Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 867203, 13 pages
doi:10.1155/2012/867203
Research Article
A Kantorovich Type of Szasz Operators Including
Brenke-Type Polynomials
Fatma Taşdelen, Rabia Aktaş, and Abdullah Altın
Department of Mathematics, Faculty of Science, Ankara University, Tandoğan, 06100 Ankara, Turkey
Correspondence should be addressed to Rabia Aktaş, raktas@science.ankara.edu.tr
Received 27 September 2012; Accepted 15 November 2012
Academic Editor: Abdelghani Bellouquid
Copyright q 2012 Fatma Taşdelen et al. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
We give a Kantorovich variant of a generalization of Szasz operators defined by means of
the Brenke-type polynomials and obtain convergence properties of these operators by using
Korovkin’s theorem. We also present the order of convergence with the help of a classical approach,
the second modulus of continuity, and Peetre’s K-functional. Furthermore, an example of Kantorovich type of the operators including Gould-Hopper polynomials is presented and Voronovskaya-type result is given for these operators including Gould-Hopper polynomials.
1. Introduction
The Szasz operators also called Szasz-Mirakyan operators which are defined by 1
∞
k
nxk
Sn f; x : e−nx
f
,
k!
n
k0
1.1
where n ∈ N, x ≥ 0, and f ∈ C0, ∞ have an important role in the approximation theory, and
their approximation properties have been investigated by many researchers.
In 2, Jakimovski and Leviatan proposed a generalization of Szasz operators by means
of the Appell polynomials pk x which have the generating functions of the form:
gtetx ∞
pk xtk ,
k0
1.2
2
Abstract and Applied Analysis
k
0 is an analytic function in the disc |z| < R, R > 1 and g1 /
0.
where gz ∞
k0 ak z a0 /
Under the assumption that pk x ≥ 0 for x ∈ 0, ∞, Jakimovski and Leviatan 2, defined the
following linear positive operators:
∞
k
e−nx .
pk nxf
Pn f; x :
g1 k0
n
1.3
After that, Ismail 3 defined another generalization of Szasz operators involving the
k
operators 1.1 and 1.3 by means of Sheffer polynomials. Let Az ∞
0 and
k0 ak z a0 /
∞
k
0
be
analytic
functions
in
the
disc
|z|
<
R,
R
>
1.
Here,
a
Hz k1 hk z h1 /
k and hk
are real. The Sheffer polynomials pk x are generated by
AtexHt ∞
pk xtk .
1.4
k0
With the help of these polynomials, Ismail constructed the following linear positive operators:
∞
k
e−nxH1 Tn f; x :
pk nxf
,
A1 k0
n
n∈N
1.5
under the assumptions
i for x ∈ 0, ∞, pk x ≥ 0,
ii A1 / 0 and H 1 1.
Later, Varma et al. 4 defined another generalization of Szasz operators by means of
the Brenke-type polynomials. Suppose that
At ∞
ar tr ,
a0 /
0,
Bt ∞
br tr ,
r0
br /
0 r ≥ 0
1.6
r0
are analytic functions. The Brenke-type polynomials 5 have generating functions of the form
AtBxt ∞
pk xtk
1.7
k0
from which the explicit form of pk x is as follows:
pk x k
ak−r br xr ,
r0
k 0, 1, 2, . . . .
1.8
Abstract and Applied Analysis
3
Under the assumptions
ak−r br
≥ 0,
A1
i
A1 /
0,
ii
B : 0, ∞ −→ 0, ∞,
0 ≤ r ≤ k, k 0, 1, 2, . . . ,
1.9
iii 1.6 and 1.7 converge for |t| < R,
R > 1,
Varma et al. introduced the linear positive operators Ln f; x via
Ln f; x :
∞
k
1
,
pk nxf
A1Bnx k0
n
1.10
where x ≥ 0 and n ∈ N.
