Journal of Inequalities in Pure and
Applied Mathematics
ON THE APPROXIMATION OF LOCALLY BOUNDED FUNCTIONS
BY OPERATORS OF BLEIMANN, BUTZER AND HAHN
volume 6, issue 1, article 4,
2005.
JESÚS DE LA CAL AND VIJAY GUPTA
Departamento de Matemática Aplicada y Estadística e Investigación Operativa
Facultad de Ciencias
Universidad del País Vasco
Apartado 644, 48080 Bilbao, Spain
EMail: mepcaagj@lg.ehu.es
School of Applied Sciences
Netaji Subhas Institute of Technology
Sector 3 Dwarka, New Delhi-110045, India
EMail: vijay@nsit.ac.in
Received 25 November, 2004;
accepted 29 December, 2004.
Communicated by: A. Lupaş
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
228-04
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Abstract
We estimate the rate of the pointwise approximation by operators of Bleimann,
Butzer and Hahn of locally bounded functions, and of functions having a locally
bounded derivative.
2000 Mathematics Subject Classification: 41A20, 41A25, 41A36.
Key words: Operators of Bleimann, Butzer and Hahn, Locally bounded function,
Function of bounded variation, Total variation, Rate of convergence, Binomial distribution.
On The Approximation Of
Locally Bounded Functions By
Operators Of Bleimann, Butzer
And Hahn
J. de la Cal was supported by the Spanish MCYT, Proyecto BFM2002-04163-C0202, and by FEDER.
Jesús de la Cal and Vijay Gupta
Contents
1
Introduction and Main Results . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Proof of Theorem 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
Proof of Theorem 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
Proof of Theorem 1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
Remarks on Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
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8
14
17
18
21
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J. Ineq. Pure and Appl. Math. 6(1) Art. 4, 2005
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1.
Introduction and Main Results
Bleimann, Butzer and Hahn [1] introduced the Bernstein type operator Ln over
the interval [0, ∞) given by
n
X
Ln (f, x) :=
f
k=0
k
n−k+1
bn,k (x),
where f is a real function on [0, ∞), and
n k n−k
x
(1.1) bn,k (x) :=
p x qx ,
px :=
,
1+x
k
x ≥ 0, n = 1, 2, . . . ,
qx := 1 − px =
1
.
1+x
On The Approximation Of
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Operators Of Bleimann, Butzer
And Hahn
Jesús de la Cal and Vijay Gupta
The approximation of uniformly continuous functions by these operators has
been considered in [1] – [4]. For other properties of Ln (preservation of global
smoothness, preservation of φ-variation, behavior of the iterates, etc.) we refer,
for instance, to [4] – [10]. In some of the mentioned works, the results are
achieved by using probabilistic methods. This comes from the fact that Ln is an
operator of probabilistic type. We can actually write
Ln (f, x) = Ef (Zn,x ),
where E denotes mathematical expectation, and Zn,x is the random variable
given by
(1.2)
Zn,x :=
Sn,x
,
n − Sn,x + 1
Sn,x := ξ1,x + · · · + ξn,x ,
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where ξ1,x , ξ2,x , . . . are independent random variables having the same Bernoulli
distribution with parameter px , i.e.,
P (ξk,x = 1) = px = 1 − P (ξk,x = 0)
(so that Sn,x has the binomial distribution with parameters n, px ). This probabilistic representation also plays a significant role in the present paper (for a
more refined representation useful for other purposes, see [5, 6]).
Here, we discuss the approximation of real functions f on the semi axis
which are locally bounded, i.e., bounded on each finite subinterval of [0, ∞). In
such a case, we set, for x > 0 and h ≥ 0,
ωx+ (f ; h) :=
ωx− (f ; h) :=
sup |f (t) − f (x)|,
Jesús de la Cal and Vijay Gupta
x≤t≤x+h
sup
|f (t) − f (x)|,
(x−h)+ ≤t≤x
ωx (f ; h) := ωx+ (f ; h) + ωx− (f ; h),
where (x−h)+ := max(x−h, 0), and we observe that these functions are (nonnegative and) nondecreasing on [0, ∞). In particular, every continuous function
is locally bounded. Also, if f is locally of bounded variation, i.e., such that
b
_
On The Approximation Of
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Operators Of Bleimann, Butzer
And Hahn
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0 ≤ a < b < ∞,
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a
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W
where ba (f ) stands for the total variation of f on the interval [a, b], then f is
locally bounded, and we obviously have
ωx (f ; h) ≤
x+h
_
(f ),
0 ≤ h ≤ x.
x−h
This kind of problem has been already considered for other Bernstein-type
operators (see, for instance, [11] – [14] and the references therein). Our main
results are stated as follows.
