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Rates of convergence of certain King-type operators for functions with derivative of bounded variation
Mathematical and Computer Modelling 52 (2010) 334–345
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Rates of convergence of certain King-type operators for functions with derivative of bounded variation M. Ali Özarslan a , Oktay Duman b,∗ , Cem Kaanoğlu c a
Eastern Mediterranean University, Faculty of Arts and Sciences, Department of Mathematics, Gazimagusa, Mersin 10, Turkey
b
TOBB Economics and Technology University, Faculty of Arts and Sciences, Department of Mathematics, Söğütözü TR-06530, Ankara, Turkey
c
Cyprus International University, Faculty of Engineering, Lefkoşa, Mersin 10, Turkey
article
abstract
info
Article history: Received 4 November 2009 Received in revised form 24 February 2010 Accepted 24 February 2010
In this paper, we estimate the rates of pointwise approximation of certain King-type positive linear operators for functions with derivative of bounded variation. We also extend our results to the statistical approximation process via the concept of statistical convergence. © 2010 Elsevier Ltd. All rights reserved.
Keywords: A-statistical convergence Statistical approximation King-type operators Functions with derivative of bounded variation Rates of convergence
1. Introduction King-type modification of positive linear operators preserves the test function e2 (x) = x2 and so provides better error estimations on some appropriate domains than the classical ones. Such a modification was first noticed by King [1] for the classical Bernstein polynomials. Later, this idea was applied to some other well-known approximating operators, such as, the Szász-Mirakjan operators [2], the Baskakov operators [3], the Meyer-König and Zeller operators [4], the BernsteinChlodovsky operators [5] and more general summation-type positive linear operators [6]. It is well known that the classical Szász-Mirakjan operators Sn and the classical Baskakov operators Vn are defined respectively by Sn (f ; x) :=
∞ X k f
k=0
n
sk,n (x),
sk,n (x) :=
vk,n (x),
vk,n (x) :=
e−nx (nx)k k!
,
and Vn (f ; x) :=
∞ X k f
k=0
n
n+k−1 k
xk (1 + x)−n−k ,
where f : [0, ∞) → R is a function such that the above series are convergent for each n. In order to obtain better error estimations we modify and combine the above approximating operators as the operators Ln given by Ln (f ; x) :=
∞ X k f
k=0
∗
n
Pk,n (αn (x)),
Corresponding author. Tel.: +90 3122924141; fax: +90 3122924324. E-mail addresses: [email protected] (M. Ali Özarslan), [email protected] (O. Duman), [email protected] (C. Kaanoğlu).
0895-7177/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2010.02.048
(1.1)
M. Ali Özarslan et al. / Mathematical and Computer Modelling 52 (2010) 334–345
where {Pk,n } ∈ {sk,n }, {vk,n } and √ −1 + 1 + 4n2 x2 2n αn (x) := p −1 + 1 + 4n(n + 1)x2 2(n + 1)
335
if Pk,n ≡ sk,n (1.2) if Pk,n ≡ vk,n .
Throughout the paper, we use
√
un (x) :=
−1 +
1 + 4n2 x2
2n It is known from [2] that if we take Pk,n ≡ sk,n
and vn (x) :=
−1 +
1 + 4n(n + 1)x2
p
2(n + 1)
.
and αn ≡ un ,
(1.3)
then the operators Ln given by (1.1) have better error estimations on the interval [0, ∞) than the classical Szász-Mirakjan operators. Furthermore, the following conditions hold provided that (1.3) holds: (i) Ln (e0 ; x) = e0 (x) = 1, (ii) Ln (e1 ; x) = un (x) with e1 (x) = x, (iii) Ln (e2 ; x) = e2 (x) = x2 . Similarly, it follows from [3] that if one takes Pk,n ≡ vk,n and αn ≡ vn , (1.4) then the operators Ln have better estimations than the classical Baskakov operators. In this case, we get (i), (iii) and also
(ii)0 Ln (e1 ; x) = vn (x) with e1 (x) = x. Now, as usual, let BV [0, ∞) be the class of all functions defined on [0, ∞) having bounded variation on every finite subinterval of [0, ∞). Now, for γ ≥ 0, we consider the following function spaces: DBVEγ [0, ∞) := {f : [0, ∞) → R : f 0 ∈ BV [0, ∞) and |f (t )| ≤ Meγ t for every t ≥ 0 and some M > 0} and DBVγ [0, ∞) := {f : [0, ∞) → R : f 0 ∈ BV [0, ∞) and |f (t )| ≤ M (1 + t γ ) for every t ≥ 0 and some M > 0}. We also use the following notations: p(x) :=
f 0 (x+ ) + f 0 (x− )
,
f 0 (x+ ) − f 0 (x− )
q(x) :=
. (1.5) 2 2 In this paper, we estimate the rates of pointwise approximation of the operators Ln on the subclasses mentioned above. Recall that such investigations were already studied by many authors for the classical approximating operators (see, for instance, [7–13]). Some other results regarding the statistical approximation process may be found in the papers [14,15,4,16] and cited therein. Now, as usual, we define the auxiliary function fx , by f (t ) − f (x− ), 0, fx (t ) = f (t ) − f (x+ ),
if 0 ≤ t < x, if t = x if x < t < ∞.
Then, we mainly obtain the following results. Theorem 1.1. Assume that (1.3) holds. Then, for every f ∈ DBVEγ [0, ∞) (γ ≥ 0), x ∈ (0, ∞) and for sufficiently large n, we have
|Ln (f ; x) − f (x)| ≤
√
2 x (|p(x)| + |q(x)|) + Me
+2 1 −
un ( x )
+ 2x 1 −
x un (x) x
γ ex
r 0 + un (x) + x f (x ) 1− x
|f (x)| + f (2x) − f (x) − xf 0 (x+ )
√ X [ n]
x+ kx
_
k =1
x− kx
n _ x fx0 + √ fx0
n
x+ √x
(1.6)
x− √x
n
where p(x) and q(x) are defined as in (1.5). Theorem 1.2. Assume that (1.4) holds. Then, for every f ∈ DBVγ [0, ∞) (γ ≥ 2), x ∈ (0, ∞) and for sufficiently large n, we have
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M. Ali Özarslan et al. / Mathematical and Computer Modelling 52 (2010) 334–345
r 0 + o vn (x) |Ln (f ; x) − f (x)| ≤ 2 x (|p(x)| + |q(x)|) + M K (x) + x f (x ) 1− x vn (x) |f (x)| + f (2x) − f (x) − xf 0 (x+ ) +2 1 − √ n
p
x
+ 2x 1 −
vn (x)
√ X [ n]
x+ kx
_
x
k=1
x− kx
n _ x fx0 + √ fx0
n
x+ √x
(1.7)
x− √x
n
for some function K (x) defined on (0, ∞). The proofs of the above theorems will be given in the last section after some auxiliary results. We also extend the above results to the statistical approximation process via the concepts of A-density and A-statistical convergence (see Section 3 for details). 2. Auxiliary results Observe that we may rewrite (1.1) as follows: Ln (f ; x) =
∞
Z
f (t ) 0
∂ {Kn (x, t )} dt , ∂t
(2.1)
where
X Pk,n (x),
Kn (x, u) :=
if 0 < u < ∞
k≤nu
0,
if u = 0.
Also let
λn (x, t ) :=
Z 0
t
∂ {Kn (x, u)}du. ∂u
(2.2)
Then observe that
λn (x, t ) ≤ 1. In this section, some results are given which are necessary to prove our main theorems. Remark 2.1. It is easily verified that Ln (t − x; x) = αn (x) − x and
Ln ((t − x)2 ; x) = 2x(x − αn (x)),
where αn (x) is defined by (1.2). Then, we have 1
Ln (|t − x| ; x) ≤ (Ln ((t − x)2 ; x)) 2 ≤
p
2x (x − αn (x)).
Lemma 2.1. For every x ∈ (0, ∞) and n ∈ N, we have
n (x)) {Kn (x, u)} du ≤ 2x((xx−α for 0 ≤ y < x, −y)2 R∞ ∂ 2x(x−αn (x)) (b) 1 − λn (x, z ) = z ∂ u {Kn (x, u)} du ≤ (x−z )2 for x < z < ∞.
(a) λn (x, y) =
Ry
∂
0 ∂u
Proof. We first prove (a). By Remark 2.1, we get, for 0 ≤ y < x, that y
Z 0
∂ {Kn (x, u)} du ≤ ∂u
Z y
≤
x−u
2
x−y
0
2x(x − αn (x))
(x − y)2
∂ {Kn (x, u)} du ∂u .
Similarly, for x < z < ∞, we conclude that ∞
Z z
∂ {Kn (x, u)} du ≤ ∂u
Z
≤ 0
which is the desired result.
