Journal of Inequalities in Pure and
Applied Mathematics
ON A HYBRID FAMILY OF SUMMATION INTEGRAL
TYPE OPERATORS
volume 7, issue 1, article 23,
2006.
VIJAY GUPTA AND ESRA ERKUŞ
School of Applied Sciences
Netaji Subhas Institute of Technology
Sector-3, Dwarka
New Delhi - 110075, India.
EMail: vijaygupta2001@hotmail.com
Çanakkale Onsekiz Mart University
Faculty of Sciences and Arts
Department of Mathematics
Terzioğlu Kampüsü
17020, Çanakkale, TURKEY.
EMail: erkus@comu.edu.tr
Received 21 November, 2005;
accepted 01 December, 2005.
Communicated by: A. Lupaş
Abstract
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c 2000 Victoria University
ISSN (electronic): 1443-5756
343-05
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Abstract
The present paper deals with the study of the mixed summation integral type
operators having Szász and Baskakov basis functions in summation and integration respectively. Here we obtain the rate of point wise convergence, a
Voronovskaja type asymptotic formula, an error estimate in simultaneous approximation. We also study some local direct results in terms of modulus of
smoothness and modulus of continuity in ordinary and simultaneous approximation.
2000 Mathematics Subject Classification: 41A25; 41A30.
Key words: Linear positive operators, Summation-integral type operators, Rate of
convergence, Asymptotic formula, Error estimate, Local direct results,
K-functional, Modulus of smoothness, Simultaneous approximation.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3
Direct Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4
Local Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
References
On a Hybrid Family of
Summation Integral Type
Operators
Vijay Gupta and Esra Erkuş
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1.
Introduction
The mixed summation-integral type operators discussed in this paper are defined as
Z ∞
Sn (f, x) =
(1.1)
Wn (x, t)f (t)dt
0
= (n − 1)
∞
X
Z
sn,ν (x)
ν=1
−nx
+e
∞
bn,ν−1 (t)f (t)dt
0
f (0),
x ∈ [0, ∞),
On a Hybrid Family of
Summation Integral Type
Operators
Vijay Gupta and Esra Erkuş
where
Wn (x, t) = (n − 1)
∞
X
sn,ν (x)bn,ν−1 (t) + e−nx δ(t),
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ν=1
δ(t) being Dirac delta function,
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sn,ν (x) = e−nx
and
bn,ν (t) =
(nx)ν
ν!
n+ν−1 ν
t (1 + t)−n−ν
ν
are respectively Szász and Baskakov basis functions. It is easily verified that the
operators (1.1) are linear positive operators, these operators were recently proposed by Gupta and Gupta in [3]. The behavior of these operators is very similar to the operators studied by Gupta and Srivastava [5], but the approximation
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properties of the operators Sn are different in comparison to the operators studied in [5]. The main difference is that the operators (1.1) are discretely defined
at the point zero. Recently Srivastava and Gupta [8] proposed a general family of summation-integral type operators Gn,c (f, x) which include some well
known operators (see e.g. [4], [7]) as special cases. The rate of convergence
for bounded variation functions was estimated in [8], Ispir and Yuksel [6] considered the Bézier variant of the operators Gn,c (f, x) and studied the rate of
convergence for bounded variation functions. We also note here that the results
analogous to [6] and [8] cannot be obtained for the mixed operators Sn (f, x)
because it is not easier to write the integration of Baskakov basis functions in
the summation form of Szász basis functions, which is necessary in the analysis
for obtaining the rate of convergence at the point of discontinuity. We propose
this as an open problem for the readers.
In the present paper we study some direct results, for the class of unbounded
functions with growth of order tγ , γ > 0, for the operators Sn we obtain a
point wise rate of convergence, asymptotic formula of Voronovskaja type, and
an error estimate in simultaneous approximation. We also estimate local direct
results in terms of modulus of smoothness and modulus of continuity in ordinary
and simultaneous approximation.
On a Hybrid Family of
Summation Integral Type
Operators
Vijay Gupta and Esra Erkuş
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2.
Auxiliary Results
We will subsequently need the following lemmas:
Lemma 2.1. For m ∈ N0 = N ∪ {0}, if the m-th order moment is defined as
Un,m (x) =
∞
X
sn,ν (x)
ν=0
ν
n
−x
m
,
then Un,0 (x) = 1, Un,1 (x) = 0 and
(1)
nUn,m+1 (x) = x Un,m
(x) + mUn,m−1 (x) .
