ABOUT A CLASS OF LINEAR POSITIVE
OPERATORS OBTAINED BY CHOOSING THE
NODES
Linear Positive Operators
Ovidiu T. Pop and Mircea D. Fărcaş
vol. 10, iss. 1, art. 30, 2009
OVIDIU T. POP AND MIRCEA D. FĂRCAŞ
National College "Mihai Eminescu"
5 Mihai Eminescu Street
Satu Mare 440014, Romania
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EMail: {ovidiutiberiu, mirceafarcas2005}@yahoo.com
Received:
15 June, 2007
Accepted:
18 March, 2009
Communicated by:
S.S. Dragomir
2000 AMS Sub. Class.:
41A10, 41A25, 41A35, 41A36.
Key words:
Linear positive operators, convergence theorem, the first order modulus of
smoothness, approximation theorem.
Abstract:
In this paper we consider the given linear positive operators (Lm )m≥1 and with
their help, we construct linear positive operators (Km )m≥1 . We study the convergence, the evaluation for the rate of convergence in terms of the first modulus
of smoothness for the operators (Km )m≥1 .
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Contents
1
Introduction
3
2
Preliminaries
7
3
Main Results
10
Linear Positive Operators
Ovidiu T. Pop and Mircea D. Fărcaş
vol. 10, iss. 1, art. 30, 2009
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1.
Introduction
In this section, we recall some notions and operators which we will use in this article.
Let N be the set of positive integers and N0 = N ∪ {0}. For m ∈ N, let Bm :
C([0, 1]) → C([0, 1]) be Bernstein operators, defined for any function f ∈ C([0, 1])
by
m
X
k
(1.1)
(Bm f )(x) =
pm,k (x)f
,
m
k=0
where pm,k (x) are the fundamental polynomials of Bernstein, defined as follows
m k
(1.2)
pm,k (x) =
x (1 − x)m−k ,
k
for any x ∈ [0, 1] and any k ∈ {0, 1, . . . , m} (see [5] or [24]). For the following
construction, see [15]. Define the natural number m0 by
(
max(1, −[β]), if β ∈ R − Z;
(1.3)
m0 =
max(1, 1 − β), if β ∈ Z,
where [x], {x} denote the integer and fractional parts respectively of a real number
x.
For the real number β, we have that
(1.4)
Ovidiu T. Pop and Mircea D. Fărcaş
vol. 10, iss. 1, art. 30, 2009
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m + β ≥ γβ
for any natural number m, m ≥ m0 , where
(
max (1 + β, {β}) , if
(1.5)
γβ = m0 + β =
max(1 + β, 1),
if
Linear Positive Operators
β ∈ R − Z;
β ∈ Z.
For the real numbers α, β, α ≥ 0, we note
(
1,
if
(1.6)
µ(α,β) =
1 + α−β
, if
γβ
α ≤ β;
α > β.
For the real numbers α and β, α ≥ 0, we have that 1 ≤ µ(α,β) and
(1.7)
0≤
k+α
≤ µ(α,β)
m+β
Linear Positive Operators
Ovidiu T. Pop and Mircea D. Fărcaş
vol. 10, iss. 1, art. 30, 2009
for any natural number m, m ≥ m0 and for any k ∈ {0, 1, . . . , m}.
For the real numbers α and β, α ≥ 0, m0 and µ(α,β) defined by (1.3) – (1.6),
(α,β)
(α,β)
let the operators
P
:
C
[0,
µ
]
→
C
[0,
1]
, defined for any function f ∈
m
(α,β)
] by
C [0, µ
m
X
k+α
(α,β)
(1.8)
Pm f (x) =
pm,k (x)f
,
m
+
β
k=0
for any natural number m, m ≥ m0 and for any x ∈ [0, 1]. These operators are
called Stancu operators, and were introduced and studied in 1969 by D.D. Stancu in
the paper [23]. In [23], the domain of definition of Stancu’s operators is C([0, 1])
and the numbers α and β verify the condition 0 ≤ α ≤ β.
