Journal of Inequalities in Pure and
Applied Mathematics
DIRECT RESULTS FOR CERTAIN FAMILY OF INTEGRAL
TYPE OPERATORS
volume 6, issue 1, article 21,
2005.
VIJAY GUPTA AND OGÜN DOĞRU
School of Applied Sciences
Netaji Subhas Institute of Technology
Sector 3 Dwarka, New Delhi-110045, India.
EMail: vijay@nsit.ac.in
Ankara University
Faculty of Science
Department of Mathematics
Tandoğan, Ankara-06100, Turkey
EMail: dogru@science.ankara.edu.tr
Received 08 September, 2004;
accepted 22 January, 2005.
Communicated by: D. Hinton
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
165-04
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Abstract
In the present paper we introduce a certain family of linear positive operators
and study some direct results which include a pointwise convergence, asymptotic formula and an estimation of error in simultaneous approximation.
2000 Mathematics Subject Classification: 41A25, 41A30.
Key words: Linear positive operators, Simultaneous approximation, Steklov mean,
Modulus of continuity.
Direct Results for Certain
Family of Integral Type
Operators
The authors are thankful to the referee for his/her useful recommendations and valuable remarks.
Vijay Gupta and Ogün Doğru
Contents
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1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3
Simultaneous Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 10
References
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1.
Introduction
We consider a certain family of integral type operators, which are defined as
∞
(1.1)
1X
Bn (f, x) =
pn,ν (x)
n ν=1
Z
∞
pn,ν−1 (t)f (t)dt
0
+ (1 + x)−n−1 f (0),
where
pn,ν (t) =
x ∈ [0, ∞),
1
tν (1 + t)−n−ν−1
B(n, ν + 1)
with B(n, ν + 1) = ν!(n − 1)!/(n + ν)! the Beta function.
Alternatively the operators (1.1) may be written as
Z ∞
Bn (f, x) =
Wn (x, t)f (t)dt,
0
where
∞
Wn (x, t) =
1X
pn,ν (x)pn,ν−1 (t) + (1 + x)−n−1 δ(t),
n ν=1
δ(t) being the Dirac delta function. The operators Bn are discretely defined
linear positive operators. It is easily verified that these operators reproduce only
the constant functions. As far as the degree of approximation is concerned these
operators are very similar to the operators considered by Srivastava and Gupta
[5], but the approximation properties of these operators are different. In this
paper, we study some direct theorems in simultaneous approximation for the
operators (1.1).
Direct Results for Certain
Family of Integral Type
Operators
Vijay Gupta and Ogün Doğru
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2.
Auxiliary Results
In this section we mention some lemmas which are necessary to prove the main
theorems.
Lemma 2.1 ([3]). For m ∈ N0 , if the m-th order moment is defined as
m
∞
1X
ν
Un,m (x) =
pn,ν (x)
−x
,
n ν=0
n+1
then
0
(n + 1)Un,m+1 (x) = x(1 + x) Un,m
(x) + mUn,m−1 (x) .
Direct Results for Certain
Family of Integral Type
Operators
Vijay Gupta and Ogün Doğru
Consequently
Um,n (x) = O n
−[(m+1)/2]
.
Lemma 2.2. Let the function µn,m (x), n > m and m ∈ N0 , be defined as
Z ∞
∞
1X
µn,m (x) =
pn,ν (x)
pn,ν−1 (t)(t − x)m dt + (−x)m (1 + x)−n−1 .
n ν=1
0
Then
µn,0 (x) = 1, µn,1 (x) =
2x
2nx(1 + x) + 2x(1 + 4x)
, µn,2 (x) =
(n − 1)
(n − 1)(n − 2)
and there holds the recurrence relation
(n − m − 1)µn,m+1 (x)
= x(1 + x) µ0n,m (x) + 2mµn,m−1 (x) + [m(1 + 2x) + 2x] µn,m (x).
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Consequently for each x ∈ [0, ∞), we have from this recurrence relation that
µn,m (x) = O n−[(m+1)/2] .