Theorem 1.1. Let g be a real locally bounded function on [0, ∞) such that
g(t) = O(tr ) (t → ∞), for some r = 1, 2, . . . . If g is continuous at x > 0,
then, for n large enough, we have
n
7(1 + x)2 X
x
1
.
(1.3)
|Ln (g, x) − g(x)| ≤
ωx g; √
+ Or,x
n
(n + 2)x k=1
k
In the following statements (and throughout the paper), we use the notations:
f ∗ (x) := f (x+) − f (x−)
f (x+) + f (x−)
,
f˜(x) :=
2
fx := (f − f (x−))1[0,x) + (f − f (x+))1(x,∞)
(1A being the indicator function of the set A), provided that the lateral limits f (x+) and f (x−) exist (such a condition is fulfilled when f is locally of
bounded variation). We also use the symbol bac to indicate the integral part of
the real number a.
On The Approximation Of
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And Hahn
Jesús de la Cal and Vijay Gupta
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Theorem 1.2. Let f be a real locally bounded function on [0, ∞) such that
f (t) = O(tr ) (t → ∞), for some r = 1, 2, . . . . If x > 0, and f (x+) and
f (x−) exist, then we have for n large enough
˜
Ln (f, x) − f (x)
≤ ∆n,x (fx ) +
1.6 + x + 2.6 x2 |f ∗ (x)| n,x (1 + x)
√
·
+ √
|f (x) − f (x−)|,
2
nx(1 + x)
2enx
where ∆n,x (fx ) is the right-hand side of (1.3) with g replaced by fx , and
(
1 if (n + 1)px ∈ {1, 2, . . . , n}
n,x :=
0 otherwise.
Theorem 1.3. Let g be a real function on [0, ∞) such that g(t) = O(tr )
∞), for some r = 1, 2, . . . , and having the form
Z t
g(t) = c +
f (u) du,
t ≥ 0,
On The Approximation Of
Locally Bounded Functions By
Operators Of Bleimann, Butzer
And Hahn
Jesús de la Cal and Vijay Gupta
(t →
0
where c is a constant and f is measurable and locally bounded on [0, ∞). If
x > 0, and f (x+) and f (x−) exist, then we have for n large enough
√
x(1
+
x)
∗
Ln (g, x) − g(x) − √
f
(x)
2πn
√
b nc
5(1 + x)2 X x ≤
ωx fx ;
+|f ∗ (x)| ox n−1/2 +Or,x (n−1 ).
n + 2 k=1
k
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The proofs of the preceding theorems are given in Sections 3 – 5. In Section
2, we collect the necessary auxiliary results. Some remarks on moments close
the paper.
On The Approximation Of
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And Hahn
Jesús de la Cal and Vijay Gupta
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2.
Auxiliary Results
In the following lemma, Φ denotes the standard normal distribution
√function,
∗
∗
and Fn,x stands for the distribution function of Sn,x := (Sn,x − npx )
npx qx ,,
where Sn,x is the same as in (1.2). Such a lemma is nothing but the application
of the well-known Berry-Esseen theorem (cf. [15]) to the situation at hand.
Lemma 2.1. We have, for x > 0 and n ≥ 1,
sup
−∞<t<∞
∗
|Fn,x
(t) − Φ(t)| ≤
0.8(1 + x2 )
0.8(p3x qx + px qx3 )
√
√
.
=
nx(1 + x)
n(px qx )3/2
Lemma 2.2. Let x > 0 and n ≥ 1. Then, we have:
On The Approximation Of
Locally Bounded Functions By
Operators Of Bleimann, Butzer
And Hahn
Jesús de la Cal and Vijay Gupta
(a)
Ln ((· − x)2 , x) = E(Zn,x − x)2 ≤
3x(1 + x)2
.
n+2
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(b)
P (Zn,x ≤ x − h) + P (Zn,x ≥ x + h) ≤
3x(1 + x)2
,
(n + 2)h2
h > 0.