In 1980, G. Bleimann, P. L. Butzer and L. Hahn introduced in [4] a sequence of
linear positive operators (Lm )m≥1 , Lm : CB ([0, ∞)) → CB ([0, ∞)), defined for any
function f ∈ CB ([0, ∞)) by
m X
1
m k
k
(1.9)
(Lm f )(x) =
x f
,
(1 + x)m k=0 k
m+1−k
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for any x ∈ [0, ∞) and any m ∈ N, where CB ([0, ∞)) = {f | f : [0, ∞) → R, f is
bounded and continuous on [0, ∞)}.
For m ∈ N, consider the operators Sm : C2 ([0, ∞)) → C ([0, ∞)) defined for
any function f ∈ C2 ([0, ∞)) by
∞
X
(mx)k
k
−mx
(1.10)
(Sm f ) (x) = e
f
,
k!
m
k=0
for any x ∈ [0, ∞), where
f (x)
C2 ([0, ∞)) = f ∈ C ([0, ∞)) : lim
exists and is finite .
x→∞ 1 + x2
The operators (Sm )m≥1 are called Mirakjan-Favard-Szász operators and were introduced in 1941 by G. M. Mirakjan in [12].
They were intensively studied by J. Favard in 1944 in [8] and O. Szász in 1950 in
[25].
For m ∈ N, the operator Vm : C2 ([0, ∞)) → C ([0, ∞)) is defined for any
function f ∈ C2 ([0, ∞)) by
k ∞ X
k
m+k−1
x
−m
f
,
(1.11)
(Vm f ) (x) = (1 + x)
1+x
m
k
k=0
for any x ∈ [0, ∞).
The operators (Vm )m≥1 are named Baskakov operators and they were introduced
in 1957 by V. A. Baskakov in [2].
W. Meyer-König and K. Zeller have introduced in [11] a sequence of linear and
positive operators. After a slight adjustment, given by E.W. Cheney and A. Sharma
Linear Positive Operators
Ovidiu T. Pop and Mircea D. Fărcaş
vol. 10, iss. 1, art. 30, 2009
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in [6], 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
(1.12)
(Zm f ) (x) =
,
k
m+k
k=0
for any m ∈ N and for any x ∈ [0, 1).
These operators are called the Meyer-König and Zeller operators.
Observe that Zm : C ([0, 1]) → C ([0, 1]), m ∈ N.
In [10], M. Ismail and C.P. May consider the operators (Rm )m≥1 .
For m ∈ N, Rm : C([0, ∞)) → C([0, ∞)) is defined for any function f ∈
C([0, ∞)) by
k
∞
k−1
kx
mx X m(m + k)
x
k
− 1+x
− 1+x
f
e
(1.13)
(Rm f )(x) = e
m
k!
1+x
k=0
for any x ∈ [0, ∞).
We consider I ⊂ R, I an interval and we shall use the following function sets:
E(I),
F (I) which are subsets of
the set of
real functions defined on I, B(I)
=
f | f : I → R, f bounded on I , C(I) = f | f : I → R, f continuous on I and
CB (I) = B(I) ∩ C(I).
If f ∈ B(I), then the first order modulus of smoothness of f is the function
ω(f ; ·) : [0, ∞) → R defined for any δ ≥ 0 by
(1.14)
ω(f ; δ) = sup {|f (x0 ) − f (x00 )| : x0 , x00 ∈ I, |x0 − x00 | ≤ δ} .
Linear Positive Operators
Ovidiu T. Pop and Mircea D. Fărcaş
vol. 10, iss. 1, art. 30, 2009
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2.
Preliminaries
For the following construction and result see [16] and [18], where pm = m for any
m ∈ N or pm = ∞ for any m ∈ N. Let I, J ⊂ [0, ∞) be intervals with I ∩ J 6= ∅.