Proof. The values of µn,0 (x), µn,1 (x) and µn,2 (x) easily follow from the definition. We prove the recurrence relation as follows
x(1 + x)µ0n,m (x)
Z ∞
∞
1X
0
=
x(1 + x)pn,ν (x)
pn,ν−1 (t)(t − x)m dt
n ν=1
0
Z ∞
∞
1X
−m
x(1 + x)pn,ν (x)
pn,ν−1 (t)(t − x)m−1 dt
n ν=1
0
m
−n−2
− (n + 1)(−x) (1 + x)
+ m(−x)m−1 (1 + x)−n−1 x(1 + x).
Now using the identities x(1 + x)p0n,ν (x) = [ν − (n + 1)x] pn,ν (x), we obtain
x(1 + x) µ0n,m (x) + mµn,m−1 (x)
∞
1X
=
[ν − (n + 1)x] pn,ν (x)
n ν=1
Z ∞
×
pn,ν−1 (t)(t − x)m dt + (n + 1)(−x)m+1 (1 + x)−n−1
0
Z ∞
∞
1X
pn,ν (x)
=
[{(ν − 1) − (n + 1)t} + (n + 1)(t − x) + 1]
n ν=1
0
× pn,ν−1 (t)(t − x)m dt + (n + 1)(−x)m+1 (1 + x)−n−1
Direct Results for Certain
Family of Integral Type
Operators
Vijay Gupta and Ogün Doğru
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∞
1X
=
pn,ν (x)
n ν=1
Z
∞
t(1 + t)p0n,ν−1 (t)(t − x)m dt
0
Z ∞
∞
1X
+ (n + 1)
pn,ν (x)
pn,ν−1 (t)(t − x)m+1 dt
n ν=1
0
Z
∞
∞
1X
pn,ν (x)
+
pn,ν−1 (t)(t − x)m dt + (n + 1)(−x)m+1 (1 + x)−n−1
n ν=1
0
Z
∞
∞
1X
=
pn,ν (x)
[(1 + 2x)(t − x)
n ν=1
0
+(t − x)2 + x(1 + x) p0n,ν−1 (t)(t − x)m dt
Z ∞
∞
1X
+ (n + 1)
pn,ν (x)
pn,ν−1 (t)(t − x)m+1 dt
n ν=1
0
Z
∞
∞
1X
+
pn,ν (x)
pn,ν−1 (t)(t − x)m dt + (n + 1)(−x)m+1 (1 + x)−n−1
n ν=1
0
= −(m + 1)(1 + 2x) µn,m (x) − (−x)m (1 + x)−n−1
− (m + 2) µn,m+1 (x) − (−x)m+1 (1 + x)−n−1
− mx(1 + x) µn,m−1 (x) − (−x)m−1 (1 + x)−n−1
+ (n + 1) µn,m+1 (x) − (−x)m+1 (1 + x)−n−1
+ µn,m (x) − (−x)m (1 + x)−n−1 + (n + 1)(−x)m+1 (1 + x)−n−1
= − [m(1 + 2x) + 2x] µn,m (x) + (n − m − 1)µn,m+1 (x) − mx(1 + x)µn,m−1 (x).
This completes the proof of recurrence relation.
Direct Results for Certain
Family of Integral Type
Operators
Vijay Gupta and Ogün Doğru
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Remark 1. It is easily verified from Lemma 2.2 and by the principle of mathematical induction, that for n > i and each x ∈ (0, ∞)
Bn (ti , x) =
(n + i)!(n − i − 1)! i
x
n!(n − 1)!
(n + i − 1)!(n − i − 1)! i−1
+ i(i − 1)
x
n!(n − 1)!
+ i(i − 1)(i − 2)O n−2 .