On a Hybrid Family of
Summation Integral Type
Operators
Vijay Gupta and Esra Erkuş
Consequently
Un,m (x) = O n
−[(m+1)/2]
.
Lemma 2.2. Let the function µn,m (x), m ∈ N0 , be defined as
Z ∞
∞
X
µn,m (x) = (n − 1)
sn,ν (x)
bn,ν−1 (t)(t − x)m dt + (−x)m e−nx .
ν=1
0
Then
µn,0 (x) = 1,
2x
,
µn,1 (x) =
n−2
nx(x + 2) + 6x2
µn,2 (x) =
,
(n − 2)(n − 3)
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also we have the recurrence relation:
(n − m − 2)µn,m+1 (x) = x µ(1)
n,m (x) + m(x + 2)µn,m−1 (x)
+ [m + 2x(m + 1)] µn,m (x);
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n > m + 2.
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Consequently for each x ∈ [0, ∞) we have from this recurrence relation that
µn,m (x) = O n−[(m+1)/2] .
Remark 1. It is easily verified from Lemma 2.2 that for each x ∈ (0, ∞)
(2.1) Sn (ti , x) =
(n − i − 2)!
(n − i − 2)!
(nx)i + i(i − 1)
(nx)i−1 + O(n−2 ).
(n − 2)!
(n − 2)!
Lemma 2.3. [5]. There exist the polynomials Qi,j,r (x) independent of n and ν
such that
X
xr Dr [sn,ν (x)] =
ni (ν − nx)j Qi,j,r (x)sn,ν (x),
On a Hybrid Family of
Summation Integral Type
Operators
Vijay Gupta and Esra Erkuş
2i+j≤r
i,j≥0
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where D ≡
d
.
dx
Lemma 2.4. Let n > r ≥ 1 and f
Section 3). Then
Contents
(i)
∈ CB [0, ∞) for i ∈ {0, 1, 2, . . . , r} (cf.
∞
Sn(r) (f, x)
X
nr
sn,ν (x)
=
(n − 2) · · · (n − r − 1) ν=0
Z
0
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∞
bn−r,ν+r−1 (t)f (r) (t)dt.
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3.
Direct Results
In this section we consider the class Cγ [0, ∞) of continuous unbounded functions, defined as
f ∈ Cγ [0, ∞) ≡ {f ∈ C[0, ∞) : |f (t)| ≤ M tγ , for some M > 0, γ > 0} .
We prove the following direct estimates:
Theorem 3.1. Let f ∈ Cγ [0, ∞), γ > 0 and f (r) exists at a point x ∈ (0, ∞),
then
lim Sn(r) (f (t), x) = f (r) (x).
(3.1)
n→∞
On a Hybrid Family of
Summation Integral Type
Operators
Vijay Gupta and Esra Erkuş
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 ,
where ε(t, x) → 0 as t → x. Thus, using the above, we have
Z ∞
(r)
Sn (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 7 of 23
0
= R1 + R2 ,
say.
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First to estimate R1 , using a binomial expansion of (t − x)m and applying
(2.1), we have
R1 =
r
i X
f (i) (x) X i
i=0
(r)
i!
ν=0
ν
i−ν
(−x)
∂r
∂xr
∞
Z
Wn (t, x)tν dt
0
(x) d
(n − r − 2)!nr r
=
x + terms containing lower powers of x
r! dxr
(n − 2)!
(n − r − 2)!nr
(r)
= f (x)
→ f (r) (x) as n → ∞.
(n − 2)!
f
r
Next using Lemma 2.3, we obtain
|R2 | ≤ (n − 1)
X
2i+j≤r
i,j≥0
∞
X
×
ν=1
ni
On a Hybrid Family of
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Operators
Vijay Gupta and Esra Erkuş
|Qi,j,r (x)|
xr
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|ν − nx|j sn,ν (x)
Z
∞
bn,ν−1 (t) |ε(t, x)| (t − x)r dt
0
+ (−n)r e−nx |ε(0, x)| (−x)r
= R3 + R4 , say.