The aim of this paper is to present a Kantorovich type of the operators given by 1.10
and to give their some approximation properties. We consider the Kantorovich version of the
operators 1.10 under the assumptions 1.9 as follows:
∞
n
Kn f; x :
pk nx
A1Bnx k0
k1/n
ftdt,
1.11
k/n
where n ∈ N, x ≥ 0, and f ∈ C0, ∞. It is easy to see that Kn defined by 1.11 is linear and
positive.
In the case of Bt et and At 1, with the help of 1.7, it follows that pk x xk /k!,
so the operators 1.11 reduce to the Szasz-Mirakyan-Kantorovich operators defined by 6
∞
nxk
Kn f; x : ne−nx
k!
k0
k1/n
ftdt.
1.12
k/n
Various approximation properties of the Szasz-Mirakyan-Kantorovich operators and their
iterates may be found in 7–13.
The case of Bt et gives the Kantorovich version of the operators 1.3.
The structure of the paper is as follows. In Section 2, the convergence of the operators
1.11 is given by means of Korovkin’s theorem. The order of approximation is obtained with
the help of a classical approach, the second modulus of continuity, and Peetre’s K-functional
in Section 3. Finally, as an example, we present a Kantorovich type of the operators including
Gould-Hopper polynomials and then we give a Voronovskaya-type theorem for the operators
including Gould-Hopper polynomials.
2. Approximation Properties of Kn Operators
In this section, we give our main theorem with the help of Korovkin theorem. We begin with
the following lemma which is necessary to prove the main result.
4
Abstract and Applied Analysis
Lemma 2.1. For all x ∈ 0, ∞, the operators Kn defined by 1.11 verify
Kn 1; x 1,
2.1
2A 1 A1
B nx
2.2
x
,
Bnx
2nA1
B nx
1
2B nxA 1 A1
A1
Kn s2 ; x A 1 2A 1 x2 x 2
.
Bnx
nA1Bnx
3
n A1
2.3
Kn s; x Proof. Using the generating function of the Brenke-typepolynomials given by 1.7, we can
write
∞
pk nx A1Bnx,
k0
∞
kpk nx A 1Bnx nxA1B nx,
2.4
k0
∞
k2 pk nx n2 x2 A1B nx nxB nx 2A 1 A1 Bnx A 1 A 1 .
k0
From these equalities, the assertions of the lemma are obtained.
Lemma 2.2. For x ∈ 0, ∞, one has
B nx − 2B nx Bnx x2
Kn s − x2 ; x Bnx
2A 1B nx − Bnx A12B nx − Bnx
x
nA1Bnx
1
A1
2
.
A 1 2A 1 3
n A1
2.5
Proof. From the linearity of Kn , we get
Kn s − x2 ; x Kn s2 ; x − 2xKn s; x x2 Kn 1; x.
2.6
Next, we apply Lemma 2.1.
Theorem 2.3. Let
fx
E : f : x ∈ 0, ∞,
is convergent as x −→ ∞ ,
1 x2
B y
B y
lim 1,
lim 1.
y→∞ B y
y→∞ B y
2.7
2.8
Abstract and Applied Analysis
5
If f ∈ C0, ∞ ∩ E, then
lim Kn f; x fx,
2.9
n→∞
and the operators Kn converge uniformly in each compact subset of 0, ∞.
Proof. Using Lemma 2.1 and taking into account the equality 2.8 we get
lim Kn si ; x xi ,
i 0, 1, 2.
n→∞
2.10
The above convergence is satisfied uniformly in each compact subset of 0, ∞. We can then
apply the universal Korovkin-type property vi of Theorem 4.1.4 in 14 to obtain the desired
result.
3. The Order of Approximation
In this section, we deal with the rates of convergence of the Kn f to f by means of a classical
approach, the second modulus of continuity, and Peetre’s K-functional.
∞. If δ > 0, the modulus of continuity of f is defined by
Let f ∈ C0,
w f; δ :
sup fx − f y ,
3.1
x,y∈0,∞
|x−y|≤δ
∞ denotes the space of uniformly continuous functions on 0, ∞. It is also well
where C0,
known that, for any δ > 0 and each x ∈ 0, ∞,
x − y
fx − f y ≤ w f; δ
δ
1 .