Corollary 2.3. Let δ be a positive number. Then for every n > γ > 0, x ∈
(0, ∞), there exists a constant M (s, x) independent of n and depending on s
and x such that
Z
∞
1X
pn,ν (x)
pn,ν−1 (t)tγ dt ≤ M (s, x)n−s , s = 1, 2, 3, . . . .
n ν=1
|t−x|>δ
Lemma 2.4 ([3]). There exist the polynomials Qi,j,r (x) independent of n and ν
such that
X
{x(1 + x)}r Dr [pn,ν (x)] =
(n + 1)i [ν − (n + 1)x]j Qi,j,r (x)pn,ν (x),
2i+j≤r
i,j≥0
where D ≡
d
.
dx
Lemma 2.5. Let f be r times differentiable on [0, ∞) such that f (r−1) is absolutely continuous with f (r−1) (t) = O(tγ ) for some γ > 0 as t → ∞. Then for
Direct Results for Certain
Family of Integral Type
Operators
Vijay Gupta and Ogün Doğru
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r = 1, 2, 3, . . . and n > γ + r we have
∞
Bn(r) (f, x)
(n + r − 1)!(n − r − 1)! X
=
pn+r,ν (x)
n!(n − 1)!
ν=0
Z
∞
pn−r,ν+r−1 (t)f (r) (t)dt.
0
Proof. It follows by simple computation the following relations:
p0n,ν (t) = n [pn+1,ν−1 (t) − pn+1,ν (t)] ,
(2.1)
where t ∈ [0, ∞).
Furthermore, we prove our lemma by mathematical induction. Using the
above identity (2.1), we have
∞
Bn0 (f, x)
1X 0
p (x)
=
n ν=1 n,ν
=
∞
X
Z
pn,ν−1 (t)f (t)dt − (n + 1)(1 + x)−n−2 f (0)
[pn+1,ν−1 (x) − pn+1,ν (x)]
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∞
pn,ν−1 (t)f (t)dt
0
− (n + 1)(1 + x)−n−2 f (0)
Z ∞
= npn+1,0 (x)
pn,0 (t)f (t)dt − (n + 1)(1 + x)−n−2 f (0)
0
ν=1
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0
ν=1
+
Vijay Gupta and Ogün Doğru
∞
Z
∞
X
Direct Results for Certain
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Z
∞
[pn,ν (t) − pn,ν−1 (t)] f (t)dt
pn+1,ν (x)
0
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−n−2
Z
∞
n(1 + t)−n−1 f (t)dt
0
Z ∞
∞
X
−1
+
pn+1,ν (x)
p0n−1,ν (t)f (t)dt
n
−
1
0
ν=1
= (n + 1)(1 + x)
− (n + 1)(1 + x)−n−2 f (0) .
Applying the integration by parts, we get
Bn0 (f, x)
= (n + 1)(1 + x)−n−2 f (0) + (n + 1)(1 + x)−n−2
Z
∞
(1 + t)−n f 0 (t)dt
0
1
+
n−1
=
1
n−1
∞
X
Z
pn+1,ν (x)
ν=1
∞
X
pn+1,ν (x)
ν=0
Vijay Gupta and Ogün Doğru
∞
pn−1,ν (t)f 0 (t)dt − (n + 1)(1 + x)−n−2 f (0)
0
Z
Direct Results for Certain
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Operators
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∞
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pn−1,ν (t)f 0 (t)dt ,
0
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which was to be proved.
If we suppose that
∞
(n + i − 1)!(n − i − 1)! X
Bn(i) (f, x) =
pn+i,ν (x)
n!(n − 1)!
ν=0
Z
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∞
pn−i,ν+i−1 (t)f (i) (t)dt
0
then by (2.1), and using a similar method to the one above it is easily verified
that the result is true for r = i + 1. Therefore by the principle of mathematical
induction the result follows.
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3.
Simultaneous Approximation
In this section we study the rate of pointwise convergence of an asymptotic
formula and an error estimation in terms of a higher order modulus of continuity
in simultaneous approximation for the operators defined by (1.1). Throughout
the section, we have Cγ [0, ∞) := {f ∈ C[0, ∞) : |f (t)| ≤ M tγ for some
M > 0, γ > 0}.
Theorem 3.1. Let f ∈ Cγ [0, ∞), γ > 0 and f (r) exists at a point x ∈ (0, ∞),
then
Bn(r) (f, x) = f (r) (x) + o(1) as n → ∞.
Proof. By Taylor’s expansion of f , we have
f (t) =
r
X
f (i) (x)
i=0
i!
(t − x)i + ε(t, x)(t − x)r ,
Direct Results for Certain
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Vijay Gupta and Ogün Doğru
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where ε(t, x) → 0 as t → x.