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Since ε(t, x) → 0 as t → x for a given ε > 0 there exists a δ > 0 such that
|ε(t, x)| < ε whenever 0 < |t − x| < δ. Further if s ≥ max {γ, r}, where s is
any integer, then we can find a constant M1 > 0 such that |ε(t, x)(t − x)r | ≤
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M1 |t − x|s , for |t − x| ≥ δ. Thus
∞
X
X
R3 ≤ M2 (n − 1)
ni
sn,ν (x)
2i+j≤r
ν=1
i,j≥0
Z
j
× |ν − nx| ε
bn,ν−1 (t) |t − x|r dt
|t−x|<δ
Z
s
+
bn,ν−1 (t)M1 |t − x| dt
|t−x|≥δ
= R5 + R6 ,
say.
Applying the Schwarz inequality for integration and summation respectively,
and using Lemma 2.1 and Lemma 2.2, we obtain
∞
X
X
i
R5 ≤ εM2 (n − 1)
n
sn,ν (x)
ν=1
2i+j≤r
i,j≥0
× |ν − nx|j
12 Z
∞
bn,ν−1 (t)dt
Z
0
≤ εM2
X
2i+j≤r
ni
∞
X
12
∞
2r
bn,ν−1 (t)(t − x) dt
0
On a Hybrid Family of
Summation Integral Type
Operators
Vijay Gupta and Esra Erkuş
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sn,ν (x)(ν − nx)2j
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ν=1
i,j≥0
×
(n − 1)
∞
X
ν=1
Z
! 21
∞
bn,ν−1 (t)(t − x) dt
sn,ν (x)
0
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2r
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X
≤ εM2
ni O nj/2 O n−r/2 = εO(1).
2i+j≤r
i,j≥0
Again using the Schwarz inequality, Lemma 2.1 and Lemma 2.2, we get
X
R6 ≤ M3 (n − 1)
n
i
∞
X
j
sn,ν (x) |ν − nx|
bn,ν−1 (t) |t − x|s dt
|t−x|≥δ
ν=1
2i+j≤r
Z
i,j≥0
X
≤ M3
2i+j≤r
ni
∞
X
! 12
On a Hybrid Family of
Summation Integral Type
Operators
sn,ν (x)(ν − nx)2j
ν=1
Vijay Gupta and Esra Erkuş
i,j≥0
×
(n − 1)
∞
X
Z
sn,ν (x)
ν=1
=
X
∞
! 21
bn,ν−1 (t)(t − x)2s dt
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0
ni O nj/2 O n−s/2
2i+j≤r
i,j≥0
= O n(r−s)/2 = o(1).
Thus due to the arbitrariness of ε > 0 it follows that R3 = o(1). Also R4 → 0
as n → ∞ and therefore R2 = o(1). Collecting the estimates of R1 and R2 , we
get (3.1).
Theorem 3.2. Let f ∈ Cγ [0, ∞), γ > 0. If f (r+2) exists at a point x ∈ (0, ∞),
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then
lim n Sn(r) (f, x) − f (r) (x)
n→∞
r(r + 3) (r)
x
f (x) + [x(2 + r) + r] f (r+1) (x) + (2 + x)f (r+2) (x).
2
2
Proof. By Taylor’s expansion of f , we have
=
f (t) =
r+2 (i)
X
f (x)
i!
i=0
(t − x)i + ε(t, x)(t − x)r+2
where ε(t, x) → 0 as t → x. Applying Lemma 2.2 and the above Taylor’s
expansion, we have
" r+2
#
X f (i) (x) Z ∞
(r)
n Sn (f (t), x) − f (r) (x) = n
Wn(r) (t, x)(t − x)i dt − f (r) (x)
i!
0
i=0
Z ∞
(r)
r+2
+ n
Wn (t, x)ε(t, x)(t − x) dt
0
= E1 + E2 , say.
E1 = n
r+2 (i)
i X
f (x) X i
i=0
=
i!
j=0
j
i−j
Z
(−x)
On a Hybrid Family of
Summation Integral Type
Operators
Vijay Gupta and Esra Erkuş
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∞
Wn(r) (t, x)tj dt − nf (r) (x)
0
f (r) (x) (r) r
n Sn (t , x) − r!
r!
f (r+1) (x) n (r + 1)(−x)Sn(r) (tr , x) + Sn(r) (tr+1 , x)
+
(r + 1)!
