Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 754217, 16 pages
doi:10.1155/2012/754217
Research Article
Approximation by the q-Szász-Mirakjan Operators
N. I. Mahmudov
Department of Mathematics, Eastern Mediterranean University, Gazimagusa,
North Cyprus, Mersin 10, Turkey
Correspondence should be addressed to N. I. Mahmudov, nazim.mahmudov@emu.edu.tr
Received 6 September 2012; Accepted 4 December 2012
Academic Editor: Behnam Hashemi
Copyright q 2012 N. I. Mahmudov. 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.
This paper deals with approximating properties of the q-generalization of the Szász-Mirakjan
operators in the case q > 1. Quantitative estimates of the convergence in the polynomial-weighted
spaces and the Voronovskaja’s theorem are given. In particular, it is proved that the rate of
approximation by the q-Szász-Mirakjan operators q > 1 is of order q−n versus 1/n for the classical
Szász-Mirakjan operators.
1. Introduction
The approximation of functions by using linear positive operators introduced via q-Calculus
is currently under intensive research. The pioneer work has been made by Lupaş 1 and
Phillips 2 who proposed generalizations of Bernstein polynomials based on the q-integers.
The q-Bernstein polynomials quickly gained the popularity, see 3–11. Other important
classes of discrete operators have been investigated by using q-Calculus in the case 0 < q < 1,
for example, q-Meyer-König operators 12–14, q-Bleimann, Butzer and Hahn operators 15–
17, q-Szász-Mirakjan operators 18–21, and q-Baskakov operators 22, 23.
In the present paper, we introduce a q-generalization of the Szász operators in the case
q > 1. Notice that different q-generalizations of Szász-Mirakjan operators were introduced
and studied by Aral and Gupta 18, 19, by Radu 20, and by Mahmudov 21 in the case
0 < q < 1. Since we define q-Szász-Mirakjan operators for q > 1, the rate of approximation
by the q-Szász-Mirakjan operators q > 1 is of order q−n , which is essentially better than 1/n
rate of approximation for the classical Szász-Mirakjan operators. Thus our q-Szász-Mirakjan
operators have better approximation properties than the classical Szász-Mirakjan operators
and the other q-Szász-Mirakjan operators.
2
Abstract and Applied Analysis
The paper is organized as follows. In Section 2, we give standard notations that will be
used throughout the paper, introduce q-Szász-Mirakjan operators, and evaluate the moments
of Mn,q . In Section 3 we study convergence properties of the q-Szász-Mirakjan operators in
the polynomial-weighted spaces. In Section 4, we give the quantitative Voronovskaja-type
asymptotic formula.
2. Construction of Mn,q and Estimation of Moments
Throughout the paper we employ the standard notations of q-calculus, see 24, 25.
q-integer and q-factorial are defined by
⎧
n
⎪
⎨ 1 − q , if q ∈ R \ {1},
1−q
nq :
⎪
⎩ n,
if q 1,
nq ! : 1q 2q . . . nq ,
for n ∈ N, 0 0,
2.1
for n ∈ N, 0! 1.
For integers 0 ≤ k ≤ n q-binomial is defined by
nq !
n
.
:
k q
kq !n − kq !
2.2
The q-derivative of a function fx, denoted by Dq f, is defined by
f qx − fx
,
Dq f x : q−1 x
x/
0,
Dq f 0 : lim Dq f x.
x→0
2.3
The formula for the q-derivative of a product and quotient are
Dq uxvx Dq uxvx u qx Dq vx.
2.4
Also, it is known that
Dq xn nq xn−1 ,
Dq Eax aE qax .
2.5
If |q| > 1, or 0 < |q| < 1 and |z| < 1/1 − q, the q-exponential function eq x was defined by
Jackson
eq z :
∞
zk
.
kq !
k0
2.6
Abstract and Applied Analysis
3
If |q| > 1, eq z is an entire function and
eq z ∞
z
1 q − 1 j1
q
j0
,
q > 1.
