f )(x) = (1 + x)
f
k
1+x
m
k=0
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I
for any x ∈ [0, ∞).
The operators (Vm )m≥1 are called Baskakov operators, introduced in 1957 by V.
A. Baskakov in the paper [5].
Lemma 1.3. For any m ∈ N, we have that
(1.7)
x(1 + x)
Vm ϕ2x (x) =
,
m
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(1.8)
Vm ϕ4x
3(m + 2)x4 + 6(m + 2)x3 + (3m + 7)x2 + x
(x) =
m3
for any x ∈ [0, ∞) and
a(1 + a)
Vm ϕ2x (x) ≤
,
m
(1.9)
Approximation of B-Continuous
and B-Differentiable Functions
Ovidiu T. Pop
(1.10)
Vm ϕ4x
a(9a3 + 18a2 + 10a + 1)
(x) ≤
m2
vol. 10, iss. 1, art. 7, 2009
for any x ∈ [0, a], where a > 0.
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W. Meyer-König and K. Zeller have introduced a sequence of linear positive operators in paper [12]. After a slight adjustment, given by E. W. Cheney and A. Sharma
in [10], these operators take the form Zm : B([0, 1)) → C([0, 1)), defined for any
function f ∈ B([0, 1)) by
∞ X
m+k
k
m+1 k
(1 − x)
x f
,
(1.11)
(Zm f )(x) =
m
+
k
k
k=0
Contents
for any m ∈ N and for any x ∈ [0, 1).
These operators are called the Meyer-König and Zeller operators.
In the following we consider Zm : C([0, 1]) → C([0, 1]), for any m ∈ N.
Lemma 1.4. For any m ∈ N and any x ∈ [0, 1], we have that
x(1 − x)2
2x
2
(1.12)
Zm ϕx (x) ≤
1+
m+1
m+1
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and
(1.13)
2
Zm ϕ2x (x) ≤
.
m
The inequality of Corollary 5 from [4], in the condition (1.14) becomes inequality
(1.15). Inequality (1.16) is demonstrated in [16].
Theorem 1.5. Let L : Cb (X × Y ) → B(X × Y ) be a linear positive operator and
U L : Cb (X × Y ) → B(X × Y ) the associated GBS operator. Supposing that the
operator L has the property
(1.14)
L(· − x)2i (∗ − y)2j (x, y) = L(· − x)2i (x, y) L(∗ − y)2j (x, y)
for any (x, y) ∈ X × Y and any i, j ∈ {1, 2}, where "·" and "∗" stand for the first
and second variable. Then:
(i) For any function f ∈ Cb (X × Y ), any (x, y) ∈ X × Y and any δ1 , δ2 > 0, we
have that
(1.15) |f (x, y) − (U Lf )(x, y)| ≤ |f (x, y)||1 − (Le00 )(x, y)|
h
p
p
+ (Le00 )(x, y)+δ1−1 (L(·−x)2 )(x, y)+δ2−1 (L(∗−y)2 )(x, y)
i
p
−1 −1
2
2
+ δ1 δ2
(L(· − x) )(x, y)(L(∗ − y) )(x, y) ωmixed (f ; δ1 , δ2 ).
Approximation of B-Continuous
and B-Differentiable Functions
Ovidiu T. Pop
vol. 10, iss. 1, art. 7, 2009
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(ii) For any f ∈ Db (X × Y ) with DB f ∈ B(X × Y ), any (x, y) ∈ X × Y and any
δ1 , δ2 > 0, we have that
(1.16) |f (x, y) − (U Lf )(x, y)|
≤ |f (x, y)||1 − (Le00 )(x, y)|
p
+ 3kDB f k∞ (L(· − x)2 )(x, y)(L(∗ − y)2 )(x, y)
hp
+
(L(· − x)2 )(x, y)(L(∗ − y)2 )(x, y)
p
+ δ1−1 (L(· − x)4 )(x, y)(L(∗ − y)2 )(x, y)
p
+ δ2−1 (L(· − x)2 )(x, y)(L(∗ − y)4 )(x, y)
i
−1 −1
2
2
+ δ1 δ2 (L(· − x) )(x, y)(L(∗ − y) )(x, y) ωmixed (DB f ; δ1 , δ2 ).
Approximation of B-Continuous
and B-Differentiable Functions
Ovidiu T. Pop
vol. 10, iss. 1, art. 7, 2009
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2.
Preliminaries
Let I, J, K ⊂ R be intervals, J ⊂ K and I ∩ J 6= ∅. We consider the sequence
of nodes ((xm,k )k∈N0 )m≥1 so that xm,k ∈ I ∩ J, k ∈ N0 , m ∈ N and the functions
ϕm,k : K → R with the property that ϕm,k (x) ≥ 0, for any k ∈ N0 , m ∈ N and
x ∈ J.