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f (r+2) (x) (r + 2)(r + 1) 2 (r) r
+
n
x Sn (t , x)
(r + 2)!
2
(r) r+1
(r) r+2
+ (r + 2)(−x)Sn (t , x) + Sn (t , x) .
Therefore, using (2.1) we have
r
n (n − r − 2)!
(r)
E1 = nf (x)
−1
(n − 2)!
r
n (n − r − 2)!
f (r+1) (x)
+n
(r + 1)(−x)r!
(r + 1)!
(n − 2)!
r+1
n (n − r − 3)!
nr (n − r − 3)!
+
(r + 1)!x + r(r + 1)
r!
(n − 2)!
(n − 2)!
f (r+2) (x) (r + 2)(r + 1)x2
nr (n − r − 2)!
+
(r!)
(r + 2)!
2
(n − 2)!
r+1
n (n − r − 3)!
nr (n − r − 3)!
+ (r + 2)(−x)
(r + 1)!x + r(r + 1)
r!
(n − 2)!
(n − 2)!
r+2
n (n − r − 4)! (r + 2)! 2
+
x
(n − 2)!
2
nr+1 (n − r − 4)!
−2
+(r + 1)(r + 2)
(r + 1)!x + O n
.
(n − 2)!
In order to complete the proof of the theorem it is sufficient to show that E2 → 0
as n → ∞, which can easily be proved along the lines of the proof of Theorem
3.1 and by using Lemma 2.1, Lemma 2.2 and Lemma 2.3.
On a Hybrid Family of
Summation Integral Type
Operators
Vijay Gupta and Esra Erkuş
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Theorem 3.3. Let f ∈ Cγ [0, ∞), γ > 0 and r ≤ m ≤ r + 2. If f (m) exists and
is continuous on (a − η, b + η) ⊂ (0, ∞), η > 0, then for n sufficiently large
Sn(r) (f, x)
−f
(r)
≤ M4 n
−1
m
X
f (i) +M5 n−1/2 w f (r+1) , n−1/2 +O n−2 ,
i=1
where the constants M4 and M5 are independent of f and n, w(f, δ) is the
modulus of continuity of f on (a − η, b + η) and k·k denotes the sup-norm on
the interval [a, b] .
Proof. By Taylor’s expansion of f , we have
f (t) =
m
X
On a Hybrid Family of
Summation Integral Type
Operators
Vijay Gupta and Esra Erkuş
if
(t − x)
i=0
(x)
f m (ξ) − f m (x)
+ (t − x)m ζ(t)
+ h(t, x) (1 − ζ(t)) ,
i!
m!
(i)
where ζ lies between t and x and ζ(t) is the characteristic function on the interval (a − η, b + η). For t ∈ (a − η, b + η), x ∈ [a, b] , we have
f (t) =
m
X
i=0
(t − x)i
f (i) (x)
f m (ξ) − f m (x)
+ (t − x)m
.
i!
m!
h(t, x) = f (t) −
i=0
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For t ∈ [0, ∞) \ (a − η, b + η), x ∈ [a, b] , we define
m
X
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if
(t − x)
(i)
(x)
.
i!
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Thus
Sn(r) (f, x) − f (r) (x) =
( m
X f (i) (x) Z
i!
i=0
∞
)
∞
Wn(r) (t, x)(t − x)i dt − f (r) (x)
0
(ξ) − f m (x)
m
+
(t − x) ζ(t)dt
m!
0
Z ∞
(r)
+
Wn (t, x)h(t, x)(1 − ζ(t))dt
Z
f
Wn(r) (t, x)
m
0
= ∆1 + ∆2 + ∆3 ,
say.
Using (3.1), we obtain
i m
X
f (i) (x) X i
r Z ∞
i−j ∂
∆1 =
(−x)
Wn (t, x)tj dt − f (r) (x)
r
i!
j
∂x
0
i=0
j=0
m
i
X
f (i) (x) X i
∂ r (n − j − 2)!
=
(−x)i−j r
(nx)j
i!
j
∂x
(n
−
2)!
i=0
j=0
(n − j − 2)!
j−1
−2
(nx) + O n
− f (r) (x).
+j(j − 1)
(n − 2)!