2.7
There is another q-exponential function which is entire when 0 < |q| < 1 and which converges
when |z| < 1/|1 − q| if |q| > 1. To obtain it we must invert the base in 2.6, that is, q → 1/q:
Eq z : e1/q z ∞ kk−1/2 k
q
z
k0
.
2.8
0 < q < 1.
2.9
kq !
We immediately obtain from 2.7 that
Eq z ∞ 1 1 − q zqj ,
j0
The q-difference equations corresponding to eq z and Eq z are
Dq eq az aeq qz ,
Dq Eq az aEq qaz ,
0.
D1/q eq z D1/q E1/q z E1/q q−1 z eq q−1 z , q /
2.10
Let Cp be the set of all real valued functions f, continuous on 0, ∞, such that wp f is
uniformly continuous and bounded on 0, ∞ endowed with the norm
f : sup wp xfx.
p
2.11
x∈0,∞
Here
w0 x : 1,
wp x : 1 xp −1 ,
if p ∈ N.
2.12
The corresponding Lipschitz classes are given for 0 < α ≤ 2 by
Δ2h fx : fx 2h − 2fx h fx,
ωp2 f; δ : sup Δ2h f ,
Lip2p α : f ∈ Cp : ωp2 f; δ 0δα , δ → 0 .
0<h≤δ
p
Now we introduce the q-parametric Szász-Mirakjan operator.
2.13
4
Abstract and Applied Analysis
Definition 2.1. Let q > 1 and n ∈ N. For f : 0, ∞ → R one defines the Szász-Mirakjan
operator based on the q-integers
Mn,q f; x :
∞
f
k0
kq
nq
nkq xk
1
qkk−1/2 kq !
eq −nq q−k x .
2.14
Similarly as a classical Szász-Mirakjan operator Sn , the operator Mn,q is linear and
positive. Furthermore, in the case of q → 1 we obtain classical Szász-Mirakjan operators.
Moments Mn,q tm ; x are of particular importance in the theory of approximation
by positive operators. From 2.14 one easily derives the following recurrence formula and
explicit formulas for moments Mn,q tm ; x, m 0, 1, 2, 3, 4.
Lemma 2.2. Let q > 1. The following recurrence formula holds
m xqj
m
j −1
t
M
;
q
x
.
Mn,q tm1 ; x n,q
j nm−j
j0
2.15
q
Proof. The recurrence formula 2.15 easily follows from the definition of Mn,q and qkq 1 k 1q as show below:
Mn,q tm1 ; x
∞ km1
q
m1
k0 nq
nkq xk
qkk−1/2 kq !
∞ km
q
1
k
nk−1
q x
nm
q
k1
qkk−1/2
k − 1q !
m
∞ qkq 1
k0
1
nm
q
eq −nq q−k x
eq −nq q−k x
1
nkq xk1
qkk1/2
kq !
eq −nq q−k q−1 x
m ∞
nkq xk1 1 1
m j j
eq −nq q−k q−1 x
q
k
q kk1/2
m
kq !
nq j0 j
q
k0
∞ kj
m nkq xk xqj 1
q
m
−k −1
−n
e
q
q
x
q
q
j nm−j k0 nj qkk−1/2 kq !qk
j0
q
q
m xqj
m
j −1
t
M
;
q
x
.
n,q
j nm−j
j0
q
2.16
Abstract and Applied Analysis
5
Lemma 2.3. The following identities hold for all q > 1, x ∈ 0, ∞, n ∈ N, and k ≥ 0:
xDq snk q; x nq
Mn,q t
m1
kq
nq
− x snk q; x ,
2.17
x
;x Dq Mn,q tm ; x xMn,q tm ; x,
nq
where snk q; x : 1/qkk−1/2 nkq xk /kq !eq −nq q−k x.
Proof. The first identitiy follows from the following simple calculations
xDq snk q; x
kq
nkq xk
1
qkk−1/2 kq !