Hence
Z ∞
(r)
Bn (f, x) =
Wn(r) (t, x)f (t)dt
0
=
Z
r
X
f (i) (x)
i=0
i!
Z
+
∞
Wn(r) (t, x)(t − x)i dt
0
∞
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Wn(r) (t, x)ε(t, x)(t − x)r dt
Page 10 of 21
0
=: R1 + R2 .
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First to estimate R1 , using the binomial expansion of (t − x)m , Lemma 2.2 and
Remark 1, we have
R1 =
r
i X
f (i) (x) X i
i=0
(r)
i!
(−x)
∂r
∂xr
Z
∞
Wn (t, x)tν dt
0
(x) ∂
Wn (t, x)tr dt
r! ∂xr 0
f (r) (x) (n + r)!(n − r − 1)!
=
r! + terms containing lower powers of x
r!
n!(n − 1)!
=
f
ν=0
r Z ∞
ν
i−ν
= f (r) (x) + o(1), n → ∞.
Direct Results for Certain
Family of Integral Type
Operators
Vijay Gupta and Ogün Doğru
Using Lemma 2.4, we obtain
Z ∞
R2 =
Wn(r) (t, x)ε(t, x)(t − x)r dt
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0
=
X
2i+j≤r
ni
Qi,j,r (x)
{x(1 + x)}r
∞
X
[ν − (n + 1)x]j
ν=1
pn,ν (x)
n
i,j≥0
Z
×
∞
pn,ν−1 (t)ε(t, x)(t − x)r dt
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0
r (n
+ r)!
+ (−1)
(1 + x)−n−r−1 ε(0, x)(−x)r
(n + 1)!
=: R3 + R4 .
Since ε(t, x) → 0 as t → x for a given ε > 0 there exists a δ > 0 such that
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|ε(t, x)| < ε whenever 0 < |t − x| < δ. Thus for some M1 > 0, we can write
|R3 | ≤ M1
X
n
i−1
∞
X
Z
pn,ν (x) |ν − (n + 1)x| ε
|t−x|<δ
ν=1
2i+j≤r
pn,ν−1 (t) |t − x|r dt
j
i,j≥0
pn,ν−1 (t)M2 t dt
Z
γ
+
|t−x|≥δ
=: R5 + R6 ,
where
Direct Results for Certain
Family of Integral Type
Operators
|Qi,j,r (x)|
r
2i+j≤r {x(1 + x)}
M1 = sup
Vijay Gupta and Ogün Doğru
i,j≥0
and M2 is independent of t.
Applying Schwarz’s inequality for integration and summation respectively,
we obtain
Z ∞
21
∞
X
X
j
i−1
R5 ≤ εM1
n
pn,ν (x) |ν − (n + 1)x|
pn,ν−1 (t)dt
×
∞
12
2r
pn,ν−1 (t)(t − x) dt
X
2i+j≤r
i,j≥0
ni
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0
≤ εM1
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i,j≥0
Z
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0
ν=1
2i+j≤r
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∞
X
ν=1
pn,ν (x)
1
n
∞
X
ν=1
! 12
pn,ν (x)[ν − (n + 1)x]2j
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∞
1X
pn,ν (x)
n ν=1
×
Z
! 12
∞
pn,ν−1 (t)(t − x)2r dt
.
0
Using Lemma 2.1 and Lemma 2.2, we get
R5 ≤ εM1 O nj/2 O n−r/2 = εO (1) .
Again using the Schwarz inequality, Lemma 2.1 and Corollary 2.3, we obtain
Z
∞
X
X
j
i−1
R 6 ≤ M2
n
pn,ν (x) |ν − (n + 1)x|
pn,ν−1 (t)tγ dt
|t−x|≥δ
ν=1
2i+j≤r
i,j≥0
X
≤ M2
n
i−1
∞
X
j
Z
pn,ν (x) |ν − (n + 1)x|
i,j≥0
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2γ
pn,ν−1 (t)t dt
Z
×
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|t−x|≥δ
X
≤ M2
2i+j≤r
ni
Vijay Gupta and Ogün Doğru
|t−x|≥δ
ν=1
2i+j≤r
12
pn,ν−1 (t)dt
Direct Results for Certain
Family of Integral Type
Operators
∞
1X
pn,ν (x)[ν − (n + 1)x]2j
n ν=1
! 12
i,j≥0
! 21
Z ∞
∞
1X
×
pn,ν (x)
pn,ν−1 (t)t2γ dt
n ν=1
0
X
=
ni O nj/2 O n−s/2
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2i+j≤r
i,j≥0
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for any s > 0.