On a Hybrid Family of
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Operators
Vijay Gupta and Esra Erkuş
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Hence
k∆1 k ≤ M4 n−1
m
X
f (i) + O n−2 ,
i=r
uniformly in x ∈ [a, b]. Next
Z ∞
|f m (ξ) − f m (x)|
|∆2 | ≤
Wn(r) (t, x)
|t − x|m ζ(t)dt
m!
0
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w f (m) , δ
≤
m!
∞
Z
Wn(r) (t, x)
0
|t − x|
1+
δ
|t − x|m dt.
Next, we shall show that for q = 0, 1, 2, ...
(n − 1)
∞
X
j
Z
sn,ν (x) |ν − nx|
∞
bn,ν−1 (t) |t − x|q dt = O n(j−q)/2 .
0
ν=1
Now by using Lemma 2.1 and Lemma 2.2, we have
(n − 1)
∞
X
j
Z
sn,ν (x) |ν − nx|
≤
bn,ν−1 (t) |t − x|q dt
!
1
2
2j
(n − 1)
sn,ν (x)(ν − nx)
ν=1
j/2
=O n
∞
Vijay Gupta and Esra Erkuş
0
ν=1
∞
X
On a Hybrid Family of
Summation Integral Type
Operators
∞
X
!
∞
Z
2q
bn,ν−1 (t) (t − x) dt
sn,ν (x)
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O n−q/2 = O n(j−q)/2 ,
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uniformly in x. Thus by Lemma 2.3, we obtain
(n − 1)
s(r)
n,ν (x)
Z
∞
"
≤ M6
X
ni (n − 1)
2i+j≤r
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bn,ν−1 (t) |t − x|q dt
0
ν=1
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0
ν=1
∞
X
1
2
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∞
X
ν=1
sn,ν (x) |ν − nx|j
Z
0
∞
#
bn,ν−1 (t) |t − x|q dt
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i,j≥0
= O n(r−q)/2 ,
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uniformly in x, where M6 = sup sup |Qi,j,r (x)| x−r . Choosing δ = n−1/2 ,
2i+j≤r
i,j≥0
x∈[a,b]
we get for any s > 0,
w f (m) , n−1/2 k∆2 k ≤
O(n(r−m)/2 ) + n1/2 O n(r−m−1)/2 + O n−s
m!
≤ M5 w f (m) , n−1/2 n−(m−r)/2 .
Since t ∈ [0, ∞) \ (a − η, b + η) , we can choose a δ > 0 in such a way that
|t − x| ≥ δ for all x ∈ [a, b] . Applying Lemma 2.3, we obtain
k∆3 k ≤ (n − 1)
∞ X
X
ν=1
ni |ν − nx|j
2i+j≤r
On a Hybrid Family of
Summation Integral Type
Operators
Vijay Gupta and Esra Erkuş
|Qi,j,r (x)|
sn,ν (x)
xr
i,j≥0
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Z
×
r −nx
bn,ν−1 (t) |h(t, x)| dt + n e
|h(0, x)| .
|t−x|≥δ
If β is any integer greater than or equal to {γ, m}, then we can find a constant
M7 such that |h(t, x)| ≤ M7 |t − x|β for |t − x| ≥ δ. Now applying Lemma 2.1
and Lemma 2.2, it is easily verified that ∆3 = O (n−q ) for any q > 0 uniformly
on [a, b]. Combining the estimates ∆1 − ∆3 , we get the required result.
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4.
Local Approximation
In this section we establish direct local approximation theorems for the operators (1.1). Let CB [0, ∞) be the space of all real valued continuous bounded
functions f on [0, ∞) endowed with the norm kf k = sup |f (x)|. The Kx≥0
functionals are defined as
2
K(f, δ) = inf kf − gk + δ kg 00 k : g ∈ W∞
,
2
where W∞
= {g ∈ CB [0, ∞) : g 0 , g 00 ∈ CB [0, ∞)}.By [1,pp 177, Th. 2.4],
∼
∼
√
there exists a constant M such that K(f, δ) ≤ M w2 f, δ , where δ > 0 and
the second order modulus of smoothness is defined as
√ w2 f, δ = sup√ sup |f (x + 2h) − 2f (x + h) + f (x)| ,
0<h≤ δ x∈[0,∞)
where f ∈ CB [0, ∞). Furthermore, let
w (f, δ) = sup
sup |f (x + h) − f (x)|
0<h≤δ x∈[0,∞)
be the usual modulus of continuity of f ∈ CB [0, ∞).
