Volume 10 (2009), Issue 1, Article 30, 9 pp.
ABOUT A CLASS OF LINEAR POSITIVE OPERATORS OBTAINED BY
CHOOSING THE NODES
OVIDIU T. POP AND MIRCEA D. FĂRCAŞ
NATIONAL C OLLEGE "M IHAI E MINESCU "
5 M IHAI E MINESCU S TREET
S ATU M ARE 440014, ROMANIA
ovidiutiberiu@yahoo.com
mirceafarcas2005@yahoo.com
Received 15 June, 2007; accepted 18 March, 2009
Communicated by S.S Dragomir
A BSTRACT. 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 .
Key words and phrases: Linear positive operators, convergence theorem, the first order modulus of smoothness, approximation theorem.
2000 Mathematics Subject Classification. 41A10, 41A25, 41A35, 41A36.
1. I NTRODUCTION
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.
201-07
2
OVIDIU T. P OP AND M IRCEA D. FĂRCA Ş
For the real number β, we have that
m + β ≥ γβ
(1.4)
for any natural number m, m ≥ m0 , where
(
max (1 + β, {β}) , if β ∈ R − Z;
(1.5)
γβ = m0 + β =
max(1 + β, 1),
if β ∈ Z.
For the real numbers α, β, α ≥ 0, we note
(
1,
(1.6)
µ(α,β) =
1+
if α ≤ β;
α−β
γβ
, if
α > β.
For the real numbers α and β, α ≥ 0, we have that 1 ≤ µ(α,β) and
0≤
(1.7)
k+α
≤ µ(α,β)
m+β
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), let the operators
(α,β)
Pm : C [0, µ(α,β) ] → C [0, 1] , defined for any function f ∈ C [0, µ(α,β) ] by
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
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].
J. Inequal. Pure and Appl. Math., 10(1) (2009), Art. 30, 9 pp.
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L INEAR P OSITIVE O PERATORS
3
For m ∈ N, the operator Vm : C2 ([0, ∞)) → C ([0, ∞)) is defined for any function f ∈
C2 ([0, ∞)) by
k ∞ X
x
k
m+k−1
−m
f
(1.11)
(Vm f ) (x) = (1 + x)
,
k
1
+
x
m
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 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
(1.12)
(Zm f ) (x) =
(1 − x)
x f
,
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
mx X m(m + k)
kx
x
k
− 1+x
− 1+x
(1.13)
(Rm f )(x) = e
e
f
k!
1+x
m
k=0
for any x ∈ [0, ∞).
We consider I ⊂ R, I an interval and we shall use the followingfunction 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} .
2. P RELIMINARIES
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 )
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 )
k=0
J. Inequal. Pure and Appl. Math., 10(1) (2009), Art. 30, 9 pp.
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4
OVIDIU T. P OP AND M IRCEA D. FĂRCA Ş
with the property that the convergence
lim (Lm f )(x) = f (x)
(2.2)
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
lim (Lm ei )(x) = ei (x),
(2.3)
m→∞
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→∞
are uniform on K and
(2.6)
|(Lm e0 )(x) − 1| ≤ um (K),
(2.7)
|(Lm e1 )(x) − x| ≤ vm (K),
(2.8)
(Lm ψx2 )(x) ≤ 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 =
|xm,k − ym,k | < ∞
sup
k∈{0,1,...,pm }∩N0
for any m ∈ N and
lim αm = 0.
(2.10)
m→∞
For m ∈ N and k ∈ {0, 1, . . . , pm } ∩ N0 we note that αm,k = xm,k − ym,k .
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].
J. Inequal. Pure and Appl. Math., 10(1) (2009), Art. 30, 9 pp.
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L INEAR P OSITIVE O PERATORS
5
3. M AIN R ESULTS
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→∞
is uniform on any compact K ⊂ I ∩ J.
Proof. For x ∈ K and m ∈ N we have that
(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)
=
=
X
k=0
k=0
k=0
pm
ϕm,k (x)(xm,k − αm,k )2
k=0
− 2x
pm
X
ϕm,k (x)(xm,k − αm,k ) + x
2
k=0
=
pm
X
pm
X
ϕm,k (x)
k=0
ϕm,k (x)x2m,k
−2
k=0
pm
X
ϕm,k (x)xm,k αm,k
k=0
+
pm
X
2
ϕm,k (x)αm,k
− 2x
k=0
+ 2x
pm
X
pm
X
ϕm,k (x)xm,k
k=0
pm
X
2
ϕm,k (x)αm,k + x
k=0
≤ (Lm ψx2 )(x) + 2αm (Lm e1 )(x) +
k=0
2
(αm +
ϕm,k (x)
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 ),
where
p
2 + 2xα )(L e )(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
δm,x =
and
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
(3.3)
pm
X
ϕm,k (x) = 1
k=0
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6
OVIDIU T. P OP AND M IRCEA D. FĂRCA Ş
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
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 uniform on
m→∞
m→∞
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]) =
k(k+1)
1
0 and wm ([0, 1]) = 4m
, m ∈ N. We consider the nodes ym,k =
, m ∈ N, k ∈
m
√1
{0, 1, . . . , m}. Then it is verified immediately that αm =
, m ∈ N and lim αm =
m+
m(m+1)
m→∞
0. In this case, the operators (Km )m≥1 have the form
(Km f )(x) =
m
X
p
k(k + 1)
m
pm,k (x)f
k=0
0
f ∈ C([0, 1]), x ∈ [0, 1], m ∈ N and δm
<
q
5
4m
+
√2
m+
m(m+1)
!