−k
1
−k
eq −nq x − xq nq
nkq qk xk
qkk−1/2 kq !
kq
kq snk q; x − xnq snk q; x nq
− x snk q; x .
nq
eq −nq q−k x
2.18
The second one follows from the first:
xDq Mn,q t ; x nq
m
∞
nq
m nq
k0
∞
kq
kq
kq
nq
m1
− x snk q; x
m
∞
kq
snk q; x − nq x
nq
k0
k0
nq Mn,q tm1 ; x − nq xMn,q tm ; x.
nq
snk q; x
2.19
Lemma 2.4. Let q > 1. One has
Mn,q 1; x 1,
Mn,q t; x x,
1
x,
Mn,q t2 ; x x2 nq
2q 2
1
Mn,q t3 ; x x3 x x,
nq
n2q
2.20
x3
x2
1
Mn,q t4 ; x x4 3 2q q2
3 3q q2
x.
2
nq
nq n3q
Proof. For a fixed x ∈ R , by the q-Taylor theorem 24, we obtain
ϕn t ∞ t − xk
1/q
k0
k1/q !
k
D1/q
ϕn x.
2.21
6
Abstract and Applied Analysis
Choosing t 0 and taking into account
k
D1/q
eq −nq x −1k q−kk−1/2 nkq eq −nq q−k x 2.22
−xk1/q −1k xk q−kk−1/2 ,
we get for ϕn x eq −nq x that
1 ϕn 0 −1k xk
k
D1/q
ϕn x
kk−1/2 k
q
!
1/q
k0
∞
−1k xk
k0
∞
∞
kq !
−1k q−kk−1/2 nkq eq −nq q−k x
nkq xk
k0 kq
eq
!qkk−1/2
2.23
−nq q−k z .
In other words Mn,q 1; x 1.
Calculation of Mn,q ti ; x, i 1, 2, 3, 4, based on the recurrence formula 2.17
or 2.15. We only calculate Mn,q t3 ; x and Mn,q t4 ; x:
Mn,q t3 ; x x
Dq Mn,q t2 ; x xMn,q t2 ; x
nq
x
1
1
2
x x x
2q x nq
nq
nq
1
n2q
x
2q 2
x x3 ,
nq
Mn,q t4 ; x x
Dq Mn,q t3 ; x xMn,q t3 ; x
nq
⎛
⎛
⎞
⎞
2q
2
q
1
x ⎝ 1
2⎠
2
3
x⎝
x
x x ⎠
2q x 3q x
nq n2q nq
nq
n2q
x2
x3
2
2
x
3
3q
q
3
2q
q
x4 .
nq
n3q
n2q
1
2.24
Abstract and Applied Analysis
7
Lemma 2.5. Assume that q > 1. For every x ∈ 0, ∞ there hold
Mn,q t − x2 ; x Mn,q t − x3 ; x Mn,q t − x4 ; x 1
n3q
x
,
nq
2.25
x2
x
q
−
1
,
nq
n2q
2.26
1
x2
2 x 3
x q2 3q − 1
q−1
.
2
nq
nq
2.27
Proof. First of all we give an explicit formula for Mn,q t − x4 ; x.
Mn,q t − x3 ; x Mn,q t3 ; x − 3xMn,q t2 ; x 3x2 Mn,q t; x − x3
2q 2
1
x
2
3x3 − x3
x x x − 3x x nq
nq
n2q
3
1
n2q
x2
x q−1
,
nq
Mn,q t − x4 ; x Mn,q t4 ; x − 4xMn,q t3 ; x 6x2 Mn,q t2 ; x − 4x3 Mn,q t; x x4
x2
x3
2
2
x
3
3q
q
3
2q
q
x4
3
2
n
nq
nq
q
1
⎛
⎞
2
q
1
x
2
3
2
2
− 4x4 x4
− 4x⎝
x
x x ⎠ 6x x nq
nq
n2q
1
n3q
x2
2 x 3
x −1 3q q2
q−1
.
2
nq
nq
2.28
Now we prove explicit formula for the moments Mn,q tm ; x, which is a q-analogue of
a result of Becker, see 26, Lemma 3.