∞
x−u
2
∂ {Kn (x, u)} du ∂u
2
∂ {Kn (x, u)} du ∂u
x−z
z
Z
≤
∞
x−u x−z
2x(x − αn (x))
(x − z )2
,
M. Ali Özarslan et al. / Mathematical and Computer Modelling 52 (2010) 334–345
337
Lemma 2.2. For every f ∈ DBV [0, ∞), x ∈ (0, ∞) and n ∈ N, we have
p |Ln (f ; x) − f (x)| ≤ {|p(x)| + |q(x)|} 2x(x − αn (x)) + An (f ; x) + Bn (f ; x), where αn (x) is given by (1.2), p(x) and q(x) are defined as in (1.5), and also An (f ; x) :=
t
Z x Z
fx0 (u)du
x
0
∂ {Kn (x, t )}dt ∂t
(2.3)
and Bn (f ; x) =:
∞
Z
t
Z
fx0 (u)du
x
x
∂ {Kn (x, t )}dt . ∂t
(2.4)
Proof. For f ∈ DBV [0, ∞) and x ∈ (0, ∞), we may write that f 0 (u) = p(x) + q(x)sgn(u − x) + fx0 (u) + f 0 (x) − p(x) δx (u),
where
δx (u) :=
1, 0,
if u = x if u 6= x.
Now using the above equality and also considering the linearity of the operators Ln , we obtain that
Z ∞ ∂ |Ln (f ; x) − f (x)| = {Kn (x, t )} dt (f (t ) − f (x)) ∂ t 0 Z ∞ Z t ∂ {Kn (x, t )} dt = f 0 (u)du ∂t 0 x Z ∞ Z ∞ ∂ ∂ |t − x| {Kn (x, t )} dt + |q(x)| {Kn (x, t )} dt |t − x| ≤ |p(x)| ∂t ∂t 0 Z ∞0Z t Z Z t 0 ∞ ∂ ∂ 0 {Kn (x, t )} dt + f (x) − p(x) {Kn (x, t )} dt . + fx (u)du δx (u)du ∂t ∂t 0 x 0 x Rt Since x δx (u)du = 0, we conclude that |Ln (f ; x) − f (x)| ≤ {|p(x)| + |q(x)|} Ln (|t − x| ; x) Z x Z t Z ∞ Z t ∂ ∂ 0 0 {Kn (x, t )} dt + {Kn (x, t )} dt . fx (u)du fx (u)du + ∂t ∂t 0 x x x Then it follows from Remark 2.1 that
p |Ln (f ; x) − f (x)| ≤ {|p(x)| + |q(x)|} 2x(x − αn (x)) + An (f ; x) + Bn (f ; x), which is the desired result.
Lemma 2.3. For every f ∈ DBV [0, ∞), x ∈ (0, ∞) and n ∈ N, we have
|An (f ; x)| ≤ 2x 1 −
αn (x) x
√ X [ n]
x _
k=1
x− kx
x _ x fx0 , fx0 + √
n
x− √x
n
where An (f ; x) is given by (2.3). Proof. Applying integration by parts we have
Z x Z t ∂ {Kn (x, t )} dt |An (f ; x)| = fx0 (u)du ∂t Z0 x x ∂ {Kn (x, t )} dt = fx (t ) ∂t Z 0x 0 = λn (x, t )fx (t )dt 0
x− √x
Z
n
≤ 0
|λn (x, t )| fx0 (t ) dt +
Z
x x− √x n
|λn (x, t )| fx0 (t ) dt ,
338
M. Ali Özarslan et al. / Mathematical and Computer Modelling 52 (2010) 334–345
and hence, by Lemma 2.1,
|An (f ; x)| ≤
x− √x
Z
x _
n
0
!
Z
|λn (x, t )| dt +
0
fx
x− √x
t
≤ 2x(x − αn (x))
x− √x
x _
n
0
≤ 2x(x − αn (x))
x− √x
Z
x _
n
0
1
(x − t )2
t
! fx0
t
|λn (x, t )| dt
Z
x
fx
x _ fx0 dt
! fx0
0
t
n
Z
!
x _
x
dt + x− √x
n
x− √x
n
x _ 1 x dt + √ fx0 . (x − t )2 n x− √x n
x
Letting u = x−t , since x− √x
Z
n
0
x _
√
! fx0
1
(x − t )2
t
dt =
1
Z
x _
x
1
√
≤
n
[ n] 1X
x k=1
fx0 du
x− ux
x _
fx0 ,
x− kx
we may write that
|An (f ; x)| ≤ 2x 1 −
αn (x)
which completes the proof.
x
√ X [ n]
x _
k=1
x− kx
x _ x fx0 + √ fx0 ,
n
x− √x
n
Lemma 2.4. Assume that (1.3) holds. Let γ ≥ 0 be fixed. Then, for every x ∈ (0, ∞) and for sufficiently large n, we have Ln e2γ t ; x ≤ e2eγ x .
Proof. By (1.3), we see that Ln (e
2γ t
; x) = = = =
=
k ∞ X nun (x)e2γ /n exp{−nun (x)} k! k=0 exp −nun (x) + nun (x)e2γ /n . exp nun (x) −1 + e2γ /n ( ) ∞ X (2γ /n)k exp nun (x) . k! k=1 ( ) ∞ X (2γ /n)k−1 exp 2γ un (x) . k! k =1
Letting n ≥ 2γ , we get
(
Ln (e
2γ t
∞ X 1 ; x) ≤ exp 2γ un (x) k ! k=0
)
= e2eγ un (x) . Since un (x) ≤ x for every x ∈ (0, ∞), the last inequality implies that Ln (e2γ t ; x) ≤ e2eγ x . Lemma 2.5. Assume that (1.4) holds. For r ∈ N3 := {3, 4, . . .}, there exist polynomials pr ,m (x) := a1,m x + a2,m x2 + · · · + ar ,m xr (m = 1, 2, . . . , r − 2) having degree r with non-negative coefficients such that
M. Ali Özarslan et al. / Mathematical and Computer Modelling 52 (2010) 334–345
Ln t r ; x = vnr (x) +
339
r r −2 v (x) + vnr −1 (x) X pr ,m (vn (x)) −1 n + . m+1
r (r − 1) 2
n
(2.5)
n
m=1
Proof. For the proof, we use mathematical induction. Under the condition (1.4), by a direct computation, we may write that Ln (t 3 ; x) = vn3 (x) +
2vn3 (x) + 2vn2 (x) n
+
3vn3 (x) + 4vn2 (x) + vn (x)
+
9vn4 (x) + 14vn3 (x) + 5vn2 (x)
n2
and Ln (t 4 ; x) = vn4 (x) +
+
5vn4 (x) + 5vn3 (x)
n 6vn4 (x) + 14vn3 (x) + 9vn2 (x) + vn (x) n3
n2
.
So, (2.5) holds true for r = 3, 4. Assume now that (2.5) holds for r > 4. We first observe that Ln (t r +1 ; x) =
vn2 (x) + vn (x) 0 r Ln (t ; x) + vn (x)Ln (t r ; x). nvn0 (x)
Then, we get from (2.5) that Ln (t r +1 ; x) =
vn2 (x) + vn (x) n
+
r vnr −1 (x) +
r (r − 1) 2
n
r −2 v (x) + vn (x) X qr −1,m (vn (x)) 2 n
n
+
r (r − 1) 2
+ vnr +1 (x)
nm+1
m=1
vnr +1 (x) + vnr (x)
−1
r −1 r vn (x) + (r − 1)vnr −2 (x) −1
n
+ v n ( x)
r −2 X pr ,m (vn (x))
nm+1
m=1
where, for m = 1, . . . , r − 2, qr −1,m (x) := p0r ,m (x) = a1,m + 2a2,m x + · · · + rar ,m xr −1 having degree r − 1 at most. Thus, the last equality may be written in the following form: Ln (t r +1 ; x) = vnr +1 (x) +
+ +
r (r − 1) 2
r (r − 1) 2
−1+r
vnr +1 (x) + vnr (x) n
v (x) + vn (x) r v
2 n
−1
nm+2
= vnr +1 (x) +
(x) + (r − 1)vnr −2 (x)
n2
r −2 X qr −1,m (vn (x)) vn2 (x) + vn (x) m=1
r −1 n
(r + 1)r 2
+
r −2 X vn (x)pr ,m (vn (x))
nm+1
m=1
r +1 r −1 ∗ pr +1,m (vn (x)) v (x) + vnr (x) X , −1 n + m+1 n
m=1
n
where p∗r +1,m (x) := a∗1,m x + a∗2,m x2 + · · · + a∗r +1,m xr +1 for appropriate coefficients
a∗1,m , a∗2,m , . . . , a∗r +1,m (m
(m = 1, 2, . . . , r − 1)
= 1, 2, . . . , r − 1). Therefore, the proof is completed.