3.2
The next result gives the rate of convergence of the sequence Kn f to f by means of
the modulus of continuity.
∞ ∩ E. The Kn operators satisfy the following inequality:
Theorem 3.1. Let f ∈ C0,
Kn f; x − fx ≤ 2w f; λn x ,
3.3
6
Abstract and Applied Analysis
where
B nx − 2B nx Bnx x2
λ : λn x Kn s − x2 ; x Bnx
2A 1B nx − Bnx A12B nx − Bnx
x
nA1Bnx
1
A1
2
.
A 1 2A 1 3
n A1
3.4
Proof. Using 2.1, 3.2, and the linearity property of Kn operators, we can write
Kn f; x − fx ≤
∞
n
Pk nx
A1Bnx k0
k1/n
fs − fxds
k/n
k1/n ∞
n
|s − x|
≤
1 w f; δ ds
Pk nx
A1Bnx k0
δ
k/n
k1/n
∞
n
≤ 1
Pk nx
|s − x|ds w f; δ .
A1Bnxδ k0
k/n
3.5
By using the Cauchy-Schwarz inequality for integration, we get
k1/n
k/n
1
|s − x|ds ≤ √
n
k1/n
1/2
3.6
2
|s − x| ds
k/n
which holds that
∞
Pk nx
k1/n
k/n
k0
∞
1 Pk nx
|s − x|ds ≤ √
n k0
k1/n
1/2
2
|s − x| ds
.
3.7
k/n
By applying the Cauchy-Schwarz inequality for summation on the right-hand side of 3.7,
we have
k1/n
∞
Pk nx
|s − x|ds ≤
k0
k/n
1/2
A1Bnx A1Bnx Kn s − x2 ; x
√
n
n
1/2
A1Bnx Kn s − x2 ; x
n
A1Bnx
λn x1/2 ,
n
3.8
Abstract and Applied Analysis
7
where λn x is given by 3.4. If we use this in 3.5, we obtain
Kn f; x − fx ≤
On choosing δ 1
1
λn x w f; δ .
δ
3.9
λn x, we arrive at the desired result.
Recall that the second modulus of continuity of f ∈ CB 0, ∞ is defined by
w2 f; δ : sup f· 2t − 2f· t f·CB ,
0<t≤δ
3.10
where CB 0, ∞ is the class of real valued functions defined on 0, ∞ which are bounded and
uniformly continuous with the norm f
CB supx∈0,∞ |fx|.
Peetre’s K-functional of the function f ∈ CB 0, ∞ is defined by
K f; δ :
inf
g∈CB2 0,∞
f − g δg 2 ,
CB
C
B
3.11
where
CB2 0, ∞ : g ∈ CB 0, ∞ : g , g ∈ CB 0, ∞ ,
3.12
and the norm g
C2 : g
CB g CB g CB see 15. It is clear that the following inequalB
ity:
K f; δ ≤ M w2 f; δ min1, δf CB ,
3.13
holds for all δ > 0. The constant M is independent of f and δ.
Theorem 3.2. Let f ∈ CB2 0, ∞. The following
Kn f; x − fx ≤ ζf CB2
3.14
holds, where
ζ : ζn x
B nx − 2B nx Bnx 2
x
2Bnx
2A 1B nx − Bnx A12n 1B nx − 2n 1Bnx
x
2nA1Bnx
1
2A 1 A1
A1
2
.
A 1 2A 1 3
2nA1
2n A1
3.15
8
Abstract and Applied Analysis
Proof. From the Taylor expansion of f, the linearity of the operators Kn and 2.1, we have
1 Kn f; x − fx f xKn s − x; x f η Kn s − x2 ; x ,
2
η ∈ x, s.