Definition 2.1. If m ∈ N, we define the operator L∗m : E(I) → F (K) by
(L∗m f ) (x)
(2.1)
=
∞
X
Approximation of B-Continuous
and B-Differentiable Functions
Ovidiu T. Pop
ϕm,k (x)f (xm,k )
vol. 10, iss. 1, art. 7, 2009
k=0
for any function f ∈ E(I) and any x ∈ K, where E(I) and F (K) are subsets of the
set of real functions defined on I, respectively on K.
Proposition 2.2. The operators
(L∗m )m≥1
are linear and positive on E(I ∩ J).
Proof. The proof follows immediately.
L∗m,n
Definition 2.3. If m, n ∈ N, the operator
: E(I × I) → F (K × K) defined
for any function f ∈ E(I × I) and any (x, y) ∈ K × K by
(2.2)
L∗m,n f
(x, y) =
∞ X
∞
X
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ϕm,k (x)ϕn,j (y)f (xm,k , xn,j )
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k=0 j=0
is called the bivariate operator of L∗ - type.
Proposition 2.4. The operators L∗m,n m,n≥1 are linear and positive on E[(I × I) ∩
(J × J)].
Proof. The proof follows immediately.
Close
Definition 2.5. If m, n ∈ N, the operator U L∗m,n : E(I × I) → F (K × K) defined
for any function f ∈ E(I × I) and any (x, y) ∈ K × K by
(2.3)
U L∗m,n f (x, y)
∞ X
∞
X
=
ϕm,k (x)ϕn,j (y) f (xm,k , y) + f (x, xn,j ) − f (xm,k , xn,j )
k=0 j=0
is called a GBS operator of L∗ - type.
Approximation of B-Continuous
and B-Differentiable Functions
Ovidiu T. Pop
vol. 10, iss. 1, art. 7, 2009
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3.
Main Results
Lemma 3.1. For any m, n ∈ N, i, j ∈ N0 and (x, y) ∈ K × K, the identity
(3.1)
L∗m,n (· − x)2i (∗ − y)2j (x, y) = L∗m (· − x)2i (x) L∗n (∗ − y)2j (y)
holds.
Approximation of B-Continuous
and B-Differentiable Functions
Proof. We have that
L∗m,n (·
2i
2j
− x) (∗ − y)
(x, y) =
=
=
∞
∞ X
X
ϕm,k (x)ϕn,j (y)(xm,k − x)2i (xn,j − y)2j
k=0 j=0
∞
X
2i
ϕm,k (x)(xm,k − x)
k=0
L∗m (·
∞
X
ϕn,j (y)(xn,j − y)2j
Ovidiu T. Pop
vol. 10, iss. 1, art. 7, 2009
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j=0
2i
− x)
(x)
L∗n (∗
Contents
− y)2j (y),
so (3.1) holds.
For the
p operators constructed in this section, we note that δm (x) =
δm,x = (L∗m ϕ4x ) (x), where x ∈ I ∩ J, m ∈ N, m 6= 0.
Then, by taking Lemma 3.1 into account, Theorem 1.5 becomes:
p
(L∗m ϕ2x ) (x),
Theorem 3.2.
(i) For any function f ∈ Cb (I × I), any (x, y) ∈ (I × I) ∩ (J × J), any m, n ∈ N,
any δ1 , δ2 > 0, we have that
(3.2) |f (x, y) − (U L∗m,n f )(x, y)|
≤ |f (x, y)||1 − (Le00 )(x, y)| + (Le00 )(x, y) + δ1−1 δm (x) + δ2−1 δn (y)
+ δ1−1 δ2−1 δm (x)δn (y) ωmixed (f ; δ1 , δ2 ).
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(ii) For any function f ∈ Db (I × I) with DB f ∈ B(I × I), any (x, y) ∈ (I × I) ∩
(J × J), any m, n ∈ N, any δ1 , δ2 > 0, we have that
(3.3) |f (x, y) − (U L∗ f )(x, y)| ≤ |f (x, y)||1 − (Le00 )(x, y)|
+ 3kDB f k∞ δm (x)δn (y) + δm (x)δn (y) + δ1−1 δm,x δn (y)
2
+ δ2−1 δm (x)δn,y + δ1−1 δ2−1 δm
(x)δn2 (y) ωmixed (DB f ; δ1 , δ2 ).
In the following, we give examples of operators and of the associated GBS operators.