Lemma 2.6. Assume that (1.3) holds. Then, for every f ∈ DBVEγ [0, ∞) (γ ≥ 0), x ∈ (0, ∞) and for sufficiently large n, we have
r 0 + un ( x ) Bn (f ; x) ≤ 2 Me + x f (x ) 1− x un ( x ) |f (x)| + f (2x) − f (x) − xf 0 (x+ ) +2 1 − √
γ ex
x
+ 2x 1 −
un ( x )
where Bn (f ; x) is given by (2.4).
x
√ X [ n]
x+ kx
_
k=1
x
x+ √x _n fx0 + √ fx0 ,
x
n
x
340
M. Ali Özarslan et al. / Mathematical and Computer Modelling 52 (2010) 334–345
Proof. By direct computation we have
∂ {Kn (x, t )} dt ∂t x x Z ∞ Z t Z ∂ {Kn (x, t )} dt + fx0 (u)du ≤ ∂t ∞
Z
Bn (f ; x) =
t
Z
fx0 (u)du
x
2x
2x
t
Z
fx0 (u)du
x
x
∂ {Kn (x, t )} dt ∂t
:= Bn,1 (f ; x) + Bn,2 (f ; x). We first estimate Bn,1 (f ; x). Observe that Bn,1 (f ; x) =
= ≤ ≤
Z ∞ Z t ∂ 0 {Kn (x, t )} dt fx (u)du ∂ t x 2x Z ∞ Z t ∂ 0 0 + {Kn (x, t )} dt (f (u) − f (x ))du ∂ t Z2x∞ x Z 0 + ∞ ∂ ∂ { { K ( x , t )} dt + K ( x , t )} dt f ( x ) ( f ( t ) − f ( x )) ( t − x ) n n ∂t ∂t 2x 2x Z ∞ Z ∞ Z ∞ ∂ ∂ ∂ |f (t )| {Kn (x, t )} dt + |f (x)| {Kn (x, t )} dt + f 0 (x+ ) {Kn (x, t )} dt . | t − x| ∂t ∂t 2x ∂ t 2x 2x
Since t ≥ 2x, we get t − x ≥ x, and also since f ∈ DBVEγ (γ ≥ 0), we conclude from (1.3) and Remark 2.1 that
Z p |f (x)| ∞ ∂ ∂ {Kn (x, t )} dt + 2 {Kn (x, t )} dt + f 0 (x+ ) 2x (x − un (x)) ( t − x) 2 x 2x ∂t x ∂t 0 Z ∞ p | M ∂ 2 f ( x )| {Kn (x, t )} dt + = eγ t (t − x) (x − un (x)) + f 0 (x+ ) 2x (x − un (x)). x 2x ∂t x
Bn,1 (f ; x) ≤
M
Z
∞
eγ t (t − x)
Now applying the Cauchy–Schwarz inequality and also using Lemma 2.4 we have
1/2 Z ∞ 1/2 ∂ ∂ {Kn (x, t )}dt (t − x)2 {Kn (x, t )}dt x ∂t ∂t 2x 2x 0 + p 2 |f (x)| + (x − un (x)) + f (x ) 2x (x − un (x)) x r r √ 0 + √ un (x) un (x) un (x) γ ex + 2 |f (x)| 1 − + 2x f (x ) 1 − ≤ 2Me 1− x x x r √ un (x) un (x) = 2 Meγ ex + x f 0 (x+ ) 1 − + 2 |f (x)| 1 − .
Bn,1 (f ; x) ≤
M
Z
∞
e2 γ t
x
x
For the estimation of Bn,2 (f ; x), if we apply integration by parts, then we have 2x
Z
Bn,2 (f ; x) =
t
Z
x
fx0 (u)du x
Z ≤ |1 − λn (x, 2x)|
x
∂ {Kn (x, t )} dt ∂t Z 2x 2x fx0 (u)du + fx0 (t ) (1 − λn (x, t )) dt .
x
Now by Lemma 2.1, we get un (x)
Z 2x 0 0 + f ( u ) − f ( x ) du x x Z Z x x+ √n 2x 0 0 + fx (t ) (1 − λn (x, t )) dt + fx (t ) (1 − λn (x, t )) dt x x+ √x n un (x) f (2x) − f (x) − xf 0 (x+ ) = 2 1− x Z Z 2x x+ √xn 0 0 + fx (t ) (1 − λn (x, t )) dt + fx (t ) (1 − λn (x, t )) dt . x x+ √x
Bn,2 (f ; x) ≤ 2 1 −
n
M. Ali Özarslan et al. / Mathematical and Computer Modelling 52 (2010) 334–345
341
Here observe that
Z Z x+ √xn x+ √x n 0 fx (t ) (1 − λn (x, t )) dt ≤ x x
t _
! fx0
x+ √x _n dt ≤ √ fx0 x
n
x
x
and
Z Z 2x 2x 0 fx (t ) (1 − λn (x, t )) dt ≤ 2x(x − un (x)) x+ √x x+ √x n
t _
! fx
(x − t )2
x
n
1
0
dt .
Letting u = t −x x in the last integral, we have
x Z Z √n x_ +u 2x 0 f ( t ) 1 − λ ( x , t )) dt fx0 du ≤ 2 ( x − u ( x )) ( n n x+ √x x 1 x n [√n] x+ x un (x) X _k 0 fx . ≤ 2x 1 − x
x
k=1
Therefore, combining the above inequalities we conclude that
Bn,2 (f ; x) ≤ 2 1 −
un (x)
x
+ 2x 1 −
f (2x) − f (x) − xf 0 (x+ )
un (x)
√ X [ n]
_
x
x+ kx
0
fx
x+ √x _n fx0 +√ x
n
x
k=1
x
and hence Bn (f ; x) ≤ Bn,1 (f ; x) + Bn,2 (f ; x)
√ ≤
2 Me
γ ex
+ 2x 1 −
r 0 + un ( x ) un ( x ) |f (x)| + f (2x) − f (x) − xf 0 (x+ ) + x f (x ) 1 − +2 1− x
un ( x ) x
which completes the proof.
√ X [ n]
x
x+ √x _ _n x fx0 + √ fx0 ,
x+ kx
n
x
k=1
x
Lemma 2.7. Assume that (1.4) holds. Then, for every f ∈ DBVγ [0, ∞) (γ ≥ 2), x ∈ (0, ∞) and for sufficiently large n, we have
r 0 + v n ( x) Bn (f ; x) ≤ 2 M K (x) + x f (x ) 1− x v n ( x) |f (x)| + f (2x) − f (x) − xf 0 (x+ ) +2 1 − √ p
x
+ 2x 1 −
vn (x) x
√ X [ n]
x+ kx
_
k=1
x
x+ √x _n x fx0 + √ fx0
n
x
for some function K (x) defined on (0, ∞). Here Bn (f ; x) is given by (2.4). Proof. As in the proof of Lemma 2.6, we get Bn (f ; x) ≤ Bn,1 (f ; x) + Bn,2 (f ; x). Since f ∈ DBVγ [0, ∞) (γ ≥ 2), for the estimation Bn,1 (f ; x), we have
∂ {Kn (x, t )} dt ∂t 2x x Z ∞ Z ∞ Z ∞ ∂ ∂ ∂ |f (t )| {Kn (x, t )} dt + |f (x)| {Kn (x, t )} dt + f 0 (x+ ) |t − x| {Kn (x, t )} dt ≤ ∂t ∂t 2x 2x ∂ t 0 Z
Bn,1 (f ; x) =
∞
Z
t
fx0 (u)du
342
M. Ali Özarslan et al. / Mathematical and Computer Modelling 52 (2010) 334–345
≤
|f (x)| ∂ (1 + t γ )(t − x) {Kn (x, t )} dt + 2 x 2x ∂t x p + f 0 (x+ ) 2x (x − vn (x)). Z
M
∞
∞
Z
(t − x)2 0
∂ {Kn (x, t )} dt ∂t
Using the Cauchy–Schwarz inequality, we get
1/2 Z ∞ 1/2 ∂ ∂ {Kn (x, t )}dt (t − x)2 {Kn (x, t )}dt x ∂t ∂t 2x 2x 0 + p 2 |f (x)| + (x − vn (x)) + f (x ) 2x (x − vn (x)) x r q √ vn (x) ≤ 2M 1 − Ln (1; x) + 2Ln (t [γ ]+1 ; x) + Ln t [2γ ]+1 ; x x r √ 0 + vn (x) v n ( x) + 2x f (x ) 1 − . + 2 |f (x)| 1 −
Bn,1 (f ; x) ≤
Z
M
∞
(1 + t γ )2
x
x
Now considering Lemma 2.5, there exist polynomials p[γ ]+1,m (x) = a1,m x + · · · + a[γ ]+1,m x[γ ]+1 (m = 1, 2, . . . , [γ ] − 1) and q[2γ ]+1,` (x) = b1,` x + · · · + b[2γ ]+1,` x[2γ ]+1 (` = 1, 2, . . . , [2γ ] − 1) such that Ln (1; x) + 2Ln (t [γ ]+1 ; x) + Ln t [2γ ]+1 ; x
= 1 + 2vn
[γ ]+1
(x) + 2
+vn[2γ ]+1 (x) +
([γ ] + 1)[γ ] 2
([2γ ] + 1)[2γ ] 2
−1
[γ ]+1
vn
[γ ]
(x) + vn (x) n
+2
[γ ]−1 X
p[γ ]+1,m (vn (x)) nm+1
m=1
[2γ ]+1 [2γ ] [2X γ ]−1 q[2γ ]+1,` (vn (x)) vn (x) + vn (x) −1 +2 `+1 n
n
`=1
:= Ψn (x).
√ √ Ψn (x) is bounded on N, say Ψn (x) ≤ K (x). Then we conclude that r r p √ 0 + vn (x) v n ( x) vn (x) Bn,1 (f ; x) ≤ 2K (x)M 1 − + 2 |f (x)| 1 − + 2x f (x ) 1 − x x x r √ p vn (x) vn (x) = 2 K (x)M + x f 0 (x+ ) 1− + 2 |f (x)| 1 − .