3.16
Since
Kn s − x; x 2A 1 A1
B nx − Bnx
x
≥0
Bnx
2nA1
3.17
for s ≥ x, by considering Lemmas 2.1 and 2.2 in 3.16, we can write that
B nx − Bnx
2A 1 A1 f x
CB
Bnx
2nA1
1 B nx − 2B nx Bnx 2
x
2
Bnx
2A 1B nx − Bnx A12B nx − Bnx
x
nA1Bnx
1
A1 f 2
A 1 2A 1 CB
3
n A1
B nx − 2B nx Bnx 2
x
≤
2Bnx
2A 1B nx − Bnx A12n 1B nx − 2n 1Bnx
x
2nA1Bnx
1
A1
2A 1 A1 f 2
2
A 1 2A 1 CB
3
2nA1
2n A1
3.18
Kn f; x − fx ≤
which completes the proof.
Theorem 3.3. Let f ∈ CB 0, ∞. Then
Kn f; x − fx ≤ 2M w2 f; δ min1, δf ,
CB
3.19
where
δ : δn x 1
ζn x
2
3.20
and M > 0 is a constant which is independent of the functions f and δ. Also, ζn x is the same as in
Theorem 3.2.
Abstract and Applied Analysis
9
Proof. Suppose that g ∈ CB2 0, ∞. From Theorem 3.2, we can write
Kn f; x − fx ≤ Kn f − g; x Kn g; x − gx gx − fx
≤ 2f − g CB ζg C2
B
2 f − g CB δg C2 .
3.21
B
The left-hand side of inequality 3.21 does not depend on the function g ∈ CB2 0, ∞, so
Kn f; x − fx ≤ 2K f; δ ,
3.22
where Kf; δ is Peetre’s K-functional defined by 3.11. By the relation between Peetre’s Kfunctional and the second modulus of smoothness given by 3.13, inequality 3.21 becomes
Kn f; x − fx ≤ 2M w2 f; δ min1, δf CB
3.23
whence we have the result.
Remark 3.4. Note that when n → ∞, then λn , ζn , and δn tend to zero in Theorems 3.1–3.3
under the assumption 2.8.
4. Special Cases and Further Properties
Gould-Hopper polynomials gd1
x, h 16, which are d-orthogonal polynomial sets of
k
Hermite type 17, are generated by
d1
eht
expxt ∞
tk
gd1
k x, h
k!
k0
4.1
from which it follows that
gd1
k x, h k/d1
m0
k!
hm xk−d1m ,
m!k − d 1m!
4.2
where, as usual, · denotes the integer part.
In 4, the authors showed that the Gould-Hopper polynomials are Brenke-type polyd1
nomials with At eht and Bt et , and the restrictions 1.9 and condition 2.8 for the
operators given by 1.10 are satisfied under the assumption h ≥ 0. These operators including
the Gould-Hopper polynomials are as follows:
L∗n
where x ∈ 0, ∞.
f; x : e
−nx−h
∞ gd1 nx, h k
k
f
,
k!
n
k0
4.3
10
Abstract and Applied Analysis
d1
The special case At eht and Bt et of 1.11 gives the following Kantorovich
version of Kn f; x including the Gould-Hopper polynomials:
∞ gd1 nx, h
k
Kn∗ f; x : ne−nx−h
k!
k0
k1/n
ftdt
4.4
k/n
under the assumption h ≥ 0.
nx, 0 nxk and the operators given by 4.4 reduce to
Remark 4.1. For h 0, we find gd1
k
the Szasz-Mirakyan-Kantorovich operators given by 1.12.
Now, we give a Voronovskaya-type theorem for the operators 4.4. In order to prove
this theorem, we need the following lemmas.
Lemma 4.2. For the operators Kn∗ , one has
Kn∗ 1; x 1,
Kn∗ s; x x 1
hd 1
,
n
2n
2
Kn∗ s2 ; x x2 hd 1 1x
n
1
1
2 hd 1{hd 1 d 2} ,
3
n
3
x
3x2
7
2
∗
3
3
2
hd 1 2 3h d 1 3hd 1d 3 Kn s ; x x n
2
2
n
1
3
1
3
3
2
2
3 h h 3d 1 hd 1 hh 1d 1 hd 1 ,
2
4
n
4.5
4x3
3x2 Kn∗ s4 ; x x4 {hd 1 2} 2 2hh 1d 12 6hd 1 5
n
n
x
3 12h2 d 12 d 2 4h3 d 13 2hd 1 2d2 10d 15 6
n
1
4 h3 h 6d 14 8h1 3hd 13 − 9hh 1d 12
n
7hd 1 2h3 d 13 4dd − 1d 12 h2
1
2
2 2
.