For any m ∈ N and k ∈ {0, 1, ..., pm } ∩ N0 consider the nodes xm,k ∈ I and the
functions ϕm,k : J → R with the property that ϕm,k (x) ≥ 0 for any x ∈ J. Let E(I)
and F (J) be subsets of the set of real functions defined on I, respectively J so that
the sum
pm
X
ϕm,k (x)f (xm,k )
Linear Positive Operators
Ovidiu T. Pop and Mircea D. Fărcaş
vol. 10, iss. 1, art. 30, 2009
k=0
exists for any f ∈ E(I), x ∈ J and m ∈ N. For any x ∈ I consider the functions
ψx : I → R, ψx (t) = t − x and ei : I → R, ei (t) = ti for any t ∈ I, i ∈ {0, 1, 2}.
In the following, we suppose that for any x ∈ I we have ψx ∈ E(I) and ei ∈ E(I),
i ∈ {0, 1, 2}.
For m ∈ N, let the given operator Lm : E(I) → F (J) defined by
(2.1)
(Lm f )(x) =
pm
X
ϕm,k (x)f (xm,k )
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k=0
with the property that the convergence
(2.2)
lim (Lm f )(x) = f (x)
m→∞
is uniform on any compact K ⊂ I ∩ J, for any f ∈ E(I) ∩ C(I).
Remark 1. From (2.2), for the operators (Lm )m≥1 we have that the following convergences
(2.3)
lim (Lm ei )(x) = ei (x),
m→∞
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i ∈ {0, 1, 2} and
lim (Lm ψx2 )(x) = 0
(2.4)
m→∞
are uniform on any compact K ⊂ I ∩ J.
Remark 2. From Remark 1 it results that for any compact K ⊂ I ∩ J the sequences
(um (K))m≥1 , (vm (K))m≥1 , (wm (K))m≥1 depending on K exist, so that the convergences
(2.5)
lim um (K) = lim vm (K) = lim wm (K) = 0
m→∞
m→∞
m→∞
Linear Positive Operators
Ovidiu T. Pop and Mircea D. Fărcaş
vol. 10, iss. 1, art. 30, 2009
are uniform on K and
|(Lm e0 )(x) − 1| ≤ um (K),
(2.6)
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|(Lm e1 )(x) − x| ≤ vm (K),
(2.7)
(Lm ψx2 )(x)
(2.8)
≤ wm (K),
for any x ∈ K and any m ∈ N.
In the following, for m ∈ N and k ∈ {0, 1, . . . , pm } ∩ N0 we consider the nodes
ym,k ∈ I so that
(2.9)
αm =
sup
|xm,k − ym,k | < ∞
k∈{0,1,...,pm }∩N0
for any m ∈ N and
(2.10)
lim αm = 0.
m→∞
For m ∈ N and k ∈ {0, 1, . . . , pm } ∩ N0 we note that αm,k = xm,k − ym,k .
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Definition 2.1. For m ∈ N, define the operator Km : E(I) → F (J) by
(2.11)
(Km f )(x) =
pm
X
ϕm,k (x)f (ym,k ),
k=0
for any x ∈ I and any f ∈ E(I).
Remark 3. Similar ideas to the construction above can be found in the recent papers
[9] and [13].
Linear Positive Operators
Ovidiu T. Pop and Mircea D. Fărcaş
vol. 10, iss. 1, art. 30, 2009
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3.
Main Results
In this section, we study the operators defined by (2.11).
Theorem 3.1. For any f ∈ E(I) ∩ C(I) we have that the convergence
lim (Km f )(x) = f (x)
(3.1)
m→∞
Linear Positive Operators
is uniform on any compact K ⊂ I ∩ J.