Observe that, for each x ∈ (0, ∞),
x
x
As in the proof of Lemma 2.6 one can show that
x+ √x √ x X +k [ n] x_ _n vn (x) v ( x ) x n fx0 + √ fx0 . Bn,2 (f ; x) ≤ 2 1 − f (2x) − f (x) − xf 0 (x+ ) + 2x 1 −
x
x
n
x
k=1
x
Thus, combining the above results, we obtain that Bn (f ; x) ≤ Bn,1 (f ; x) + Bn,2 (f ; x)
r p
≤ M 2K (x) 1 −
vn (x) x
+ 2 |f (x)| 1 −
v n ( x) x
r 0 + vn (x) 2x f (x ) 1 −
√
+
x
x+ √x √ x X +k [ n] x_ _n vn (x) x v ( x ) n +2 1 − f (2x) − f (x) − xf 0 (x+ ) + √ fx0 + 2x 1 − fx0 .
x
n
x
x
x
k=1
3. Proof of the main results and some extensions Proof of Theorem 1.1. Combining Lemmas 2.2, 2.3 and 2.6 one can obtain, for every f ∈ DBVEγ [0, ∞), x ∈ (0, ∞) and for sufficiently large n, that
p |Ln (f ; x) − f (x)| ≤ {|p(x)| + |q(x)|} 2x(x − un (x)) + An (f ; x) + Bn (f ; x) p un ( x ) ≤ {|p(x)| + |q(x)|} 2x(x − un (x)) + 2x 1 −
x
√ X [ n]
x
_
k=1
x− kx
x
x _ 0 fx0 + √ fx n √x x−
n
M. Ali Özarslan et al. / Mathematical and Computer Modelling 52 (2010) 334–345
√ +
2 Meγ ex + x f 0 (x+ )
+2 1 −
un ( x )
x
+ 2x 1 −
un (x)
r
un ( x ) x
|f (x)| + f (2x) − f (x) − xf 0 (x+ )
√ X [ n]
x+ kx
_
x
1−
343
x+ √x _n fx0 + √ fx0 .
x
n
x
k =1
x
Now using the fact that b _
f =
a
c _
f +
b _
a
for c ∈ (a, b),
f
c
we get
|Ln (f ; x) − f (x)| ≤
√
2 x (|p(x)| + |q(x)|) + Me
un ( x )
γ ex
r 0 + un (x) + x f (x ) 1− x
|f (x)| + f (2x) − f (x) − xf 0 (x+ ) +2 1 − x x √ x x+ √ X +k [ n] x_ n un (x) x _ 0 0 + 2x 1 − fx + √ fx , x
which completes the proof.
x− kx
k =1
n
x− √x
n
Proof of Theorem 1.2. Combining Lemmas 2.2, 2.3, 2.7 and also using the idea in the proof of Theorem 1.1, the proof follows immediately. Now we extend the above results to the statistical approximation process via the concepts of A-density and A-statistical convergence. We first recall these concepts. Let T be a subset of N. Then, the (asymptotic) density of T is defined by
δ(T ) := lim
# {n ≤ j : n ∈ T } j
j
provided the limit exists, where the symbol # {B} denotes the cardinality of a set B. Using this density Fast [17] introduced the notion of statistical convergence of number sequences as follows: x = (xn )n∈N is statistically convergent to a number L, denoted by st − lim x = L, if, for every ε > 0, the set {n ∈ N : |xn − L| ≥ ε} has density zero, i.e.,
δ ({n ∈ N : |xn − L| ≥ ε}) = lim j
# {n ≤ j : |xn − L| ≥ ε} j
= 0.
Now let A = [ajnP ] (j, n ∈ N) be an infinite summability matrix. Then, the A-transform of x, denoted Ax := ((Ax)j ), is ∞ given by (Ax)j = n=1 ajn xn , provided the series converges for each j. We say that A is regular if limj (Ax)j = L whenever limj xj = L [18]. Assume now that A is a non-negative regular summability matrix and T is a subset of N. The A-density of T is defined by
δA (T ) := lim j
X
ajn
n∈T
provided the limit exists. Observe that if we take A = C1 = [cjn ], the Cesáro matrix of order one, defined by cjn =
1
, j 0,
if 1 ≤ n ≤ j otherwise
then δC1 (T ) = δ(T ) for any subset T of N. With the help of the A-density, Freedman and Sember [19] introduced the notion of A-statistical convergence, which is a more general method of statistical convergence. Recall that the sequence x = (xn )n∈N is said to be A-statistically convergent to L if, for every ε > 0, δA {n ∈ N : |xn − L| ≥ ε} = 0; or equivalently lim j
X
ajn = 0.
n:|xn −L|≥ε
This limit is denoted by stA − lim x = L. It is not hard to see that if we take A = C1 , then C1 -statistical convergence coincides with the statistical convergence mentioned above, i.e., stC1 − lim x = st − lim x. If A is replaced by the identity matrix,
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M. Ali Özarslan et al. / Mathematical and Computer Modelling 52 (2010) 334–345
then we get the ordinary convergence of number sequences. It is well known that if A = (ajn ) is any non-negative regular summability matrix for which limj maxn {ajn } = 0, then A-statistical convergence is stronger than convergence. Actually, every convergent sequence is A-statistically convergent to the same value for any non-negative regular matrix A, but its converse is not always true. Some other results regarding statistical and A-statistical convergences may be found in the papers [20,21]. Using the above terminology, by Theorem 1.1, we immediately obtain the next result. Theorem 3.1. Let A = [ajn ] be a non-negative regular summability matrix and let T be any subset of N such that δA (T ) = 1. Assume that Pk,n ≡ sk,n and αn ≡ u∗n , where u∗n (x) =
√ −1 + 1 + 4n2 x2 2n
0,
,
if n ∈ T
(3.1)
if n 6∈ T .
Then, for every f ∈ DBVEγ [0, ∞) (γ ≥ 0), x ∈ (0, ∞) and for sufficiently large n ∈ T , (1.6) also holds when un (x) is replaced with u∗n (x). In a similar manner we can obtain the next result from Theorem 1.2. Theorem 3.2. Let A = [ajn ] be a non-negative regular summability matrix and let T be any subset of N such that δA (T ) = 1. Assume that Pk,n ≡ sk,n and αn ≡ vn∗ , where
p −1 + 1 + 4n(n + 1)x2 , v n ( x) = 2(n + 1) 0, ∗
if n ∈ T
(3.2)
if n 6∈ T .
Then, for every f ∈ DBVγ [0, ∞) (γ ≥ 2), x ∈ (0, ∞) and for sufficiently large n ∈ T , (1.7) also holds when vn (x) is replaced with vn∗ (x). By (3.1), observe that stA − lim u∗n (x) = x, which yields
stA − lim 1 −
u∗n (x) x
= 0.
So, if we replace un (x) with u∗n (x) in the R.H.S. of (1.6), then stA − lim (R.H.S. of (1.6)) = 0. Then, by Theorem 3.1, we have stA − lim |Ln (f ; x) − f (x)| = 0 for every f ∈ DBVEγ [0, ∞) (γ ≥ 0) and x ∈ (0, ∞) in the case of Pk,n ≡ sk,n and αn ≡ u∗n . Hence we get the (pointwise) statistical approximation to a function f ∈ DBVEγ [0, ∞) (γ ≥ 0) by means of the positive linear operators Ln . Here if we take A = C1 , the Cesáro matrix, and T = {n 6= m2 : m ∈ N}, then δC1 (T ) = 1. Also, by (3.1), we get stC1 − lim u∗n (x) = st − lim u∗n (x) = x but the sequence (u∗n (x)) is non-convergent (in the usual sense). This application clearly shows that it is possible to find positive linear operators satisfying the conditions of statistical approximation process but not the classical one. Similar idea is also valid for Theorem 3.2, but we omit the details. Acknowledgements The authors would like to thank the referees for carefully reading the manuscript. The present investigation was supported, in part, by the Ministry of National Education of TRNC under Project MEKB-09-01. References [1] [2] [3] [4] [5] [6] [7] [8]
J.P. King, Positive linear operators which preserve x2 , Acta. Math. Hungar. 99 (2003) 203–208. O. Duman, M.A. Özarslan, Szász-Mirakjan type operators providing a better error estimation, Appl. Math. Lett. 20 (2007) 1184–1188. L. Rempulska, K. Tomczak, Approximation by certain linear operators preserving x2 , Turkish J. Math. 33 (2009) 273–281. M.A. Özarslan, O. Duman, MKZ type operators providing a better estimation on [1/2, 1), Canad. Math. Bull. 50 (2007) 434–439. O. Agratini, Linear operators that preserve some test functions, Int. J. Math. Math. Sci. (2006) 11 Art. ID 94136. O. Agratini, On the iterates of a class of summation-type linear positive operators, Comput. Math. Appl. 55 (2008) 1178–1180. N.K. Govil, V. Gupta, Convergence rate for generalized Baskakov type operators, Nonlinear Anal. 69 (2008) 3795–3801. S.-S. Guo, M.K. Khan, On the rate of convergence of some operators on functions of bounded variation, J. Approx. Theory 58 (1989) 90–101.