Choosing s > r we get R6 = o(1). Thus, due to arbitrariness of ε > 0,
it follows that R3 = o(1). Also R4 → 0 as n → ∞ and hence R2 = o(1).
Collecting the estimates of R1 and R2 , we get the required result.
The following result holds.
Theorem 3.2. Let f ∈ Cγ [0, ∞), γ > 0. If f (r+2) exists at a point x ∈ (0, ∞),
then
lim
n→∞
n[Bn(r) (f, x)
−f
(r)
Direct Results for Certain
Family of Integral Type
Operators
(x)]
= r(r + 1)f (r) (x) + [2x(1 + r) + r]f (r+1) (x) + x(1 + x)f (r+2) (x).
Vijay Gupta and Ogün Doğru
Proof. Using Taylor’s expansion of f , we have
f (t) =
r+2 (i)
X
f (x)
i=0
i!
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(t − x)i + ε(t, x)(t − x)r+2 ,
β
where ε(t, x) → 0 as t → x and ε(t, x) = O (t − x) , t → ∞ for some
β > 0. Applying Lemma 2.2, we have
" r+2
#
X f (i) (x) Z ∞
(r)
(r)
(r)
i
(r)
n[Bn (f, x) − f (x)] = n
Wn (x, t)(t − x) dt − f (x)
i!
0
i=0
Z ∞
(r)
r+2
+ n
Wn (x, t)ε(t, x)(t − x) dt
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0
=: E1 + E2 .
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E1 = n
r+2 (i)
i X
f (x) X i
i=0
=
i!
j=0
j
i−j
Z
(−x)
∞
Wn(r) (x, t)tj dt − nf (r) (x)
0
f (r+1) (x) f (r) (x) (r) r
n Bn (t , x) − r! +
n (r + 1)(−x)Bn(r) (tr , x)
r!
(r + 1)!
(r+2)
f
(x) (r + 2)(r + 1) 2 (r) r
(r) r+1
+Bn (t , x) +
n
x Bn (t , x)
(r + 2)!
2
+ (r + 2)(−x)Bn(r) (tr+1 , x) + Bn(r) (tr+2 , x) .
Therefore by applying Remark 1, we get
(n + r)!(n − r − 1)!
E1 = nf (x)
−1
n!(n − 1)!
f (r+1) (x)
(n + r)!(n − r − 1)!
+n
(r + 1)(−x)r!
(r + 1)!
n!(n − 1)!
(n + r)!(n − r − 2)!
(n + r + 1)!(n − r − 2)!
(r + 1)!x + r(r + 1)
r!
+
n!(n − 1)!
n!(n − 1)!
f (r+2) (x) (r + 2)(r + 1)x2
(n + r)!(n − r − 1)!
+n
· r!
(r + 2)!
2
n!(n − 1)!
(n + r + 1)!(n − r − 2)!
+ (r + 2)(−x)
(r + 1)!x
n!(n − 1)!
(n + r)!(n − r − 2)!
r!
+ r(r + 1)
n!(n − 1)!
(r)
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+
(n + r + 2)!(n − r − 3)! (r + 2)! 2
x
n!(n − 1)!
2
(n + r + 1)!(n − r − 3)!
+(r + 1)(r + 2)
(r + 1)!x
+ O n−2 .
n!(n − 1)!
In order to complete the proof of the theorem it is sufficient to show that E2 → 0
as n → ∞, which can be easily proved along the lines of the proof of Theorem
3.1 and by using Lemma 2.1, Lemma 2.2 and Lemma 2.4.