Ovidiu T. Pop and Mircea D. Fărcaş
Proof. For x ∈ K and m ∈ N we have that
vol. 10, iss. 1, art. 30, 2009
(Km ψx2 )(x) = (Km e2 )(x) − 2x(Km e1 )(x) + x2 (Km e0 )(x)
pm
pm
pm
X
X
X
2
2
=
ϕm,k (x)ym,k − 2x
ϕm,k (x)ym,k + x
ϕm,k (x)
k=0
pm
=
X
k=0
k=0
− 2x
ϕm,k (x)(xm,k − αm,k ) + x
=
X
2
pm
X
ϕm,k (x)
k=0
k=0
pm
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k=0
ϕm,k (x)(xm,k − αm,k )2
pm
X
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pm
ϕm,k (x)x2m,k − 2
k=0
X
ϕm,k (x)xm,k αm,k
k=0
+
pm
X
2
ϕm,k (x)αm,k
− 2x
k=0
+ 2x
pm
X
k=0
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pm
X
ϕm,k (x)xm,k
k=0
pm
X
2
ϕm,k (x)αm,k + x
k=0
ϕm,k (x)
2
≤ (Lm ψx2 )(x) + 2αm (Lm e1 )(x) + (αm
+ 2xαm )(Lm e0 )(x).
Taking Remark 1 and Remark 2 into account, it results that (3.1) holds.
Theorem 3.2. If f ∈ E(I ∩ J) ∩ C(I ∩ J), then for any x ∈ K = [a, b] ⊂ I ∩ J
and any m ∈ N, we have that
(3.2) |(Km f )(x) − f (x)| ≤ |f (x)| |(Lm e0 (x)) − 1| + ((Lm e0 )(x) + 1)ω(f ; δm,x )
≤ M um (K) + (2 + um (K))ω(f ; δm ),
Linear Positive Operators
Ovidiu T. Pop and Mircea D. Fărcaş
where
vol. 10, iss. 1, art. 30, 2009
p
2 + 2xα )(L e )(x)],
δm,x = (Lm e0 )(x)[(Lm ψx2 )(x) + 2αm (Lm e1 )(x) + (αm
m
m 0
p
2 + 2bα )(1 + u (K))]
δm = (1 + um (K))[wm (K) + 2αm (b + vm (K) + (αm
m
m
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M = sup{|f (x)| : x ∈ K}.
Proof. We apply the Shisha-Mond Theorem (see [22] or [24]) for the operator Km
and taking the inequality from the proof of the Theorem 3.1 into account verified by
(Km ψx2 )(x) and Remark 2, the inequality (3.2) follows.
Corollary 3.3. If
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pm
X
(3.3)
ϕm,k (x) = 1
k=0
for any x ∈ J, then for any f ∈ E(I ∩ J) ∩ C(I ∩ J), any x ∈ K = [a, b] ⊂ I ∩ J
and any m ∈ N we have that
0
|(Km f )(x) − f (x)| ≤ 2ω(f ; δm,x ) ≤ 2ω(f ; δm
)
(3.4)
0
where δm
=
p
2 + 4bα .
wm (K) + 2αm vm (K) + αm
m
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Proof. It results from Theorem 3.2, because (Lm e0 )(x) = 1, for any m ∈ N and
x ∈ J, so um (K) = 0, for any m ∈ N.
Remark 4. From the conditions of Theorem 3.2 we have that
|(Km f )(x) − f (x)| ≤ M um (K) + (2 + um (K))ω(f ; δm )
and because lim δm = 0, it results that the convergence lim (Km f )(x) = f (x) is
m→∞
m→∞
uniform on K.