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[9] V. Gupta, R.N. Mohapatra, Approximation by Durrmeyer–Bezier operators, Nonlinear Anal. RWA 9 (2008) 1491–1498. [10] H. Karsli, Rate of convergence of new Gamma type operators for functions with derivatives of bounded variation, Math. Comput. Modelling 45 (2007) 617–624. [11] H. Karsli, V. Gupta, Some approximation properties of q-Chlodowsky operators, Appl. Math. Comput. 195 (2008) 220–229. [12] H. Karsli, V. Gupta, Rate of convergence of nonlinear integral operators for functions of bounded variation, Calcolo 45 (2008) 87–98. [13] M.K. Khan, On the rate of convergence of Bernstein power series for functions of bounded variation, J. Approx. Theory 57 (1989) 90–103. [14] O. Duman, Regular matrix transformations and rates of convergence of positive linear operators, Calcolo 44 (2007) 159–164. [15] E. Erkuş-Duman, O. Duman, Integral-type generalizations of operators obtained from certain multivariate polynomials, Calcolo 45 (2008) 53–67. [16] M.A. Özarslan, O. Duman, O. Doğru, Rates of A-statistical convergence of approximating operators, Calcolo 42 (2005) 93–104. [17] H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951) 241–244. [18] G.H. Hardy, Divergent Series, Oxford Univ. Press, London, 1949. [19] A.R. Freedman, J.J. Sember, Densities and summability, Pacific J. Math. 95 (1981) 293–305. [20] J.A. Fridy, On statistical convergence, Analysis 5 (1985) 301–313. [21] H.I. Miller, A measure theoretical subsequence characterization of statistical convergence, Trans. Amer. Math. Soc. 347 (1995) 1811–1819.
Contents lists available at ScienceDirect
Mathematical and Computer Modelling journal homepage: www.elsevier.com/locate/mcm
Rates of convergence of certain King-type operators for functions with derivative of bounded variation M. Ali Özarslan a , Oktay Duman b,∗ , Cem Kaanoğlu c a
Eastern Mediterranean University, Faculty of Arts and Sciences, Department of Mathematics, Gazimagusa, Mersin 10, Turkey
b
TOBB Economics and Technology University, Faculty of Arts and Sciences, Department of Mathematics, Söğütözü TR-06530, Ankara, Turkey
c
Cyprus International University, Faculty of Engineering, Lefkoşa, Mersin 10, Turkey
article
abstract
info
Article history: Received 4 November 2009 Received in revised form 24 February 2010 Accepted 24 February 2010
In this paper, we estimate the rates of pointwise approximation of certain King-type positive linear operators for functions with derivative of bounded variation. We also extend our results to the statistical approximation process via the concept of statistical convergence. © 2010 Elsevier Ltd. All rights reserved.
Keywords: A-statistical convergence Statistical approximation King-type operators Functions with derivative of bounded variation Rates of convergence
1. Introduction King-type modification of positive linear operators preserves the test function e2 (x) = x2 and so provides better error estimations on some appropriate domains than the classical ones. Such a modification was first noticed by King [1] for the classical Bernstein polynomials. Later, this idea was applied to some other well-known approximating operators, such as, the Szász-Mirakjan operators [2], the Baskakov operators [3], the Meyer-König and Zeller operators [4], the BernsteinChlodovsky operators [5] and more general summation-type positive linear operators [6]. It is well known that the classical Szász-Mirakjan operators Sn and the classical Baskakov operators Vn are defined respectively by Sn (f ; x) :=
∞ X k f
k=0
n
sk,n (x),
sk,n (x) :=
vk,n (x),
vk,n (x) :=
e−nx (nx)k k!
,
and Vn (f ; x) :=
∞ X k f
k=0
n
n+k−1 k
xk (1 + x)−n−k ,
where f : [0, ∞) → R is a function such that the above series are convergent for each n. In order to obtain better error estimations we modify and combine the above approximating operators as the operators Ln given by Ln (f ; x) :=
∞ X k f
k=0
∗
n
Pk,n (αn (x)),
Corresponding author. Tel.: +90 3122924141; fax: +90 3122924324. E-mail addresses: [email protected] (M. Ali Özarslan), [email protected] (O. Duman), [email protected] (C. Kaanoğlu).
0895-7177/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2010.02.048
(1.1)
M. Ali Özarslan et al. / Mathematical and Computer Modelling 52 (2010) 334–345
where {Pk,n } ∈ {sk,n }, {vk,n } and √ −1 + 1 + 4n2 x2 2n αn (x) := p −1 + 1 + 4n(n + 1)x2 2(n + 1)
335
if Pk,n ≡ sk,n (1.2) if Pk,n ≡ vk,n .
Throughout the paper, we use
√
un (x) :=
−1 +
1 + 4n2 x2
2n It is known from [2] that if we take Pk,n ≡ sk,n
and vn (x) :=
−1 +
1 + 4n(n + 1)x2
p
2(n + 1)
.
and αn ≡ un ,
(1.3)
then the operators Ln given by (1.1) have better error estimations on the interval [0, ∞) than the classical Szász-Mirakjan operators. Furthermore, the following conditions hold provided that (1.3) holds: (i) Ln (e0 ; x) = e0 (x) = 1, (ii) Ln (e1 ; x) = un (x) with e1 (x) = x, (iii) Ln (e2 ; x) = e2 (x) = x2 . Similarly, it follows from [3] that if one takes Pk,n ≡ vk,n and αn ≡ vn , (1.4) then the operators Ln have better estimations than the classical Baskakov operators. In this case, we get (i), (iii) and also
(ii)0 Ln (e1 ; x) = vn (x) with e1 (x) = x. Now, as usual, let BV [0, ∞) be the class of all functions defined on [0, ∞) having bounded variation on every finite subinterval of [0, ∞). Now, for γ ≥ 0, we consider the following function spaces: DBVEγ [0, ∞) := {f : [0, ∞) → R : f 0 ∈ BV [0, ∞) and |f (t )| ≤ Meγ t for every t ≥ 0 and some M > 0} and DBVγ [0, ∞) := {f : [0, ∞) → R : f 0 ∈ BV [0, ∞) and |f (t )| ≤ M (1 + t γ ) for every t ≥ 0 and some M > 0}. We also use the following notations: p(x) :=
f 0 (x+ ) + f 0 (x− )
,
f 0 (x+ ) − f 0 (x− )
q(x) :=
. (1.5) 2 2 In this paper, we estimate the rates of pointwise approximation of the operators Ln on the subclasses mentioned above. Recall that such investigations were already studied by many authors for the classical approximating operators (see, for instance, [7–13]). Some other results regarding the statistical approximation process may be found in the papers [14,15,4,16] and cited therein. Now, as usual, we define the auxiliary function fx , by f (t ) − f (x− ), 0, fx (t ) = f (t ) − f (x+ ),
if 0 ≤ t < x, if t = x if x < t < ∞.
Then, we mainly obtain the following results. Theorem 1.1. Assume that (1.3) holds. Then, for every f ∈ DBVEγ [0, ∞) (γ ≥ 0), x ∈ (0, ∞) and for sufficiently large n, we have
|Ln (f ; x) − f (x)| ≤
√
2 x (|p(x)| + |q(x)|) + Me
+2 1 −
un ( x )
+ 2x 1 −
x un (x) x
γ ex
r 0 + un (x) + x f (x ) 1− x
|f (x)| + f (2x) − f (x) − xf 0 (x+ )
√ X [ n]
x+ kx
_
k =1
x− kx
n _ x fx0 + √ fx0
n
x+ √x
(1.6)
x− √x
n
where p(x) and q(x) are defined as in (1.5). Theorem 1.2. Assume that (1.4) holds. Then, for every f ∈ DBVγ [0, ∞) (γ ≥ 2), x ∈ (0, ∞) and for sufficiently large n, we have
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M. Ali Özarslan et al. / Mathematical and Computer Modelling 52 (2010) 334–345
r 0 + o vn (x) |Ln (f ; x) − f (x)| ≤ 2 x (|p(x)| + |q(x)|) + M K (x) + x f (x ) 1− x vn (x) |f (x)| + f (2x) − f (x) − xf 0 (x+ ) +2 1 − √ n
p
x
+ 2x 1 −
vn (x)
√ X [ n]
x+ kx
_
x
k=1
x− kx
n _ x fx0 + √ fx0
n
x+ √x
(1.7)
x− √x
n
for some function K (x) defined on (0, ∞). The proofs of the above theorems will be given in the last section after some auxiliary results. We also extend the above results to the statistical approximation process via the concepts of A-density and A-statistical convergence (see Section 3 for details). 2. Auxiliary results Observe that we may rewrite (1.1) as follows: Ln (f ; x) =
∞
Z
f (t ) 0
∂ {Kn (x, t )} dt , ∂t
(2.1)
where
X Pk,n (x),
Kn (x, u) :=
if 0 < u < ∞
k≤nu
0,
if u = 0.
Also let
λn (x, t ) :=
Z 0
t
∂ {Kn (x, u)}du. ∂u
(2.2)
Then observe that
λn (x, t ) ≤ 1. In this section, some results are given which are necessary to prove our main theorems. Remark 2.1. It is easily verified that Ln (t − x; x) = αn (x) − x and
Ln ((t − x)2 ; x) = 2x(x − αn (x)),
where αn (x) is defined by (1.2). Then, we have 1
Ln (|t − x| ; x) ≤ (Ln ((t − x)2 ; x)) 2 ≤
p
2x (x − αn (x)).