Our first theorem in this section is in ordinary approximation which involves
second order and ordinary moduli of smoothness:
Theorem 4.1. Let f ∈ CB [0, ∞). Then there exists an absolute constant M8 >
0 such that
!
r
x(1 + x)
x
|Sn (f, x) − f (x)| ≤ M8 w2 f,
+ w f,
,
n−2
n−2
On a Hybrid Family of
Summation Integral Type
Operators
Vijay Gupta and Esra Erkuş
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for every x ∈ [0, ∞) and n = 3, 4, ... .
∧
Proof. We define a new operator S n : CB [0, ∞) → CB [0, ∞) as follows
∧
nx
(4.1)
S n (f, x) = Sn (f, x) − f (x) + f
.
n−2
∧
Then by Lemma 2.2, we obtain S n (t − x, x) = 0. Now, let x ∈ [0, ∞) and
2
g ∈ W∞
. From Taylor’s formula
Z t
0
g(t) = g(x) + g (x)(t − x) +
(t − u)g 00 (u)du, t ∈ [0, ∞)
On a Hybrid Family of
Summation Integral Type
Operators
Vijay Gupta and Esra Erkuş
x
we get
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∧
(4.2)
∧
Z
t
(t − u)g 00 (u)du, x
Z xt
00
= Sn
(t − u)g (u)du, x
x
Z nx/(n−2) n
+
x − u g 00 (u)du.
n
−
2
x
S n (g, x) − g(x) = S n
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On the other hand,
Z
(4.3)
x
Page 18 of 23
t
(t − u)g 00 (u)du ≤ (t − x)2 kg 00 k
J. Ineq. Pure and Appl. Math. 7(1) Art. 23, 2006
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and
Z
nx/(n−2)
(4.4)
x
2
nx
n
00
x − u g (u)du ≤
− x kg 00 k
n−2
n−2
4x2
00
2 kg k
(n − 2)
4x(1 + x) 00
≤
kg k .
(n − 2)2
≤
Thus by (4.2), (4.3), (4.4) and by the positivity of Sn , we obtain
∧
4x(1 + x) 00
kg k .
S n (g, x) − g(x) ≤ Sn (t − x)2 , x kg 00 k +
(n − 2)2
Hence in view of Lemma 2.2, we have
∧
2nx + (n + 6)x2 4x(1 + x)
+
(4.5)
S n (g, x) − g(x) ≤
kg 00 k
(n − 2)(n − 3)
(n − 2)2
n
1
4x(1 + x) 00
≤
+
kg k
n−3 n−2
n−2
18
≤
x(1 + x) kg 00 k .
n−2
|Sn (f, x)| ≤ (n − 1)
ν=1
Vijay Gupta and Esra Erkuş
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Again applying Lemma 2.2
∞
X
On a Hybrid Family of
Summation Integral Type
Operators
Z
Page 19 of 23
∞
−nx
bn,ν−1 (t) |f (t)| dt + e
sn,ν (x)
0
|f (0)| ≤ kf k .
J. Ineq. Pure and Appl. Math. 7(1) Art. 23, 2006
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This means that Sn is a contraction, i.e. kSn f k ≤ kf k , f ∈ CB [0, ∞). Thus
by (4.2)
∧
(4.6)
Sn f ≤ kSn f k + 2 kf k ≤ 3 kf k ,
f ∈ CB [0, ∞).
Using (4.1), (4.5) and (4.6), we obtain
∧
|Sn (f, x) − f (x)| ≤ Sn (f − g, x) − (f − g)(x)
∧
nx
n−2
18
nx
00
≤ 4 kf − gk +
x(1 + x) kg k + f (x) − f
n−2
n−2
x(1 + x) 00
x
≤ 18 kf − gk +
kg k + w f,
.
n−2
n−2
+ Sn (g, x) − g(x) + f (x) − f
2
and using (4.1)
Now taking the infimum on the right hand side over all g ∈ W∞
we arrive at the assertion of the theorem.