In the following, by particularisation of the sequence ym,k , m ∈ N, k ∈ {0, 1,
. . . , pm } ∩N0 and applying Theorem 3.1 and Corollary 3.3, we can obtain a convergence and approximation theorem for the new operators. In Applications 1 – 2, let
pm = m, ϕm,k (x) = pm,k (x), where m ∈ N, k ∈ {0, 1, . . . , m} and K = [0, 1].
k
Application 1. If I = J = [0, 1], E(I) = F (J) = C([0, 1]), xm,k = m
, m ∈ N,
k ∈ {0, 1, . . . , m}, we obtain the Bernstein operators. We have that um ([0,√
1]) = 0,
vm ([0, 1]) =
m ∈ N, k ∈
k(k+1)
1
0 and wm ([0, 1]) = 4m
, m ∈ N. We consider the nodes ym,k =
,
m
1
√
{0, 1, . . . , m}. Then it is verified immediately that αm =
,
m+ m(m+1)
m ∈ N and lim αm = 0. In this case, the operators (Km )m≥1 have the form
m→∞
(Km f )(x) =
m
X
p
pm,k (x)f
k=0
0
f ∈ C([0, 1]), x ∈ [0, 1], m ∈ N and δm
<
q
5
4m
+
k(k + 1)
m
!
m(m+1)
Ovidiu T. Pop and Mircea D. Fărcaş
vol. 10, iss. 1, art. 30, 2009
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,
√2
m+
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<
3
√
,
2 m
m ∈ N.
Application 2. We study a particular case of the Stancu operators. Let α = 10 and
β = − 12 . We obtain I = [0, 22] and for any f ∈ C([0, 22]), x ∈ [0, 1] and m ∈ N
m
X
2k + 20
(10,−1/2)
Pm
f (x) =
pm,k (x)f
.
2m − 1
k=0
We consider the nodes ym,k =
the form
(4k+40)m
.
(2m−1)2
(Km f )(x) =
m
X
k=0
In this case, the operators (Km )m≥1 have
Linear Positive Operators
Ovidiu T. Pop and Mircea D. Fărcaş
pm,k (x)f
m(4k + 40)
(2m − 1)2
0
where f ∈ C([0, 22]), x ∈ [0, 1], m ∈ N and δm
<
m ∈ N.
√
vol. 10, iss. 1, art. 30, 2009
,
36m3 +2220m2 −399m+81
(2m−1)2
<
√ 45 ,
2m−1
Contents
Application 3. If I = J = [0, ∞), E(I) = C2 ([0, ∞)), F (J) = C([0, ∞)), K =
k
k
[0, b], pm = ∞, xm,k = m
, ϕm,k (x) = e−mx (mx)
, m ∈ N, k ∈ N0 , we obtain the
k!
Mirakjan-Favard-Szász operators and we have that um (K) = 0, vm (K) = 0 and
2k(k+1)
wm (K) = mb , m ∈ N. We consider the nodes ym,k = m(2k+1)
, m ∈ N, k ∈ N0 and
1
we have that αm = 2m , m ∈ N. In this case, the operators (Km )m≥1 have the form
∞
X
(mx)k
2k(k + 1)
−mx
(Km f )(x) = e
f
,
k!
m(2k
+
1)
k=0
0
where f ∈ C2 ([0, ∞)), x ∈ [0, ∞), m ∈ N and δm
=
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q
3b
m
+
1
,
4m2
m ∈ N.
Application 4. Let I = J = [0, ∞), E(I) = C2 ([0, ∞)), F (J) = C([0, ∞)),
x k
k
K = [0, b], pm = ∞, xm,k = m
, ϕm,k (x) = (1 + x)−m m+k−1
, m ∈ N, k ∈
k
1+x
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N0 . In this case, we obtain the Baskakov operators and we have that um√(K) = 0,
2 +4k+2
vm (K) = 0 and wm (K) = b(1+b)
, m ∈ N. We consider the nodes ym,k = 4k 2m
,
2m
m ∈ N, k ∈ N0 and we have that αm = m1√2 . The operators (Km )m≥1 have the form
!
√
k
∞ 2 + 4k + 2
X
x
m
+
k
−
1
4k
(Km f )(x) = (1 + x)−m
f
,
2m
k
1+x
k=0
Linear Positive Operators
0
=
where f ∈ C2 ([0, ∞)), x ∈ [0, ∞), m ∈ N and δm
q
√
b(b+1+2 2)
m
+
1
,
2m2
m ∈ N.