Lemma 2.1. For every x ∈ (0, ∞) and n ∈ N, we have
n (x)) {Kn (x, u)} du ≤ 2x((xx−α for 0 ≤ y < x, −y)2 R∞ ∂ 2x(x−αn (x)) (b) 1 − λn (x, z ) = z ∂ u {Kn (x, u)} du ≤ (x−z )2 for x < z < ∞.
(a) λn (x, y) =
Ry
∂
0 ∂u
Proof. We first prove (a). By Remark 2.1, we get, for 0 ≤ y < x, that y
Z 0
∂ {Kn (x, u)} du ≤ ∂u
Z y
≤
x−u
2
x−y
0
2x(x − αn (x))
(x − y)2
∂ {Kn (x, u)} du ∂u .
Similarly, for x < z < ∞, we conclude that ∞
Z z
∂ {Kn (x, u)} du ≤ ∂u
Z
≤ 0
which is the desired result.
∞
x−u
2
∂ {Kn (x, u)} du ∂u
2
∂ {Kn (x, u)} du ∂u
x−z
z
Z
≤
∞
x−u x−z
2x(x − αn (x))
(x − z )2
,
M. Ali Özarslan et al. / Mathematical and Computer Modelling 52 (2010) 334–345
337
Lemma 2.2. For every f ∈ DBV [0, ∞), x ∈ (0, ∞) and n ∈ N, we have
p |Ln (f ; x) − f (x)| ≤ {|p(x)| + |q(x)|} 2x(x − αn (x)) + An (f ; x) + Bn (f ; x), where αn (x) is given by (1.2), p(x) and q(x) are defined as in (1.5), and also An (f ; x) :=
t
Z x Z
fx0 (u)du
x
0
∂ {Kn (x, t )}dt ∂t
(2.3)
and Bn (f ; x) =:
∞
Z
t
Z
fx0 (u)du
x
x
∂ {Kn (x, t )}dt . ∂t
(2.4)
Proof. For f ∈ DBV [0, ∞) and x ∈ (0, ∞), we may write that f 0 (u) = p(x) + q(x)sgn(u − x) + fx0 (u) + f 0 (x) − p(x) δx (u),
where
δx (u) :=
1, 0,
if u = x if u 6= x.
Now using the above equality and also considering the linearity of the operators Ln , we obtain that
Z ∞ ∂ |Ln (f ; x) − f (x)| = {Kn (x, t )} dt (f (t ) − f (x)) ∂ t 0 Z ∞ Z t ∂ {Kn (x, t )} dt = f 0 (u)du ∂t 0 x Z ∞ Z ∞ ∂ ∂ |t − x| {Kn (x, t )} dt + |q(x)| {Kn (x, t )} dt |t − x| ≤ |p(x)| ∂t ∂t 0 Z ∞0Z t Z Z t 0 ∞ ∂ ∂ 0 {Kn (x, t )} dt + f (x) − p(x) {Kn (x, t )} dt . + fx (u)du δx (u)du ∂t ∂t 0 x 0 x Rt Since x δx (u)du = 0, we conclude that |Ln (f ; x) − f (x)| ≤ {|p(x)| + |q(x)|} Ln (|t − x| ; x) Z x Z t Z ∞ Z t ∂ ∂ 0 0 {Kn (x, t )} dt + {Kn (x, t )} dt . fx (u)du fx (u)du + ∂t ∂t 0 x x x Then it follows from Remark 2.1 that
p |Ln (f ; x) − f (x)| ≤ {|p(x)| + |q(x)|} 2x(x − αn (x)) + An (f ; x) + Bn (f ; x), which is the desired result.
Lemma 2.3. For every f ∈ DBV [0, ∞), x ∈ (0, ∞) and n ∈ N, we have
|An (f ; x)| ≤ 2x 1 −
αn (x) x
√ X [ n]
x _
k=1
x− kx
x _ x fx0 , fx0 + √
n
x− √x
n
where An (f ; x) is given by (2.3). Proof. Applying integration by parts we have
Z x Z t ∂ {Kn (x, t )} dt |An (f ; x)| = fx0 (u)du ∂t Z0 x x ∂ {Kn (x, t )} dt = fx (t ) ∂t Z 0x 0 = λn (x, t )fx (t )dt 0
x− √x
Z
n
≤ 0
|λn (x, t )| fx0 (t ) dt +
Z
x x− √x n
|λn (x, t )| fx0 (t ) dt ,
338
M. Ali Özarslan et al. / Mathematical and Computer Modelling 52 (2010) 334–345
and hence, by Lemma 2.1,
|An (f ; x)| ≤
x− √x
Z
x _
n
0
!
Z
|λn (x, t )| dt +
0
fx
x− √x
t
≤ 2x(x − αn (x))
x− √x
x _
n
0
≤ 2x(x − αn (x))
x− √x
Z
x _
n
0
1
(x − t )2
t
! fx0
t
|λn (x, t )| dt
Z
x
fx
x _ fx0 dt
! fx0
0
t
n
Z
!
x _
x
dt + x− √x
n
x− √x
n
x _ 1 x dt + √ fx0 . (x − t )2 n x− √x n
x
Letting u = x−t , since x− √x
Z
n
0
x _
√
! fx0
1
(x − t )2
t
dt =
1
Z
x _
x
1
√
≤
n
[ n] 1X
x k=1
fx0 du
x− ux
x _
fx0 ,
x− kx
we may write that
|An (f ; x)| ≤ 2x 1 −
αn (x)
which completes the proof.
x
√ X [ n]
x _
k=1
x− kx
x _ x fx0 + √ fx0 ,
n
x− √x
n
Lemma 2.4. Assume that (1.3) holds. Let γ ≥ 0 be fixed. Then, for every x ∈ (0, ∞) and for sufficiently large n, we have Ln e2γ t ; x ≤ e2eγ x .
Proof. By (1.3), we see that Ln (e
2γ t
; x) = = = =
=
k ∞ X nun (x)e2γ /n exp{−nun (x)} k! k=0 exp −nun (x) + nun (x)e2γ /n . exp nun (x) −1 + e2γ /n ( ) ∞ X (2γ /n)k exp nun (x) . k! k=1 ( ) ∞ X (2γ /n)k−1 exp 2γ un (x) . k! k =1
Letting n ≥ 2γ , we get
(
Ln (e
2γ t
∞ X 1 ; x) ≤ exp 2γ un (x) k ! k=0
)
= e2eγ un (x) . Since un (x) ≤ x for every x ∈ (0, ∞), the last inequality implies that Ln (e2γ t ; x) ≤ e2eγ x . Lemma 2.5. Assume that (1.4) holds. For r ∈ N3 := {3, 4, . . .}, there exist polynomials pr ,m (x) := a1,m x + a2,m x2 + · · · + ar ,m xr (m = 1, 2, . . . , r − 2) having degree r with non-negative coefficients such that
M. Ali Özarslan et al. / Mathematical and Computer Modelling 52 (2010) 334–345
Ln t r ; x = vnr (x) +
339
r r −2 v (x) + vnr −1 (x) X pr ,m (vn (x)) −1 n + . m+1
r (r − 1) 2
n
(2.5)
n
m=1
Proof. For the proof, we use mathematical induction. Under the condition (1.4), by a direct computation, we may write that Ln (t 3 ; x) = vn3 (x) +
2vn3 (x) + 2vn2 (x) n
+
3vn3 (x) + 4vn2 (x) + vn (x)
+
9vn4 (x) + 14vn3 (x) + 5vn2 (x)
n2
and Ln (t 4 ; x) = vn4 (x) +
+
5vn4 (x) + 5vn3 (x)
n 6vn4 (x) + 14vn3 (x) + 9vn2 (x) + vn (x) n3
n2
.
So, (2.5) holds true for r = 3, 4. Assume now that (2.5) holds for r > 4. We first observe that Ln (t r +1 ; x) =
vn2 (x) + vn (x) 0 r Ln (t ; x) + vn (x)Ln (t r ; x). nvn0 (x)
Then, we get from (2.5) that Ln (t r +1 ; x) =
vn2 (x) + vn (x) n
+
r vnr −1 (x) +
r (r − 1) 2
n
r −2 v (x) + vn (x) X qr −1,m (vn (x)) 2 n
n
+
r (r − 1) 2
+ vnr +1 (x)
nm+1
m=1
vnr +1 (x) + vnr (x)
−1
r −1 r vn (x) + (r − 1)vnr −2 (x) −1
n
+ v n ( x)
r −2 X pr ,m (vn (x))
nm+1
m=1
where, for m = 1, . . . , r − 2, qr −1,m (x) := p0r ,m (x) = a1,m + 2a2,m x + · · · + rar ,m xr −1 having degree r − 1 at most. Thus, the last equality may be written in the following form: Ln (t r +1 ; x) = vnr +1 (x) +
+ +
r (r − 1) 2
r (r − 1) 2
−1+r
vnr +1 (x) + vnr (x) n
v (x) + vn (x) r v
2 n
−1
nm+2
= vnr +1 (x) +
(x) + (r − 1)vnr −2 (x)
n2
r −2 X qr −1,m (vn (x)) vn2 (x) + vn (x) m=1
r −1 n
(r + 1)r 2
+
r −2 X vn (x)pr ,m (vn (x))
nm+1
m=1
r +1 r −1 ∗ pr +1,m (vn (x)) v (x) + vnr (x) X , −1 n + m+1 n
m=1
n
where p∗r +1,m (x) := a∗1,m x + a∗2,m x2 + · · · + a∗r +1,m xr +1 for appropriate coefficients
a∗1,m , a∗2,m , . . . , a∗r +1,m (m
(m = 1, 2, . . . , r − 1)
= 1, 2, . . . , r − 1). Therefore, the proof is completed.