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(c)
r
|P (Zn,x > x) − P (Zn,x ≤ x)| ≤
2
x
1.6(1 + x )
+√
.
n
nx(1 + x)
(d)
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Ln ((· − x), x) = E(Zn,x − x) =
−xpnx
−1
= ox (n ),
(n → ∞).
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(e)
√
Ln (| · −x|, x) = E|Zn,x − x| =
2x(1 + x)
√
+ ox (n−1/2 ),
πn
(n → ∞).
Proof. Part (a) was shown in [10]. Part (b) follows from (a) and the fact that, by
Markov’s inequality,
2
P (Zn,x ≤ x − h) + P (Zn,x ≥ x + h) = P (|Zn,x − x| ≥ h) ≤
E(Zn,x − x)
.
h2
To show (c), observe that
|P (Zn,x > x) − P (Zn,x ≤ x)|
= |1 − 2P (Zn,x ≤ x)|
= |1 − 2P (Sn,x ≤ (n + 1)px )|
r x ∗
= 1 − 2Fn,x
n r r r x
x
x ∗
+ 1 − 2Φ
≤ 2 Φ
− Fn,x
.
n
n
n Thus, the conclusion in part (c) follows from Lemma 2.1 and the fact that (cf.
[16])
2 1/2
0 < 2Φ(t) − 1 ≤ 1 − e−t
≤ t,
(t > 0).
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Part (d) is immediate. Finally, to show (e), let m := b(n + 1)px c. We have
Ln (| · −x|, x) − Ln ((· − x), x)
m X
k
=2
x−
bn,k (x)
n
−
k
+
1
k=0
= 2x
m
X
k=0
= 2x
m
X
k=0
bn,k (x) − 2
m
X
n!
pk q n−k
(k − 1)!(n − k + 1)! x x
k=1
m−1
X
bn,k (x) − 2x
On The Approximation Of
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And Hahn
bn,k (x)
k=0
= 2x bn,m (x)
√
2x(1 + x)
√
=
+ ox (n−1/2 ),
πn
Jesús de la Cal and Vijay Gupta
(n → ∞),
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the last equality by [13, Lemma 1], and the conclusion follows from (d).
Lemma 2.3. Let x > 0 and r = 1, 2, . . . . Then, we have for all integers n such
that (n + 1)(p2x − p3x/2 ) ≥ r,
X
k∈K
r n o s−1
X
kr
r x (1 + x)r−s+2
n!
b
(x)≤
12
r!
·
n,k
(n − k + 1)r
s
n + r − s + 2 (n + r − s)!
s=1
= Or,x (n−1 ),
(n → ∞),
where the rs are the Stirling numbers of the second kind, and K is the set of
all integers k such that n ≥ k > (n − k + 1)2x (i.e., n ≥ k > (n + 1)p2x ).
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Proof. Using the well known identity
r
a =
r n o
X
r
s=1
s
a(a − 1) · · · (a − s + 1),
we can write
r
(2.1)
X
k∈K
X nro
kr
As ,
bn,k (x) =
(n − k + 1)r
s
s=1
where
As :=
X k(k − 1) · · · (k − s + 1)
(n − k + 1)r
k∈K
=
X
k∈K
bn,k (x)
n!
1
·
pk q n−k .
r
(n − k + 1) (k − s)!(n − k)! x x
Since
r Y
1
n−k+i
1
=
(n − k + 1)r i=1 n − k + i n − k + 1
r Y
1
i−1
=
1+
n−k+i
n−k+1
i=1
≤
r
Y
i=1
i
r!(n − k)!
=
,
n−k+i
(n − k + r)!
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we have
As ≤ r!
X
k∈K
n!
pk q n−k
(k − s)!(n − k + r)! x x
n!
n−l−s
pl+s
x qx
l!(n + r − s − l)!
l∈Ks
xl
r!n!psx qx−r X n + r − s
=
(n + r − s)! l∈K
l
(1 + x)n+r−s
s
r!n!psx qx−r X n + r − s
xl
≤
,
(n + r − s)! l∈K 0
l
(1 + x)n+r−s
= r!