The following error estimation is in terms of ordinary modulus of continuity
in simultaneous approximation:
(i)
Theorem 4.2. Let n > r + 3 ≥ 4 and f ∈ CB [0, ∞) for i ∈ {0, 1, 2, ..., r}.
Then
r
n (n − r − 2)!
nr (n − r − 2)!
(r)
(r)
(r)
Sn (f, x) − f (x) ≤
−1 f
+
(n − 2)!
(n − 2)!
On a Hybrid Family of
Summation Integral Type
Operators
Vijay Gupta and Esra Erkuş
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r
×
1+
[n + (r + 1)(r + 2)] x2 + 2 [n + r(r + 2)] x + r(r + 1)
n−r−3
!
× w f (r) , (n − r − 2)−1/2
where x ∈ [0, ∞).
Proof. Using Lemma 2.4 and taking into account the well known property
w f (r) , λδ ≤ (1 + λ)w f (r) , δ , λ ≥ 0,
we obtain
On a Hybrid Family of
Summation Integral Type
Operators
Vijay Gupta and Esra Erkuş
(4.7)
Sn(r) (f, x)
−f
(r)
(x)
Z ∞
∞
n (n − r − 1)! X
≤
sn,ν (x)
bn−r,ν+r−1 (t) f (r) (t) − f (r) (x) dt
(n − 2)!
0
r ν=0
n (n − r − 2)!
+
− 1 f (r) (x)
(n − 2)!
∞
r
n (n − r − 1)! X
≤
sn,ν (x)
(n − 2)!
Z ∞ ν=0
×
bn−r,ν+r−1 (t) 1 + δ −1 |t − x| w f (r) , δ dt
0 r
n (n − r − 1)!
+
− 1 f (r) .
(n − 2)!
r
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Further, using Cauchy’s inequality, we have
(4.8) (n − r − 1)
∞
X
Z
bn−r,ν+r−1 (t) |t − x| dt
sn,ν (x)
0
ν=0
(
≤
∞
(n − r − 1)
∞
X
ν=0
Z
sn,ν (x)
) 12
∞
bn−r,ν+r−1 (t) (t − x)2 dt
.
0
By direct computation
(4.9) (n − r − 1)
∞
X
ν=0
Z
sn,ν (x)
On a Hybrid Family of
Summation Integral Type
Operators
∞
bn−r,ν+r−1 (t) (t − x)2 dt
2n + 2r(r + 2)
n + (r + 1)(r + 2)
x2 +
x
(n − r − 3)(n − r − 2)
(n − r − 3)(n − r − 2)
r(r + 1)
.
+
(n − r − 3)(n − r − 2)
√
Thus by combining (4.7), (4.8) and (4.9) and choosing δ −1 = n − r − 2, we
obtain the desired result.
=
Vijay Gupta and Esra Erkuş
0
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References
[1] R.A. DEVORE AND G.G. LORENTZ, Constructive Approximation,
Springer-Verlag, Berlin Heidelberg, New York, 1993.
[2] Z. FINTA AND V. GUPTA, Direct and inverse estimates for Phillips type
operators, J. Math Anal Appl., 303(2) (2005), 627–642.
[3] V. GUPTA AND M.K. GUPTA, Rate of convergence for certain families of
summation-integral type operators, J. Math. Anal. Appl., 296 (2004), 608–
618.
[4] V. GUPTA, M.K. GUPTA AND V. VASISHTHA, Simultaneous approximation by summation integral type operators, J. Nonlinear Functional Analysis
and Applications, 8(3) (2003), 399–412.
[5] V. GUPTA AND G.S. SRIVASTAVA, On convergence of derivatives by
Szász-Mirakyan-Baskakov type operators, The Math Student, 64 (1-4)
(1995), 195–205.
[6] N. ISPIR AND I. YUKSEL, On the Bezier variant of Srivastava-Gupta operators, Applied Mathematics E Notes, 5 (2005), 129–137.
[7] C.P. MAY, On Phillips operators, J. Approx. Theory, 20 (1977), 315–322.
[8] H.M. SRIVASTAVA AND V. GUPTA, A certain family of summation integral type operators, Math. Comput. Modelling, 37 (2003), 1307–1315.
On a Hybrid Family of
Summation Integral Type
Operators
Vijay Gupta and Esra Erkuş
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