Lemma 2.6. For q > 1, m ∈ N there holds
Mn,q tm ; x m
Sq m, j
j1
xj
m−j
nq
,
2.29
8
Abstract and Applied Analysis
where
Sq m 1, j j Sq m, j Sq m, j − 1 ,
Sq 0, 0 1,
Sq m, 0 0,
m ≥ 0, j ≥ 1,
Sq m, j 0, m < j.
m > 0,
2.30
In particular Mn,q tm ; x is a polynomial of degree m without a constant term.
Proof. Because of Mn,q t; x x, Mn,q t2 ; x x2 x/nq , the representation 2.29 holds true
for m 1, 2 with Sq 2, 1 1, Sq 1, 1 1.
Now assume 2.29 to be valued for m then by Lemma 2.3 we have
Mn,q tm1 ; x x
Dq Mn,q tm ; x xMn,q tm ; x
nq
m
m
xj−1
xj
x x
S
j q Sq m, j
m,
j
q
m−j
m−j
nq j1
nq
nq
j1
m
j q Sq m, j
j1
xj
m−j1
nq
m
xj1
Sq m, j
m−j
nq
j1
2.31
x
Sq m, 1 xm1 Sq m, m
nm
m j q Sq m, j Sq m, j − 1
j2
xj
m−j1
nq
.
Remark 2.7. Notice that Sq m, j are Stirling numbers of the second kind introduced by
Goodman et al. in 8. For q 1 the formulae 2.30 become recurrence formulas satisfied
by Stirling numbers of the second type.
3. Mn,q in Polynomial-Weighted Spaces
Lemma 3.1. Let p ∈ N ∪ {0} and q ∈ 1, ∞ be fixed. Then there exists a positive constant K1 q, p
such that
Mn,q 1/wp ; x ≤ K1 q, p ,
p
n ∈ N.
3.1
n ∈ N.
3.2
Moreover for every f ∈ Cp one has
Mn,q f ≤ K1 q, p f ,
p
p
Thus Mn,q is a linear positive operator from Cp into Cp for any p ∈ N ∪ {0}.
Abstract and Applied Analysis
9
Proof. The inequality 3.1 is obvious for p 0. Let p ≥ 1. Then by 2.29 we have
p
xj
wp xMn,q 1/wp ; x wp x wp x Sq p, j
≤ K1 q, p ,
p−j
nq
j1
3.3
K1 q, p is a positive constant depending on p and q. From this follows 3.1. On the other
hand
1 Mn,q f ≤ f M
,
p
p
n,q wp 3.4
p
for every f ∈ Cp . By applying 3.1, we obtain 3.2.
Lemma 3.2. Let p ∈ N ∪ {0} and q ∈ 1, ∞ be fixed. Then there exists a positive constant K2 q, p
such that
K2 q, p
t − ·2
;· ≤
,
Mn,q
wp t
nq
n ∈ N.
3.5
p
Proof. The formula 2.25 imply 3.5 for p 0. We have
Mn,q
t − x2
;x
wp t
Mn,q t − x2 ; x Mn,q t − x2 tp ; x ,
3.6
for p, n ∈ N. If p 1 then we get
Mn,q t − x2 1 t; x Mn,q t − x2 ; x Mn,q t − x2 t; x
3
2
Mn,q t − x ; x 1 xMn,q t − x ; x ,
which by Lemma 2.5 yields 3.5 for p 1.
Let p ≥ 2. By applying 2.29, we get
wp xMn,q t − x2 tp ; x
wp x Mn,q tp2 ; x − 2xMn,q tp1 ; x x2 Mn,q tp ; x
⎛
wp x⎝xp2 p1
Sq p 2, j
j1
xj
p2−j
nq
⎞
p−1
j2
x
⎠
xp2 Sq p, j
p−j
nq
j1
p
xj1
− 2xp2 − 2 Sq p 1, j
p1−j
nq
j1
3.7
10
Abstract and Applied Analysis
⎛
wp x⎝
p
xj1
Sq p 2, j − 2Sq p 1, j Sq p, j
p1−j
nq
j2
Sq p 2, 1
wp x
x
p1
nq
⎞
x2
⎠
Sq p 2, 2 − 2Sq p 2, 1
p
nq
x
Pp q; x ,
nq
3.8
where Pp q; x is a polynomial of degree p. Therefore one has
x
.
wp xMn,q t − x2 tp ; x ≤ K2 q, p
nq
3.9
Our first main result in this section is a local approximation property of Mn,q stated
below.