Application 1. If I = J = K = [0, ∞), E(I) = C2 ([0, ∞)), F (K) = C([0, ∞)),
k
k
ϕm,k (x) = e−mx (mx)
, xm,k = m
, x ∈ [0, ∞), m, k ∈ N0 , m 6= 0, then we obtain
k!
the Mirakjan-Favard-Szász operators.
Theorem 3.3. Let a, b ∈ R, a > 0 and b > 0. Then:
(i) For any function f ∈ C([0, ∞) × [0, ∞)), any (x, y) ∈ [0, a] × [0, b] and
m, n ∈ N, we have that
Approximation of B-Continuous
and B-Differentiable Functions
Ovidiu T. Pop
vol. 10, iss. 1, art. 7, 2009
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(3.4)
|f (x, y) − (U Sm,n f )(x, y)|
√ √ 1
1
≤ 1+ a 1+ b ωmixed f ; √ , √
.
m
n
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(ii) For any function f ∈ Db ([0, ∞) × [0, ∞)) ∩ C([0, ∞) × [0, ∞)) with DB f ∈
B([0, a] × [0, b]), any (x, y) ∈ [0, a] × [0, b], any m, n ∈ N, we have that
"
√
√
(3.5) |f (x, y) − (U Sm,n f )(x, y)| ≤ ab 3kDB f k∞ + 1 + 3a + 1
+
√
#
√ 1
1
1
√
3b + 1 + ab ωmixed DB f ; √ , √
.
m
n
mn
Proof. It results from Theorem 3.2, by choosing δ1 =
1.2.
√1
m
, δ2 =
√1
n
and Lemma
Approximation of B-Continuous
and B-Differentiable Functions
Ovidiu T. Pop
vol. 10, iss. 1, art. 7, 2009
Theorem 3.4. If f ∈ C([0, ∞) × [0, ∞)), then the convergence
(3.6)
lim (U Sm,n f )(x, y) = f (x, y)
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m,n→∞
is uniform on any compact [0, a] × [0, b], where a, b > 0.
Proof. It results from Theorem 1.1 and Theorem 3.3.
Application 2. If I = J = K = [0, ∞), E(I) = C2 ([0, ∞)), F (K) = C([0, ∞)),
x k
k
ϕm,k (x) = (1 + x)−m m+k−1
, xm,k = m
, x ∈ [0, ∞), m, k ∈ N0 , m 6= 0,
k
1+x
then we obtain the Baskakov operators.
Theorem 3.5. Let a, b ∈ R, a > 0 and b > 0. Then:
(i) For any function f ∈ C([0, ∞) × [0, ∞)), any (x, y) ∈ [0, a] × [0, b] and any
m, n ∈ N, we have that
(3.7) |f (x, y) − (U Vm,n f )(x, y)|
p
p
1
1
≤ 1 + a(1 + a) 1 + b(1 + b) ωmixed f ; √ , √
.
m
n
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(ii) For any function f ∈ Db ([0, ∞) × [0, ∞)) ∩ C([0, ∞) × [0, ∞)) with DB f ∈
B([0, a] × [0, b]), any (x, y) ∈ [0, a] × [0, b], any m, n ∈ N, we have that
(
p
(3.8) |f (x, y) − (U Vm,n f )(x, y)| ≤ ab(1 + a)(1 + b) 3kDB k∞
h
√
√
+ 1 + 9a3 + 18a2 + 10a + 1 + 9b3 + 18b2 + 10b + 1
)
i
p
1
1
1
√
.
+ ab(1 + a)(1 + b) ωmixed DB f ; √ , √
mn
n
m
Proof. It results from Theorem 3.2, by choosing δ1 =
1.3.
√1
m
, δ2 =
√1
n
and Lemma
Approximation of B-Continuous
and B-Differentiable Functions
Ovidiu T. Pop
vol. 10, iss. 1, art. 7, 2009
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Contents
Theorem 3.6. If f ∈ C([0, ∞) × [0, ∞)), then the convergence
(3.9)
lim (U Vm,n f )(x, y) = f (x, y)
m,n→∞
is uniform on any compact [0, a] × [0, b], where a, b > 0.
Proof. It results from Theorem 1.1 and Theorem 3.5.
Application
3. If I = J = K = [0, 1], E(I) = F (K) = C([0, 1]), ϕm,k (x) =
m+k
k
(1 − x)m+1 xk , xm,k = m
, x ∈ [0, 1], m, k ∈ N0 , m 6= 0, then we obtain the
k
Meyer-König and Zeller operators.