3h d d 1 d − 2d − 1dd 1h 5
Proof. From the generating function 4.1 for the Gould-Hopper polynomials, one can easily
find the above equalities.
Abstract and Applied Analysis
11
Lemma 4.3. For x ∈ 0, ∞, one has
x
1
1
Kn∗ s − x2 ; x 2 hd 1{hd 1 d 2} n n
3
3x2
x
Kn∗ s − x4 ; x 2 3 6hh 2d 12 2hd 1−3d 2 5
n
n
1
4 h3 h 6d 14 2h h2 12h 4 d 13
n
4.6
7hd 1 − 9hh 1d 12 4dd − 1d 12 h2
1
2
2 2
.
3h d d 1 d − 2d − 1dd 1h 5
Proof. It is enough to use Lemma 4.2 to obtain above equalities.
Theorem 4.4. Let f ∈ C2 0, a. Then one has
xf x
1
.
lim n Kn∗ f; x − fx f x hd 1 n→∞
2
2!
4.7
Proof. By Taylor’s theorem, we get
fs fx s − xf x s − x2 f x s − x2 ηs; x,
2!
4.8
where ηs; x ∈ C0, a and lims → x ηs; x 0. If we apply the operator Kn∗ to the both sides
of 4.8, we obtain
Kn∗ f; x fx f xKn∗ s − x; x
f x ∗ Kn s − x2 ; x Kn∗ s − x2 ηs; x; x .
2!
4.9
In view of Lemmas 4.2 and 4.3, the equality 4.9 can be written in the form
1
hd 1
n Kn∗ f; x − fx n
f x
n
2n
1
x
1 f x
2 hd 1{hd 1 d 2} n
n n
3
2!
nKn∗ s − x2 ηs; x; x ,
4.10
12
Abstract and Applied Analysis
where
∞ gd1 nx, h k1/n
k
Kn∗ s − x2 ηs; x; x ne−nx−h
s − x2 ηs; xds.
k!
k/n
k0
4.11
Applying Cauchy-Schwarz inequality, we get
nKn∗ s − x2 ηs; x; x
2 −nx−h
≤n e
∞ gd1 nx, h
k
k0
k1/n
k!
1/2 k1/n
4
s − x ds
k/n
1/2
η s; xds
2
4.12
.
k/n
If we use Cauchy-Schwarz inequality again on the right-hand side of the inequality above,
then we conclude that
nKn∗
2
3 −nx−h
s − x ηs; x; x ≤
n e
∞ gd1 nx, h k1/n
k
k0
·
ne
−nx−h
k!
1/2
s − x ds
k/n
∞ gd1 nx, h k1/n
k
k0
k!
4
1/2
η s; xds
2
4.13
k/n
n2 Kn∗ s − x4 ; x
Kn∗ η2 s; x; x .
In view of Lemma 4.3,
lim n2 Kn∗ s − x4 ; x 3x2
n→∞
4.14
holds. On the other hand, since ηs; x ∈ C0, a and lims → x ηs; x 0, then it follows from
Theorem 2.3 that
lim Kn∗ η2 s; x; x η2 x; x 0.
n→∞
4.15
Considering 4.13, 4.14, and 4.15, we immediately see that
lim nKn∗ s − x2 ηs; x; x 0.
n→∞
4.16
Then, taking limit as n → ∞ in 4.10 and using 4.16, we have
xf x
1
lim n Kn∗ f; x − fx f x hd 1 n→∞
2
2!
which completes the proof.
4.17
Abstract and Applied Analysis
13
Remark 4.5. Getting h 0 in Theorem 4.4 gives a Voronovskaya-type result for the SzaszMirakyan-Kantorovich operators given by 1.12.
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Optimization
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014