Theorem 3.3. There exists an absolute constant C > 0 such that
x
wp xMn,q g; x − gx ≤ K3 q, p g ,
nq
3.10
where g ∈ Cp2 , q > 1 and x ∈ 0, ∞.
Proof. Using the Taylor formula
gt gx g xt − x t s
x
x
g udu ds,
g ∈ Cp2 ,
we obtain that
t s
wp x Mn,q g; x − gx wp xMn,q
g udu ds; x x x
t s
≤ wp xMn,q g udu ds; x
x x
t s
≤ wp xMn,q g p 1 um du ds; x
x x
1 ≤ wp x g p Mn,q t − x2 1/wp x 1/wp t ; x
2
3.11
Abstract and Applied Analysis
11
1
g Mn,q t − x2 ; x wp xMn,q t − x2 wp t; x
p
2
x
.
≤ K3 q, x g p
nq
≤
3.12
Now we consider the modified Steklov means
4
fh x : 2
h
h/2
2fx s t − fx 2s t ds dt.
3.13
0
fh x has the following properties:
4
fx − fh x 2
h
h/2
0
fh x h−2 8Δ2h/2 fx − Δ2h fx
Δ2st fxds dt,
3.14
and therefore
f − fh ≤ ωp2 f; h ,
p
f ≤ 1 ωp2 f; h .
h p
2
9h
3.15
We have the following direct approximation theorem.
Theorem 3.4. For every p ∈ N ∪ {0}, f ∈ Cp and x ∈ 0, ∞, q > 1, one has
⎛ ⎞
q−1 x
x
Mp ωp2 ⎝f; n
wp xMn,q f; x − fx ≤ Mp ωp2 f;
⎠.
nq
q −1
3.16
Particularly, if Lip2p α for some α ∈ 0, 2, then
wp xMn,q f; x − fx ≤ Mp
x
nq
α/2
3.17
.
Proof. For f ∈ Cp and h > 0
Mn,q f; x − fx ≤ Mn,q f − fh ; x − f − fh x Mn,q fh ; x − fh x
3.18
and therefore
wp x Mn,q f; x − fx ≤ f − fh p wp xMn,q
x
.
K3 q, p fh p
nq
1
;x
wp t
1
3.19
12
Abstract and Applied Analysis
Since wp xMn,q 1/wp t; x ≤ K1 q, p, we get that
!
2
wp x Mn,q f; x − fx ≤ M q, p ωp f; h 1 Thus, choosing h #
"
x
.
nq h2
3.20
x/nq , the proof is completed.
Corollary 3.5. If p ∈ N ∪ {0}, f ∈ Cp , q > 1 and x ∈ 0, ∞, then
lim Mn,q f; x fx.
n→∞
3.21
This converegnce is uniform on every a, b, 0 ≤ a < b.
Remark 3.6. Theorem 3.4 shows the rate of approximation by the q-Szász-Mirakjan operators
q > 1 is of order q−n versus 1/n for the classical Szász-Mirakjan operators.
4. Convergence of q-Szász-Mirakjan Operators
An interesting problem is to determine the class of all continuous functions f such that
Mn,q f converges to f uniformly on the whole interval 0, ∞ as n → ∞. This problem
was investigated by Totik 27, Theorem 1 and de la Cal and Cárcamo 28, Theorem 1. The
following result is a q-analogue of Theorem 1 28.
Theorem 4.1. Assume that f : 0, ∞ → R is bounded or uniformly continuous. Let
f ∗ z f z2 ,
z ∈ 0, ∞.