Lemma 2.6. Assume that (1.3) holds. Then, for every f ∈ DBVEγ [0, ∞) (γ ≥ 0), x ∈ (0, ∞) and for sufficiently large n, we have
r 0 + un ( x ) Bn (f ; x) ≤ 2 Me + x f (x ) 1− x un ( x ) |f (x)| + f (2x) − f (x) − xf 0 (x+ ) +2 1 − √
γ ex
x
+ 2x 1 −
un ( x )
where Bn (f ; x) is given by (2.4).
x
√ X [ n]
x+ kx
_
k=1
x
x+ √x _n fx0 + √ fx0 ,
x
n
x
340
M. Ali Özarslan et al. / Mathematical and Computer Modelling 52 (2010) 334–345
Proof. By direct computation we have
∂ {Kn (x, t )} dt ∂t x x Z ∞ Z t Z ∂ {Kn (x, t )} dt + fx0 (u)du ≤ ∂t ∞
Z
Bn (f ; x) =
t
Z
fx0 (u)du
x
2x
2x
t
Z
fx0 (u)du
x
x
∂ {Kn (x, t )} dt ∂t
:= Bn,1 (f ; x) + Bn,2 (f ; x). We first estimate Bn,1 (f ; x). Observe that Bn,1 (f ; x) =
= ≤ ≤
Z ∞ Z t ∂ 0 {Kn (x, t )} dt fx (u)du ∂ t x 2x Z ∞ Z t ∂ 0 0 + {Kn (x, t )} dt (f (u) − f (x ))du ∂ t Z2x∞ x Z 0 + ∞ ∂ ∂ { { K ( x , t )} dt + K ( x , t )} dt f ( x ) ( f ( t ) − f ( x )) ( t − x ) n n ∂t ∂t 2x 2x Z ∞ Z ∞ Z ∞ ∂ ∂ ∂ |f (t )| {Kn (x, t )} dt + |f (x)| {Kn (x, t )} dt + f 0 (x+ ) {Kn (x, t )} dt . | t − x| ∂t ∂t 2x ∂ t 2x 2x
Since t ≥ 2x, we get t − x ≥ x, and also since f ∈ DBVEγ (γ ≥ 0), we conclude from (1.3) and Remark 2.1 that
Z p |f (x)| ∞ ∂ ∂ {Kn (x, t )} dt + 2 {Kn (x, t )} dt + f 0 (x+ ) 2x (x − un (x)) ( t − x) 2 x 2x ∂t x ∂t 0 Z ∞ p | M ∂ 2 f ( x )| {Kn (x, t )} dt + = eγ t (t − x) (x − un (x)) + f 0 (x+ ) 2x (x − un (x)). x 2x ∂t x
Bn,1 (f ; x) ≤
M
Z
∞
eγ t (t − x)
Now applying the Cauchy–Schwarz inequality and also using Lemma 2.4 we have
1/2 Z ∞ 1/2 ∂ ∂ {Kn (x, t )}dt (t − x)2 {Kn (x, t )}dt x ∂t ∂t 2x 2x 0 + p 2 |f (x)| + (x − un (x)) + f (x ) 2x (x − un (x)) x r r √ 0 + √ un (x) un (x) un (x) γ ex + 2 |f (x)| 1 − + 2x f (x ) 1 − ≤ 2Me 1− x x x r √ un (x) un (x) = 2 Meγ ex + x f 0 (x+ ) 1 − + 2 |f (x)| 1 − .
Bn,1 (f ; x) ≤
M
Z
∞
e2 γ t
x
x
For the estimation of Bn,2 (f ; x), if we apply integration by parts, then we have 2x
Z
Bn,2 (f ; x) =
t
Z
x
fx0 (u)du x
Z ≤ |1 − λn (x, 2x)|
x
∂ {Kn (x, t )} dt ∂t Z 2x 2x fx0 (u)du + fx0 (t ) (1 − λn (x, t )) dt .
x
Now by Lemma 2.1, we get un (x)
Z 2x 0 0 + f ( u ) − f ( x ) du x x Z Z x x+ √n 2x 0 0 + fx (t ) (1 − λn (x, t )) dt + fx (t ) (1 − λn (x, t )) dt x x+ √x n un (x) f (2x) − f (x) − xf 0 (x+ ) = 2 1− x Z Z 2x x+ √xn 0 0 + fx (t ) (1 − λn (x, t )) dt + fx (t ) (1 − λn (x, t )) dt . x x+ √x
Bn,2 (f ; x) ≤ 2 1 −
n
M. Ali Özarslan et al. / Mathematical and Computer Modelling 52 (2010) 334–345
341
Here observe that
Z Z x+ √xn x+ √x n 0 fx (t ) (1 − λn (x, t )) dt ≤ x x
t _
! fx0
x+ √x _n dt ≤ √ fx0 x
n
x
x
and
Z Z 2x 2x 0 fx (t ) (1 − λn (x, t )) dt ≤ 2x(x − un (x)) x+ √x x+ √x n
t _
! fx
(x − t )2
x
n
1
0
dt .
Letting u = t −x x in the last integral, we have
x Z Z √n x_ +u 2x 0 f ( t ) 1 − λ ( x , t )) dt fx0 du ≤ 2 ( x − u ( x )) ( n n x+ √x x 1 x n [√n] x+ x un (x) X _k 0 fx . ≤ 2x 1 − x
x
k=1
Therefore, combining the above inequalities we conclude that
Bn,2 (f ; x) ≤ 2 1 −
un (x)
x
+ 2x 1 −
f (2x) − f (x) − xf 0 (x+ )
un (x)
√ X [ n]
_
x
x+ kx
0
fx
x+ √x _n fx0 +√ x
n
x
k=1
x
and hence Bn (f ; x) ≤ Bn,1 (f ; x) + Bn,2 (f ; x)
√ ≤
2 Me
γ ex
+ 2x 1 −
r 0 + un ( x ) un ( x ) |f (x)| + f (2x) − f (x) − xf 0 (x+ ) + x f (x ) 1 − +2 1− x
un ( x ) x
which completes the proof.
√ X [ n]
x
x+ √x _ _n x fx0 + √ fx0 ,
x+ kx
n
x
k=1
x
Lemma 2.7. Assume that (1.4) holds. Then, for every f ∈ DBVγ [0, ∞) (γ ≥ 2), x ∈ (0, ∞) and for sufficiently large n, we have
r 0 + v n ( x) Bn (f ; x) ≤ 2 M K (x) + x f (x ) 1− x v n ( x) |f (x)| + f (2x) − f (x) − xf 0 (x+ ) +2 1 − √ p
x
+ 2x 1 −
vn (x) x
√ X [ n]
x+ kx
_
k=1
x
x+ √x _n x fx0 + √ fx0
n
x
for some function K (x) defined on (0, ∞). Here Bn (f ; x) is given by (2.4). Proof. As in the proof of Lemma 2.6, we get Bn (f ; x) ≤ Bn,1 (f ; x) + Bn,2 (f ; x). Since f ∈ DBVγ [0, ∞) (γ ≥ 2), for the estimation Bn,1 (f ; x), we have
∂ {Kn (x, t )} dt ∂t 2x x Z ∞ Z ∞ Z ∞ ∂ ∂ ∂ |f (t )| {Kn (x, t )} dt + |f (x)| {Kn (x, t )} dt + f 0 (x+ ) |t − x| {Kn (x, t )} dt ≤ ∂t ∂t 2x 2x ∂ t 0 Z
Bn,1 (f ; x) =
∞
Z
t
fx0 (u)du
342
M. Ali Özarslan et al. / Mathematical and Computer Modelling 52 (2010) 334–345
≤
|f (x)| ∂ (1 + t γ )(t − x) {Kn (x, t )} dt + 2 x 2x ∂t x p + f 0 (x+ ) 2x (x − vn (x)). Z
M
∞
∞
Z
(t − x)2 0
∂ {Kn (x, t )} dt ∂t
Using the Cauchy–Schwarz inequality, we get
1/2 Z ∞ 1/2 ∂ ∂ {Kn (x, t )}dt (t − x)2 {Kn (x, t )}dt x ∂t ∂t 2x 2x 0 + p 2 |f (x)| + (x − vn (x)) + f (x ) 2x (x − vn (x)) x r q √ vn (x) ≤ 2M 1 − Ln (1; x) + 2Ln (t [γ ]+1 ; x) + Ln t [2γ ]+1 ; x x r √ 0 + vn (x) v n ( x) + 2x f (x ) 1 − . + 2 |f (x)| 1 −
Bn,1 (f ; x) ≤
Z
M
∞
(1 + t γ )2
x
x
Now considering Lemma 2.5, there exist polynomials p[γ ]+1,m (x) = a1,m x + · · · + a[γ ]+1,m x[γ ]+1 (m = 1, 2, . . . , [γ ] − 1) and q[2γ ]+1,` (x) = b1,` x + · · · + b[2γ ]+1,` x[2γ ]+1 (` = 1, 2, . . . , [2γ ] − 1) such that Ln (1; x) + 2Ln (t [γ ]+1 ; x) + Ln t [2γ ]+1 ; x
= 1 + 2vn
[γ ]+1
(x) + 2
+vn[2γ ]+1 (x) +
([γ ] + 1)[γ ] 2
([2γ ] + 1)[2γ ] 2
−1
[γ ]+1
vn
[γ ]
(x) + vn (x) n
+2
[γ ]−1 X
p[γ ]+1,m (vn (x)) nm+1
m=1
[2γ ]+1 [2γ ] [2X γ ]−1 q[2γ ]+1,` (vn (x)) vn (x) + vn (x) −1 +2 `+1 n
n
`=1
:= Ψn (x).