Let us assume that 0 < a < a1 < b1 < b < ∞, for sufficiently small δ > 0,
the m-th order Steklov mean fm,δ (t) corresponding to f ∈ Cγ [0, ∞) is defined
by
Z δ Z δ Z δ
m
2
2
2 Y
−m
m
...
f (t) − ∆η f (t)
fm,δ (t) = δ
dti
− 2δ
1
m
− 2δ
− 2δ
i=1
Pm
m
where η =
i=1 ti , t ∈ [a, b] and ∆η f (t) is the m−th forward difference
with step length η.
It is easily checked (see e. g. [1], [4]) that
(i) fm,δ has continuous derivatives up to order m on [a, b];
(r) (ii) fm,δ ≤ M1 δ −r ωr (f, δ, a1 , b1 ), r = 1, 2, 3, . . . , m;
C[a1 ,b1 ]
(iii) kf − fm,δ kC[a1 ,b1 ] ≤ M2 ωm (f, δ, a, b);
(iv) kfm,δ kC[a1 ,b1 ] ≤ M3 kf kγ ,
Direct Results for Certain
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Vijay Gupta and Ogün Doğru
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where Mi , for i = 1, 2, 3 are certain unrelated constants independent of f and δ.
The r−th order modulus of continuity ωr (f, δ, a, b) for a function f continuous
on the interval [a, b] is defined by:
ωr (f, δ, a, b) = sup {|∆rh f (x)| : |h| ≤ δ; x, x + h ∈ [a, b]} .
For r = 1, ω1 (f, δ) is written simply ωf (δ) or ω(f, δ).
The following error estimation is in terms of higher order modulus of continuity:
Theorem 3.3. Let f ∈ Cγ [0, ∞), γ > 0 and 0 < a < a1 < b1 < b < ∞. Then
for all n sufficiently large
n
o
(r)
(r)
−1/2
−1
Bn (f, ∗) − f (r) (x)
≤
max
M
ω
(f
,
n
,
a,
b),
M
n
kf
k
3 2
4
γ
C[a1 ,b1 ]
where M3 = M3 (r), M4 = M4 (r, f ).
C[a1 ,b1 ]
By property (iii) of the Steklov mean, we have
A3 ≤ C1 ω2 (f (r) , δ, a, b).
Vijay Gupta and Ogün Doğru
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Proof. First by the linearity property, we have
(r)
Bn (f, ∗) − f (r) ≤ Bn(r) ((f − f2,δ ), ∗)C[a1 ,b1 ]
C[a1 ,b1 ]
(r) + Bn(r) (f2,δ , ∗) − f2,δ =: A1 + A2 + A3 .
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(r) + f (r) − f2,δ C[a1 ,b1 ]
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Next using Theorem 3.2, we have
A2 ≤ C2 n
−(k+1)
r+2 X
(j) f2,δ j=r
.
C[a,b]
By applying the interpolation property due to Goldberg and Meir [2] for each
j = r, r + 1, r + 2, we have
(j) (r+2) ≤ C3 kf2,δ kC[a,b] + f2,δ .
f2,δ C[a,b]
C[a,b]
Therefore by applying properties (ii) and (iv) of the Steklov mean, we obtain
n
o
−1
−2
(r)
kf kγ + δ ω2 (f , δ) .
A2 ≤ C4 n
∗
Direct Results for Certain
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Vijay Gupta and Ogün Doğru
∗
Finally we estimate A1 , choosing a , b satisfying the condition 0 < a <
a < a1 < b1 < b∗ < b < ∞. Also let ψ(t) denote the characteristic function
of the interval [a∗ , b∗ ], then
A1 ≤ Bn(r) (ψ(t)(f (t) − f2,δ (t)), ∗)C[a1 ,b1 ]
+ Bn(r) ((1 − ψ(t))(f (t) − f2,δ (t)), ∗)C[a1 ,b1 ]
∗
=: A4 + A5 .
We may note here that to estimate A4 and A5 , it is enough to consider their
expressions without the linear combinations. By Lemma 2.5, we have
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Bn(r) (ψ(t)(f (t) − f2,δ (t)), x)
∞
(n − r − 1)!(n + r − 1)! X
pn+r,ν (x)
=
n!(n − 1)!
ν=0
Z
0
Page 18 of 21
∞
pn−1,ν+r−1 (t)f (r) (t)dt.