X
On The Approximation Of
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Jesús de la Cal and Vijay Gupta
0
where Ks := {k − s : k ∈ K}, and K stands for the set of all integers l such
that n ≥ l > (n − l + 1)(3x/2) (observe that, by the assumption on n, we have
Ks ⊂ K 0 ). The probabilistic interpretation of the last sum together with Lemma
2.2(b) yield
r!n!xs (1 + x)r−s
3x
As ≤
P Zn+r−s,x >
(n + r − s)!
2
s−1
r−s+2
12 r!n!x (1 + x)
(2.2)
≤
,
(n + r − s)!(n + r − s + 2)
and the conclusion follows from (2.1) and (2.2).
Remark 1. The same procedure as in the preceding proof leads to the following
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upper bound for the integral moments of Ln (or Zn,x ):
Ln (tr , x) = E(Zn,x )r
n
X
kr
=
bn,k (x)
(n − k + 1)r
k=0
r n o
X
r n!xs (1 + x)r−s
≤ r!
.
s
(n
+
r
−
s)!
s=1
On The Approximation Of
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3.
Proof of Theorem 1.1
Without loss of generality, we assume that g(x) = 0. Denote by Kn,x the
distribution function of Zn,x , i.e.,
X
Kn,x (t) := P (Zn,x ≤ t) =
bn,k (x)
t ≥ 0.
k≤(n−k+1)t
We can write Ln (g, x) as the Lebesgue-Stieltjes integral
Z
Ln (g, x) = Eg(Zn,x ) =
g(t) dKn,x (t) =
[0,∞)
where
4 Z
X
j=1
g(t) dKn,x (t),
Ij
x
x
x
I2 := x − √ , x + √ ,
I1 := 0, x − √ ,
n
n
n
x
I3 := x + √ , 2x
and
I4 := (2x, ∞).
n
We obviously have
Z
Z
x
|g(t)| dKn,x (t)≤ ωx g; √
dKn,x (t)
n
I2
I2
x
≤ ωx g; √
n
n
1X
x
ωx g; √
(3.1)
≤
.
n k=1
k
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On the other hand, from the asymptotic assumption on g, we have
|g(t)| ≤ M tr ,
t ≥ α,
for some constants M > 0 and α ≥ 2x. Therefore,
Z
|g(t)| dKn,x (t)
I4
Z
Z
=
+
|g(t)| dKn,x (t)
(2x,α]
(α,∞)
≤ ωx+ (g; α − x)P (Zn,x > 2x) + M
X
k>(n−k+1)α
r
k
bn,k (x).
(n − k + 1)r
By Lemma 2.2(b) and Lemma 2.3, this shows that
Z
(3.2)
|g(t)| dKn,x (t) = Or,x (n−1 )
(n → ∞).
I4
Finally, using Lemma 2.2(b) and integration by parts (follow the same procedure as in the proof of Theorem 1 in [13]), we obtain
Z
Z
|g(t)| dKn,x (t)≤
ωx− (g; x − t) dKn,x (t)
I1
I1
#
"
Z x−x/√n −
ωx (g; x − t)
3x(1 + x)2 ωx− (g; x)
≤
dt
+2
(x − t)3
(n + 2)
x2
0
n
6(1 + x)2 X −
x
≤
ω
g; √
(3.3)
,
(n + 2)x k=1 x
k
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and, analogously,
n
x
6(1 + x)2 X +
|g(t)| dKn,x (t) ≤
ω
g; √
.
(n + 2)x k=1 x
k
I3
Z
(3.4)
The conclusion follows from (3.1) – (3.4).
On The Approximation Of
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4.
Proof of Theorem 1.2
We can write, for t ≥ 0,
(4.1)
f ∗ (x)
f (t) − f˜(x) = fx (t) +
σx (t) + (f (x) − f˜(x))δx (t),
2
where σx := −1[0,x) + 1(x,∞) , and δx := 1{x} is Dirac’s delta at x (this is the so
called Bojanic-Vuilleumier-Cheng decomposition).