Application 5. If I = J = [0, ∞), E(I) = F (J) = C([0, ∞)), K = [0, b], pm =
k
∞, xm,k = m
,
k
−(k+m)x
m(m + k)k−1
x
m ∈ N, k ∈ N0 ,
ϕm,k (x) =
e 1+x ,
k!
1+x
we obtain the Ismail-May operators and we have that um (K)
√ = 0, vm (K) = 0 and
3
b(1+b)2
,
m
k2 (k+1)
,
m
wm (K) =
m ∈ N. We consider the nodes ym,k =
m ∈ N, k ∈ N0
1
and we have that αm = 3m . In this case, the operators (Km )m≥1 have the form
!
p
k
∞
3
k−1
2 (k + 1)
−mx X m(m + k)
kx
k
x
(Km f )(x) = e 1+x
e− 1+x f
,
k!
1+x
m
k=0
Ovidiu T. Pop and Mircea D. Fărcaş
vol. 10, iss. 1, art. 30, 2009
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0
where f ∈ C([0, ∞)), m ∈ N and δm
=
q
b(7+6b+3b2 )
3m
+
1
,
9m2
m ∈ N.
Application 6. We consider I = J = [0, ∞), E(I) = F (J)
k = CB ([0, ∞)),
m
k
1
K = [0, b], pm = m, xm,k = m+1−k
, ϕm,k (x) = (1+x)
m k x , m ∈ N, k ∈
{0, 1, . . . , m}. In this case we obtain the Bleimann-Butzer-Hahn operators and we
m
2
b
and wm (K) = 4b(1+b)
have that um (K) = 0, vm (K) = b 1+b
, m ∈ N. We
m+2
βm k
consider the nodes ym,k = m+1−k , m ∈ N, k ∈ {0, 1, . . . , m}, where (βm )m≥1 is
a sequence of positive real numbers such that lim m(1 − βm ) = 0 and we have
m→∞
αm = m|1 − βm |, m ∈ N. The operators (Km )m≥1 have the form
m X
m k
βm k
−m
(Km f )(x) = (1 + x)
x f
,
k
m+1−k
k=0
Linear Positive Operators
Ovidiu T. Pop and Mircea D. Fărcaş
where x ∈ [0, ∞), m ∈ N, f ∈ CB ([0, ∞)).
vol. 10, iss. 1, art. 30, 2009
Application 7. If I = J = [0, 1], E(I) = B([0,
(J) = C([0, 1]), K =
1]), F
m+k
k
m+1 k
[0, 1], pm = ∞, xm,k = m+k , ϕm,k (x) = k (1 − x)
x , m ∈ N, k ∈ N0 ,
we obtain the Meyer-König and Zeller operators and we have that um ([0, 1]) = 0,
1
vm ([0, 1]) = 0 and wm ([0, 1]) = 4(m+1)
, m ∈ N. We consider the nodes ym,k =
k+βm
, m ∈ N, k ∈ N0 , where (βm )m≥1 is a sequence of positive real numbers so
m+k+βm
βm
βm
that lim m+β
= 0. Then it is verified immediately that αm = m+β
, m ∈ N and
m
m
m→∞
the operators (Km )m≥1 have the form
∞ X
m+k
k + βm
m+1 k
(Km f )(x) =
(1 − x)
x f
,
k
m
+
k
+
β
m
k=0
0
where f ∈ B([0, 1]), x ∈ [0, 1], m ∈ N and δm
=
q
1
4(m+1)
+
βm (4m+5βm )
,
(m+βm )2
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m ∈ N.
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References
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Linear Positive Operators
Ovidiu T. Pop and Mircea D. Fărcaş
vol. 10, iss. 1, art. 30, 2009
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