√ √ Ψn (x) is bounded on N, say Ψn (x) ≤ K (x). Then we conclude that r r p √ 0 + vn (x) v n ( x) vn (x) Bn,1 (f ; x) ≤ 2K (x)M 1 − + 2 |f (x)| 1 − + 2x f (x ) 1 − x x x r √ p vn (x) vn (x) = 2 K (x)M + x f 0 (x+ ) 1− + 2 |f (x)| 1 − .
Observe that, for each x ∈ (0, ∞),
x
x
As in the proof of Lemma 2.6 one can show that
x+ √x √ x X +k [ n] x_ _n vn (x) v ( x ) x n fx0 + √ fx0 . Bn,2 (f ; x) ≤ 2 1 − f (2x) − f (x) − xf 0 (x+ ) + 2x 1 −
x
x
n
x
k=1
x
Thus, combining the above results, we obtain that Bn (f ; x) ≤ Bn,1 (f ; x) + Bn,2 (f ; x)
r p
≤ M 2K (x) 1 −
vn (x) x
+ 2 |f (x)| 1 −
v n ( x) x
r 0 + vn (x) 2x f (x ) 1 −
√
+
x
x+ √x √ x X +k [ n] x_ _n vn (x) x v ( x ) n +2 1 − f (2x) − f (x) − xf 0 (x+ ) + √ fx0 + 2x 1 − fx0 .
x
n
x
x
x
k=1
3. Proof of the main results and some extensions Proof of Theorem 1.1. Combining Lemmas 2.2, 2.3 and 2.6 one can obtain, for every f ∈ DBVEγ [0, ∞), x ∈ (0, ∞) and for sufficiently large n, that
p |Ln (f ; x) − f (x)| ≤ {|p(x)| + |q(x)|} 2x(x − un (x)) + An (f ; x) + Bn (f ; x) p un ( x ) ≤ {|p(x)| + |q(x)|} 2x(x − un (x)) + 2x 1 −
x
√ X [ n]
x
_
k=1
x− kx
x
x _ 0 fx0 + √ fx n √x x−
n
M. Ali Özarslan et al. / Mathematical and Computer Modelling 52 (2010) 334–345
√ +
2 Meγ ex + x f 0 (x+ )
+2 1 −
un ( x )
x
+ 2x 1 −
un (x)
r
un ( x ) x
|f (x)| + f (2x) − f (x) − xf 0 (x+ )
√ X [ n]
x+ kx
_
x
1−
343
x+ √x _n fx0 + √ fx0 .
x
n
x
k =1
x
Now using the fact that b _
f =
a
c _
f +
b _
a
for c ∈ (a, b),
f
c
we get
|Ln (f ; x) − f (x)| ≤
√
2 x (|p(x)| + |q(x)|) + Me
un ( x )
γ ex
r 0 + un (x) + x f (x ) 1− x
|f (x)| + f (2x) − f (x) − xf 0 (x+ ) +2 1 − x x √ x x+ √ X +k [ n] x_ n un (x) x _ 0 0 + 2x 1 − fx + √ fx , x
which completes the proof.
x− kx
k =1
n
x− √x
n
Proof of Theorem 1.2. Combining Lemmas 2.2, 2.3, 2.7 and also using the idea in the proof of Theorem 1.1, the proof follows immediately. Now we extend the above results to the statistical approximation process via the concepts of A-density and A-statistical convergence. We first recall these concepts. Let T be a subset of N. Then, the (asymptotic) density of T is defined by
δ(T ) := lim
# {n ≤ j : n ∈ T } j
j
provided the limit exists, where the symbol # {B} denotes the cardinality of a set B. Using this density Fast [17] introduced the notion of statistical convergence of number sequences as follows: x = (xn )n∈N is statistically convergent to a number L, denoted by st − lim x = L, if, for every ε > 0, the set {n ∈ N : |xn − L| ≥ ε} has density zero, i.e.,
δ ({n ∈ N : |xn − L| ≥ ε}) = lim j
# {n ≤ j : |xn − L| ≥ ε} j
= 0.
Now let A = [ajnP ] (j, n ∈ N) be an infinite summability matrix. Then, the A-transform of x, denoted Ax := ((Ax)j ), is ∞ given by (Ax)j = n=1 ajn xn , provided the series converges for each j. We say that A is regular if limj (Ax)j = L whenever limj xj = L [18]. Assume now that A is a non-negative regular summability matrix and T is a subset of N. The A-density of T is defined by
δA (T ) := lim j
X
ajn
n∈T
provided the limit exists. Observe that if we take A = C1 = [cjn ], the Cesáro matrix of order one, defined by cjn =
1
, j 0,
if 1 ≤ n ≤ j otherwise
then δC1 (T ) = δ(T ) for any subset T of N. With the help of the A-density, Freedman and Sember [19] introduced the notion of A-statistical convergence, which is a more general method of statistical convergence. Recall that the sequence x = (xn )n∈N is said to be A-statistically convergent to L if, for every ε > 0, δA {n ∈ N : |xn − L| ≥ ε} = 0; or equivalently lim j
X
ajn = 0.
n:|xn −L|≥ε
This limit is denoted by stA − lim x = L. It is not hard to see that if we take A = C1 , then C1 -statistical convergence coincides with the statistical convergence mentioned above, i.e., stC1 − lim x = st − lim x. If A is replaced by the identity matrix,
344
M. Ali Özarslan et al. / Mathematical and Computer Modelling 52 (2010) 334–345
then we get the ordinary convergence of number sequences. It is well known that if A = (ajn ) is any non-negative regular summability matrix for which limj maxn {ajn } = 0, then A-statistical convergence is stronger than convergence. Actually, every convergent sequence is A-statistically convergent to the same value for any non-negative regular matrix A, but its converse is not always true. Some other results regarding statistical and A-statistical convergences may be found in the papers [20,21]. Using the above terminology, by Theorem 1.1, we immediately obtain the next result. Theorem 3.1. Let A = [ajn ] be a non-negative regular summability matrix and let T be any subset of N such that δA (T ) = 1. Assume that Pk,n ≡ sk,n and αn ≡ u∗n , where u∗n (x) =
√ −1 + 1 + 4n2 x2 2n
0,
,
if n ∈ T
(3.1)
if n 6∈ T .
Then, for every f ∈ DBVEγ [0, ∞) (γ ≥ 0), x ∈ (0, ∞) and for sufficiently large n ∈ T , (1.6) also holds when un (x) is replaced with u∗n (x). In a similar manner we can obtain the next result from Theorem 1.2. Theorem 3.2. Let A = [ajn ] be a non-negative regular summability matrix and let T be any subset of N such that δA (T ) = 1. Assume that Pk,n ≡ sk,n and αn ≡ vn∗ , where
p −1 + 1 + 4n(n + 1)x2 , v n ( x) = 2(n + 1) 0, ∗
if n ∈ T
(3.2)
if n 6∈ T .
Then, for every f ∈ DBVγ [0, ∞) (γ ≥ 2), x ∈ (0, ∞) and for sufficiently large n ∈ T , (1.7) also holds when vn (x) is replaced with vn∗ (x). By (3.1), observe that stA − lim u∗n (x) = x, which yields
stA − lim 1 −
u∗n (x) x
= 0.
So, if we replace un (x) with u∗n (x) in the R.H.S. of (1.6), then stA − lim (R.H.S. of (1.6)) = 0. Then, by Theorem 3.1, we have stA − lim |Ln (f ; x) − f (x)| = 0 for every f ∈ DBVEγ [0, ∞) (γ ≥ 0) and x ∈ (0, ∞) in the case of Pk,n ≡ sk,n and αn ≡ u∗n . Hence we get the (pointwise) statistical approximation to a function f ∈ DBVEγ [0, ∞) (γ ≥ 0) by means of the positive linear operators Ln . Here if we take A = C1 , the Cesáro matrix, and T = {n 6= m2 : m ∈ N}, then δC1 (T ) = 1. Also, by (3.1), we get stC1 − lim u∗n (x) = st − lim u∗n (x) = x but the sequence (u∗n (x)) is non-convergent (in the usual sense). This application clearly shows that it is possible to find positive linear operators satisfying the conditions of statistical approximation process but not the classical one. Similar idea is also valid for Theorem 3.2, but we omit the details. Acknowledgements The authors would like to thank the referees for carefully reading the manuscript. The present investigation was supported, in part, by the Ministry of National Education of TRNC under Project MEKB-09-01. References [1] [2] [3] [4] [5] [6] [7] [8]
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