By Theorem 1.1, we have
(4.2)
|Ln (fx , x)| ≤ ∆n,x (fx ),
On The Approximation Of
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And Hahn
where ∆n,x (fx ) is the right-hand side of (1.2) with g replaced by fx . Moreover,
Jesús de la Cal and Vijay Gupta
Ln (σx , x)= P (Zn,x > x) − P (Zn,x < x)
= (P (Zn,x > x) − P (Zn,x ≤ x)) + P (Zn,x = x),
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(4.3)
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and
(4.4)
Ln (δx , x) = P (Zn,x = x).
Using Lemma 2.2(c) and the fact that (cf. [17, Theorem 1])
  n pkx qxn−k ≤ √(1+x) if (n + 1)px = k ∈ {1, 2, . . . , n}
k
2enx
P (Zn,x = x) =
 0
otherwise,
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the conclusion readily follows from (4.1) – (4.4).
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5.
Proof of Theorem 1.3
Using the decomposition (4.1), it is easily checked that
Ln (g, x) − g(x) =
(5.1)
4
X
Ai (n, x),
i=1
where
f ∗ (x)
A1 (n, x) := f˜(x)Ln ((· − x), x) +
Ln (| · −x|, x),
2
Z x
Z
A2 (n, x) :=
fx (u) du dKn,x (t),
[0,x]
Z
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t
t
Z
A3 (n, x) :=
Jesús de la Cal and Vijay Gupta
fx (u) du dKn,x (t),
(x,2x]
Z
x
Z
A4 (n, x) :=
(2x,∞)
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fx (u) du dKn,x (t),
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t
x
and Kn,x (t) is the same as in the preceding proofs.
From Lemma 2.2(d,e), we have
√
x(1 + x) ∗
(5.2) A1 (n, x) = √
f (x)+f ∗ (x) ox (n−1/2 )+ox (n−1 ),
2πn
JJ
J
(n → ∞).
Next, we estimate A2 (n, x). By Fubini’s theorem,
Z
A2 (n, x) =
√
x
Z
Kn,x (u) fx (u) du =
0
x−x/ n
Z
+
0
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Kn,x (u) fx (u) du.
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It is clear that
Z
Z x
≤
K
(u)
f
(u)
du
|fx (u)| du
n,x
x
√
√
x−x/ n
x−x/ n
Z x
≤
ωx− (fx ; x − u) du
√
x−x/ n
x
x −
≤ √ ωx fx ; √
n
n
√
b nc
2x X − x ω fx ;
,
≤
n k=1 x
k
x
and, using Lemma 2.2(b),
Z
x−x/√n
3x(1 + x)2 Z x−x/√n |f (u)|
x
Kn,x (u) fx (u) du≤
du
0
(n + 2)
(x
− u)2
0
√
Z
3x(1 + x)2 x−x/ n ωx− (fx ; x − u)
du
≤
(x − u)2
(n + 2)
0
Z √n
x
3(1 + x)2
−
dt
≤
ωx fx ;
t
(n + 2) 1
√
b nc
3(1 + x)2 X − x ≤
ω fx ;
.
n + 2 k=1 x
k
On The Approximation Of
Locally Bounded Functions By
Operators Of Bleimann, Butzer
And Hahn
Jesús de la Cal and Vijay Gupta
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We therefore conclude that
√
b nc
5(1 + x)2 X − x ω fx ;
.
|A2 (n, x)| ≤
n + 2 k=1 x
k
(5.3)
Similarly,
√
b nc
5(1 + x)2 X + x |A3 (n, x)| ≤
ω fx ;
.
n + 2 k=1 x
k
(5.4)
Finally,
Z
Jesús de la Cal and Vijay Gupta
Z
g(t) dKn,x (t) −
A4 (n, x) =
(2x,∞)
[g(x) + f (x+)(t − x)] dKn,x (t),
(2x,∞)
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and, by the asymptotic assumption on g, Lemma 2.2(b) and Lemma 2.3, we
obtain
(5.5)
On The Approximation Of
Locally Bounded Functions By
Operators Of Bleimann, Butzer
And Hahn
−1
|A4 (n, x)| = Or,x (n ),
The conclusion follows from (5.1) – (5.5).
(n → ∞).
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6.