<
,
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 =
(4k+40)m
.
(2m−1)2
In this case, the operators (Km )m≥1 have the form
m
X
m(4k + 40)
(Km f )(x) =
pm,k (x)f
,
(2m − 1)2
k=0
0
where f ∈ C([0, 22]), x ∈ [0, 1], m ∈ N and δm
<
√
36m3 +2220m2 −399m+81
(2m−1)2
<
√ 45 ,
2m−1
m ∈ N.
Application 3. If I = J = [0, ∞), E(I) = C2 ([0, ∞)), F (J) = C([0, ∞)), K = [0, b],
k
k
pm = ∞, xm,k = m
, ϕm,k (x) = e−mx (mx)
, m ∈ N, k ∈ N0 , we obtain the Mirakjan-Favardk!
Szász operators and we have that um (K) = 0, vm (K) = 0 and wm (K) = mb , m ∈ N. We
2k(k+1)
1
consider the nodes ym,k = m(2k+1)
, m ∈ N, k ∈ N0 and 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
J. Inequal. Pure and Appl. Math., 10(1) (2009), Art. 30, 9 pp.
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L INEAR P OSITIVE O PERATORS
0
where f ∈ C2 ([0, ∞)), x ∈ [0, ∞), m ∈ N and δm
=
q
3b
m
7
+
1
,
4m2
m ∈ N.
Application 4. Let I = J = [0, ∞), E(I) = C2 ([0, ∞)), F (J) = C([0, ∞)), K = [0, b],
x k
k
, ϕm,k (x) = (1 + x)−m m+k−1
, m ∈ N, k ∈ N0 . In this case, we
pm = ∞, xm,k = m
k
1+x
obtain the Baskakov operators and we have
that um (K) = 0, vm (K) = 0 and wm (K) = b(1+b)
,
2m
√
4k2 +4k+2
1
m ∈ N. We consider the nodes ym,k =
, m ∈ N, k ∈ N0 and we have that αm = m√2 .
2m
The operators (Km )m≥1 have the form
!
√
k
∞ 2 + 4k + 2
X
m
+
k
−
1
x
4k
(Km f )(x) = (1 + x)−m
f
,
k
2m
1
+
x
k=0
q
√
2)
0
where f ∈ C2 ([0, ∞)), x ∈ [0, ∞), m ∈ N and δm = b(b+1+2
+ 2m1 2 , m ∈ N.
m
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
ϕm,k (x) =
e 1+x ,
m ∈ N, k ∈ N0 ,
k!
1+x
we obtain the Ismail-May operators and we have √
that um (K) = 0, vm (K) = 0 and wm (K) =
3
b(1+b)2
,
m
k2 (k+1)
m ∈ N. We consider the nodes ym,k =
, m ∈ N, k ∈ N0 and we have that
m
1
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
q
2)
0
where f ∈ C([0, ∞)), m ∈ N and δm
= b(7+6b+3b
+ 9m1 2 , m ∈ N.
3m
Application 6. We consider I = J = [0,
E(I) = F (J) = CB ([0, ∞)), K = [0, b], pm =
∞),
m
k
1
k
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
m
b
the Bleimann-Butzer-Hahn operators and we have that um (K) = 0, vm (K) = b 1+b
and
2
βm k
, m ∈ N. We consider the nodes ym,k = m+1−k
, m ∈ N, k ∈ {0, 1, . . . , m},
wm (K) = 4b(1+b)
m+2
where (βm )m≥1 is a sequence of positive real numbers such that lim m(1 − βm ) = 0 and we
m→∞
have 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
where x ∈ [0, ∞), m ∈ N, f ∈ CB ([0, ∞)).
Application 7. If I = J = [0, 1], E(I) = B([0, 1]), F (J) = C([0, 1]), K = [0, 1], pm = ∞,
k
xm,k = m+k
, ϕm,k (x) = m+k
(1 − x)m+1 xk , m ∈ N, k ∈ N0 , we obtain the Meyer-König
k
1
and Zeller operators and we have that um ([0, 1]) = 0, vm ([0, 1]) = 0 and wm ([0, 1]) = 4(m+1)
,
k+βm
m ∈ N. We consider the nodes ym,k = m+k+βm , m ∈ N, k ∈ N0 , where (βm )m≥1 is a
βm
sequence of positive real numbers so that lim m+β
= 0. Then it is verified immediately that
m
m→∞
αm =
βm
,
m+βm
m ∈ N and 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 + βm
k=0
J. Inequal. Pure and Appl. Math., 10(1) (2009), Art. 30, 9 pp.
http://jipam.vu.edu.au/
8
OVIDIU T. P OP AND M IRCEA D. FĂRCA Ş
0
where f ∈ B([0, 1]), x ∈ [0, 1], m ∈ N and δm
=
q
1
4(m+1)
+
βm (4m+5βm )
,
(m+βm )2
m ∈ N.
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