Theorem 3.7. For any function f ∈ C([0, 1] × [0, 1]), any (x, y) ∈ [0, 1] × [0, 1] and
any m, n ∈ N, we have that
√
1
1
(3.10)
|f (x, y) − (U Zm,n f )(x, y)| ≤ (3 + 2 2)ωmixed f ; √ , √
.
m
n
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Proof. It results from Theorem 3.2, by choosing δ1 =
1.4.
√1
m
, δ2 =
√1
n
and Lemma
Theorem 3.8. If f ∈ C([0, 1] × [0, 1]), then the convergence
(3.11)
lim (U Zm,n f )(x, y) = f (x, y)
m,n→∞
is uniform on [0, 1] × [0, 1].
Proof. It results from Theorem 1.1 and Theorem 3.7.
Approximation of B-Continuous
and B-Differentiable Functions
Ovidiu T. Pop
vol. 10, iss. 1, art. 7, 2009
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References
[1] I. BADEA, Modulul de continuitate în sens Bögel şi unele aplicaţii în aproximarea printr-un operator Bernstein, Studia Univ. Babeş-Bolyai, Ser. Math.Mech., 4(2) (1973), 69–78 (Romanian).
[2] C. BADEA, I. BADEA AND H.H. GONSKA, A test function theorem and approximation by pseudopolynomials, Bull. Austral. Math. Soc., 34 (1986), 53–
64.
[3] C. BADEA, I. BADEA, C. COTTIN AND H.H. GONSKA, Notes on the degree
of approximation of B-continuous and B-differentiable functions, J. Approx.
Theory Appl., 4 (1988), 95–108.
[4] C. BADEA AND C. COTTIN, Korovkin-type theorems for generalized boolean
sum operators, Colloquia Mathematica Societatis János Bolyai, 58 (1990), Approximation Theory, Kecskemét (Hungary), 51–67.
[5] V.A. BASKAKOV, An example of a sequence of linear positive operators in the
space of continuous functions, Dokl. Acad. Nauk, SSSR, 113 (1957), 249–251.
[6] D. BĂRBOSU, On the approximation by GBS operators of Mirakjan type,
BAM-1794/2000, XCIV, 169–176.
[7] M. BECHER AND R.J. NESSEL, A global approximation theorem for MeyerKönig and Zeller operators, Math. Zeitschr., 160 (1978), 195–206.
[8] K. BÖGEL, Mehrdimensionale Differentiation von Funktionen mehrer Veränderlicher, J. Reine Angew. Math., 170 (1934), 197–217.
[9] K. BÖGEL, Über mehrdimensionale Differentiation, Integration und
beschränkte Variation, J. Reine Angew. Math., 173 (1935), 5–29.
Approximation of B-Continuous
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[10] E.W. CHENEY AND A. SHARMA, Bernstein power series, Canadian J. Math.,
16 (1964), 241–252.
[11] J. FAVARD, Sur les multiplicateur d’interpolation, J. Math. Pures Appl., 23(9)
(1944), 219–247.
[12] W. MEYER-KÖNIG AND K. ZELLER, Bernsteinsche Potenzreihen, Studia
Math., 19 (1960), 89–94.
[13] G.M. MIRAKJAN, Approximation of continuous functions with the aid of
polynomials, Dokl. Acad. Nauk SSSR, 31 (1941), 201–205.
Approximation of B-Continuous
and B-Differentiable Functions
Ovidiu T. Pop
vol. 10, iss. 1, art. 7, 2009
[14] M.W. MÜLLER, Die Folge der Gammaoperatoren, Dissertation Stuttgart,
1967.
[15] M. NICOLESCU, Analiză matematică, II, Editura Didactică şi Pedagogică,
Bucureşti, 1980 (Romanian).
Title Page
Contents
[16] O.T. POP, Approximation of B-differentiable functions by GBS operators,
Anal. Univ. Oradea, Fasc. Matematica, Tom XIV (2007), 15–31.
JJ
II
[17] O.T. POP, Approximation of B-continuous and B-differentiable functions by
GBS operators defined by finite sum, Facta Universitatis (Niš), Ser. Math. Inform., 22(1) (2007), 33–41.
J
I
Page 17 of 17
[18] O.T. POP, About some linear and positive operators defined by infinite sum,
Dem. Math., 39 (2006).
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[19] D.D. STANCU, GH. COMAN, O. AGRATINI AND R. TRÎMBIŢAŞ, Analiză
numerică şi teoria aproximării, I, Presa Universitară Clujeană, Cluj-Napoca,
2001 (Romanian).
[20] O. SZÁSZ, Generalization of S. Bernstein’s polynomials to the infinite interval,
J. Research National Bureau of Standards, 45 (1950), 239–245.
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