J. Ineq. Pure and Appl. Math. 6(1) Art. 21, 2005
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Hence
(r)
(r)
(r) Bn (ψ(t)(f (t) − f2,δ (t)), ∗)
≤ C5 f − f2,δ C[a,b]
C[a∗ ,b∗ ]
.
Now for x ∈ [a1 , b1 ] and t ∈ [0, ∞)\[a∗ , b∗ ], we choose a δ1 > 0 satisfying
|t − x| ≥ δ1 . Therefore by Lemma 2.4 and the Schwarz inequality, we have
I = Bn(r) ((1 − ψ(t))(f (t) − f2,δ (t)), x)
X
|Qi,j,r (x)|
≤
ni
xr
2i+j≤r
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i,j≥0
∞
1X
×
pn,ν (x) |ν − (n + 1)x|j
n ν=1
Z
Vijay Gupta and Ogün Doğru
∞
pn,ν−1 (t)(1 − ψ(t)) |f (t) − f2,δ (t)| dt
0
+ (1 + x)−n−1 |(−n − 1)(−n) · · · (−n − r)|(1 − ψ(0)) |f (0) − f2,δ (0)|


∞
 X
X
i−1
≤ C6 kf kγ
n
pn,ν (x) |ν − (n + 1)x|j

2i+j≤r
ν=1
i,j≥0
Z
−n−1
×
pn,ν−1 (t)dt + (1 + x)
|(−n − 1)(−n) · · · (−n − r)|
|t−x|≥δ1


Z ∞
21
∞

X
X
j
i−1
−2s
pn,ν (x) |ν − (n + 1)x|
pn,ν−1 (t)dt
n
≤ C6 kf kγ δ1

0

ν=1
2i+j≤r
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i,j≥0
J. Ineq. Pure and Appl. Math. 6(1) Art. 21, 2005
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Z
∞
×
)
21
pn,ν−1 (t)(t − x)4s dt
+ (1 + x)−n−1 |(−n − 1)(−n) · · · (−n − r)|
0
≤ C6 kf kγ δ1−2s
( ∞
) 12
X
X
1
×
pn,ν (x) [ν − (n + 1)x]2j − (1 + x)−n−1 − {−(n + 1)x}2j
ni
n ν=0
2i+j≤r
i,j≥0
(
×
∞
1X
pn,ν (x)
n ν=0
−n−1
− (1 + x)
∞
Z
pn,ν−1 (t)(t − x)4s dt − (1 + x)−n−1 (−x)4s
0
4s
(−x)
12
+ C6 kf kγ (1 + x)−n−1 |(−n − 1)(−n) · · · (−n − r)|.
Hence by Lemma 2.1 and Lemma 2.2, we have
j
r
I ≤ C7 kf kγ δ1−2s O n(i+ 2 −s) ≤ C7 n−q kf kγ , q = s − ,
2
where the last term vanishes as n → ∞. Now choosing q satisfying q ≥ 1, we
obtain
I ≤ C7 n−1 kf kγ .
Therefore by property (iii) of the Steklov mean, we get
(r)
(r) A1 ≤ C8 f − f2,δ + C7 n−1 kf kγ
C[a∗ ,b∗ ]
≤ C9 ω2 (f
(r)
, δ, a, b) + C7 n
−1
kf kγ .
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Choosing δ = n−1/2 , the theorem follows.
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References
[1] G. FREUD AND V. POPOV, On approximation by spline functions, Proc.
Conf. on Constructive Theory Functions, Budapest (1969), 163–172.
[2] S. GOLDBERG AND V. MEIR, Minimum moduli of ordinary differential
operators, Proc. London Math. Soc., 23 (1971), 1–15.
[3] V. GUPTA AND G.S. SRIVASTAVA, Convergence of derivatives by
summation-integral type operators, Revista Colombiana de Matematicas,
29 (1995), 1–11.
[4] E. HEWITT AND K. STROMBERG, Real and Abstract Analysis, McGraw
Hill, New York, 1956.
[5] H.M. SRIVASTAVA AND V. GUPTA, A certain family of summation integral type operators, Mathematical and Computer Modelling, 37 (2003),
1307–1315.
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Vijay Gupta and Ogün Doğru
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