4.1
One has, for all t > 0 and x ≥ 0,
1
Mn,q f; x − fx ≤ 2ω f ∗ ;
.
nq
4.2
Therefore, Mn,q f; x converges to f uniformly on 0, ∞ as n → ∞, whenever f ∗ is uniformly
continuous.
Proof. By the definition of f ∗ we have
√ Mn,q f; x Mn,q f ∗ · ; x .
4.3
Abstract and Applied Analysis
13
Thus we can write
√ √ Mn,q f; x − fx Mn,q f ∗ · ; x − f ∗ x ⎛ ⎛
⎞
⎞
kq
∞
√
⎠ − f ∗ x ⎠sn,k q; x ⎝f ∗ ⎝
nq
k0
⎛ ⎛
⎞
⎞
∞ kq
√ ∗
∗
⎝f ⎝
⎠sn,k q; x
⎠−f
≤
x
n
q
k0 ⎞
∞
kq √ ≤
ω⎝f ∗ ; − x⎠sn,k q; x
n
q
k0
⎛
4.4
#
⎞
⎛
√ ∞
kq /nq − x
√
√
⎟
⎜
≤
ω⎝f ∗ ;
√ √ Mn,q · − x; x ⎠sn,k q; x .
Mn,q
· − x ;x
k0
Finally, from the inequality
ω f ∗ ; αδ ≤ 1 αω f ∗ ; δ ,
α, δ ≥ 0,
4.5
we obtain
#
⎛
√ ⎞
∞
kq /nq − x
∗
√ √ ⎟
⎜
Mn,q f; x − fx ≤ ω f ; Mn,q · − x; x
√ √ ⎠sn,k q; x
⎝1 Mn,q · − x; x
k0
√ √ 2ω f ∗ ; Mn,q · − x; x .
4.6
In order to complete the proof we need to show that we have for all t > 0 and x > 0,
√ √ Mn,q · − x; x ≤
1
.
nq
4.7
14
Abstract and Applied Analysis
Indeed we obtain from the Cauchy-Schwarz inequality:
∞ kq √ √ √ Mn,q
· − x ;x n − xsn,k q; x
q
k0 ∞
∞ k
/n
−
x
k
q
q
1 q
s
−
x
q;
x
≤
√
#
sn,k q; x
n,k
√
x k0 nq
kq /nq x
k0
2
&
∞ k
1 1
q
≤ √
− x sn,k q; x √
Mn,q · − x2 ; x
x k0 nq
x
1
√
x
1
x
nq
4.8
1
nq
showing 4.2, and completing the proof.
Next we prove Voronovskaja type result for q-Szász-Mirakjan operators.
Theorem 4.2. Assume that q ∈ 1, ∞. For any f ∈ Cp2 the following equality holds
1
lim nq Mn,q f; x − fx f xx,
2
n→∞
4.9
for every x ∈ 0, ∞.
Proof. Let x ∈ 0, ∞ be fixed. By the Taylor formula we may write
1
ft fx f xt − x f xt − x2 rt; xt − x2 ,
2
4.10
where rt; x is the Peano form of the remainder, r·; x ∈ Cp , and limt → x rt; x 0. Applying
Mn,q to 4.10 we obtain
n Mn,q f; x − fx f xnq Mn,q t − x; x
1
f xnq Mn,q t − x2 ; x nq Mn,q rt; xt − x2 ; x .
2
4.11
By the Cauchy-Schwartz inequality, we have
Mn,q
&
#
2
rt; xt − x ; x ≤ Mn,q r t; x; x Mn,q t − x4 ; x .
2
4.12
Abstract and Applied Analysis
15
Observe that r 2 x; x 0. Then it follows from Corollary 3.5 that
lim Mn,q r 2 t; x; x r 2 x; x 0.
n→∞
4.13
Now from 4.12, 4.13, and Lemma 2.5 we get immediately
lim nq Mn,q rt; xt − x2 ; x 0.
n→∞
4.14
The proof is completed.
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