Remarks on Moments
Fix x > 0, and let g(·) := | · −x|β , with β > 2. Since
ωx (g, h) = 2 hβ ,
and
n
X
0 ≤ h ≤ x,
k −β/2 = O(1),
(n → ∞),
k=1
On The Approximation Of
Locally Bounded Functions By
Operators Of Bleimann, Butzer
And Hahn
we conclude from Theorem 1.1 that
Ln (| · −x|β , x) = Or,x (n−1 ),
(n → ∞).
Jesús de la Cal and Vijay Gupta
In the case that 0 < β ≤ 2, we have, by Jensen’s inequality (or Hölder’s
inequality) and Lemma 2.2(a),
β
β
2 β/2
Ln (| · −x| , x) = E|Zn,x − x| ≤ E(Zn,x − x)
for all n ≥ 1.
≤
3x(1 + x)2
n+2
β2
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References
[1] G. BLEIMANN, P.L. BUTZER AND L. HAHN, A Bernstein-type operator
approximating continuous functions on the semi axis, Indag. Math., 42
(1980), 255–262.
[2] V. TOTIK, Uniform approximation by Bernstein-type operators, Indag.
Math., 46 (1984), 87–93.
[3] R.A. KHAN, A note on a Bernstein type operator of Bleimann, Butzer and
Hahn, J. Approx. Theory, 53 (1988), 295–303.
[4] R.A. KHAN, Some properties of a Bernstein type operator of Bleimann,
Butzer and Hahn, In Progress in Approximation Theory, (Edited by P.
Nevai and A. Pinkus), pp. 497–504, Academic Press, New York (1991).
[5] J.A. ADELL AND J. DE LA CAL, Preservation of moduli of continuity
for Bernstein-type operators, In Approximation, Probability, and Related
Fields, (Edited by G. Anastassiou and S.T. Rachev), pp. 1–18, Plenum
Press, New York (1994).
[6] J.A. ADELL AND J. DE LA CAL, Bernstein-type operators diminish the
φ-variation, Constr. Approx., 12 (1996), 489–507.
[7] J.A. ADELL, F.G. BADíA AND J. DE LA CAL, On the iterates of some
Bernstein-type operators, J. Math. Anal. Appl., 209 (1997), 529–541.
[8] B. DELLA VECCHIA, Some properties of a rational operator of Bernstein
type, In Progress in Approximation Theory, (Edited by P. Nevai and A.
Pinkus), pp. 177–185, Academic Press, New York (1991).
On The Approximation Of
Locally Bounded Functions By
Operators Of Bleimann, Butzer
And Hahn
Jesús de la Cal and Vijay Gupta
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[9] U. ABEL AND M. IVAN, Some identities for the operator of BleimannButzer-Hahn involving divided differences, Calcolo, 36 (1999), 143–160.
[10] U. ABEL AND M. IVAN, Best constant for a Bleimann-Butzer-Hahn moment estimation, East J. Approx., 6 (2000), 1–7.
[11] S. GUO AND M. KHAN, On the rate of convergence of some operators on
functions of bounded variation, J. Approx. Theory, 58 (1989), 90–101.
[12] V. GUPTA AND R.P. PANT, Rate of convergence for the modified SzászMirakyan operators on functions of bounded variation, J. Math. Anal.
Appl., 233 (1999), 476–483.
[13] X.M. ZENG AND F. CHENG, On the rates of approximation of Bernstein
type operators, J. Approx. Theory, 109 (2001), 242–256.
[14] X.M. ZENG AND V. GUPTA, Rate of convergence of Baskakov-Bézier
type operators for locally bounded functions, Computers. Math. Applic.,
44 (2002), 1445–1453.
[15] A.N. SHIRYAYEV, Probability, Springer, New York (1984).
[16] N.L. JOHNSON, S. KOTZ AND N. BALAKRISHNAN, Continuous Univariate Distributions, Vol. 1, 2nd Edition, Wiley, New York (1994).
[17] X.M. ZENG, Bounds for Bernstein basis functions and Meyer-König and
Zeller basis functions, J. Math. Anal. Appl., 219 (1998), 364–376.
On The Approximation Of
Locally Bounded Functions By
Operators Of Bleimann, Butzer
And Hahn
Jesús de la Cal and Vijay Gupta
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