Journal of Inequalities in Pure and
Applied Mathematics
SIMULTANEOUS APPROXIMATION BY LUPAŞ MODIFIED
OPERATORS WITH WEIGHTED FUNCTION OF
SZASZ OPERATORS
volume 5, issue 4, article 113,
2004.
NAOKANT DEO
Department of Applied Mathematics
Delhi College of Engineering
Bawana Road, Delhi - 110042, India.
Received 20 August, 2004;
accepted 01 September, 2004.
Communicated by: A. Lupaş
EMail: dr_naokant_deo@yahoo.com
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
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Abstract
In the present paper, we consider a new modification of the Lupaş operators
with the weight function of Szasz operators and study simultaneous approximation. Here we obtain a Voronovskaja type asymptotic formula and an estimate
of error in simultaneous approximation for these Lupaş-Szasz operators.
Simultaneous Approximation by
Lupaş Modified Operators with
Weighted Function of Szasz
Operators
2000 Mathematics Subject Classification: 41A28, 41A36.
Key words: Simultaneous approximation, Lupaş operators, Szasz operators.
Contents
Naokant Deo
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Basic Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
3
5
9
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1.
Introduction
Lupaş proposed a family of linear positive operators mapping C[0, ∞) into
C[0, ∞), the class of all bounded and continuous functions on [0, ∞), namely,
(Ln f ) (x) =
∞ X
n+k−1
k
k=0
xk
k
f
,
n+k
n
(1 + x)
where x ∈ [0, ∞).
Motivated by the integral modification of Bernstein polynomials by Derriennic [1], Sahai and Prasad [3] modified the operators Ln for functions integrable
on C[0, ∞) as
Simultaneous Approximation by
Lupaş Modified Operators with
Weighted Function of Szasz
Operators
Naokant Deo
(Mn f ) (x) = (n − 1)
∞
X
∞
Z
Pn,k (x)
Pn,k (y)f (y)dy,
0
k=0
Contents
where
Pn,k (t) =
n+k−1
k
tk
n+k
(1 + t)
.
Integral modification of Szasz-Mirakyan operators were studied by Gupta [2].
Now we consider another modification of Lupaş operators with the weight function of Szasz operators, which are defined as
(1.1)
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(Bn f ) (x) = n
∞
X
k=0
Z
Pn,k (x)
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∞
Sn,k (y)f (y)dy
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J. Ineq. Pure and Appl. Math. 5(4) Art. 113, 2004
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where
Pn,k (x) =
n+k−1
k
and
Sn,k (y) =
xk (1 + x)−n−k
e−ny (ny)k
.
k!
Simultaneous Approximation by
Lupaş Modified Operators with
Weighted Function of Szasz
Operators
Naokant Deo
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2.
Basic Results
The following lemmas are useful for proving the main results.
Lemma 2.1. Let m ∈ N 0 , n ∈ N, if we define
Tn,m (x) = n
∞
X
Z
∞
Sn,k+r (y)(y − x)m dy
Pn+r,k (x)
0
k=0
then
(i) Tn,0 (x) = 1, Tn,1 (x) =
(2.1)
1+r(1+x)
,
n
Simultaneous Approximation by
Lupaş Modified Operators with
Weighted Function of Szasz
Operators
and
rx(1 + x) + 1 + [1 + r(1 + x)]2 + nx(2 + x)
Tn,2 (x) =
n2
Naokant Deo
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(ii) For all x ≥ 0,
Tn,m (x) = O
n[
m+1
2
Contents
1
]
.
(iii)
(1)
nTn,m+1 (x) = x(1+x)Tn,m+1 (x)+[m+1+r(1+x)]Tn,m (x)+mxTn,m−1 (x)
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Proof. The value of Tn,0 (x), Tn,1 (x) easily follows from the definition, we give
the proof of (iii) as follows.
(1)
(x)
x(x + 1)Tn,m
Z
∞
X
(1)
=n
x(1 + x)Pn+r,k (x)
∞
Sn,k+r (y)(y − x)m dy
0
k=0
− mn
∞
X
Z
x(1 + x)Pn+r,k (x)
∞
Sn,k+r (y)(y − x)m−1 dy.
0
k=0
Now using the identities
(1)
ySn,k (y) = (k − ny)Sn,k (y),
Simultaneous Approximation by
Lupaş Modified Operators with
Weighted Function of Szasz
Operators
Naokant Deo
(1)
and x(1 + x)Pn,k (x) = (k − nx)Pn,k (x), we get
(1)
(x)
x(1 + x)Tn,m
Z
∞
X
=n
[k − (n + r)x]Pn+r,k (x)
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∞
Sn,k+r (y)(y − x)m dy
0
k=0
− mx(1 + x)Tn,m−1 (x).
Therefore,
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(1)
x(1 + x)[Tn,m
(x) + mTn,m−1 (x)]
Z ∞
∞
X
Pn+r,k (x)
[(k + r − ny) + n(y − x) − r(1 + x)]Sn,k+r (y)(y − x)m dy
=n
k=0
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Z
∞
X
= n Pn+r,k (x)
=n
k=0
∞
X
∞
(1)
ySn,k+r (y)(y − x)m dy + nTn,m+1 (x) − r(1 + x)Tn,m (x)
0
Z
∞
Pn+r,k (x)
(1)
ySn,k+r (y)(y − x)m+1 dy
0
k=0
+ nx
∞
X
Z
Pn+r,k (x)
k=0
∞
(1)
Sn,k+r (y)(y − x)m dy
0
+ nTn,m+1 (x) − r(1 + x)Tn,m (x)
= −(m + 1)Tn,m (x) − mxTn,m−1 (x) + nTn,m+1 (x) − r(1 + x)Tn,m (x)
This leads to proof of (iii).
Naokant Deo
Corollary 2.2. Let α and δ be positive numbers, then for every m ∈ N and
x ∈ [0, ∞), there exists a positive constant Cm,x depending on m and x such
that
Z
∞
X
Pn,k (x)
Sn,k (t)eαt dt ≤ Cm,x n−m .
n
k=0
Simultaneous Approximation by
Lupaş Modified Operators with
Weighted Function of Szasz
Operators
|t−x|≥δ
Lemma 2.3. If f is differentiable r times (r = 1, 2, 3, . . . ) on [0, ∞), then we
have
Z ∞
∞
(n + r − 1)! X
(r)
(2.2) (Bn f )(x) = r−1
Pn+r,k (x)
Sn,k+r (y)f (r) (y)dy.
n (n − 1)! k=0
0
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Proof. Applying Leibniz’s theorem in (1.1) we have
(Bn(r) f )(x)
r X
∞ X
r (n + k + r − i − 1)!
=n
(−1)r−i xk−i (1 + x)−n−k−r+i
i
(n − 1)!k!
i=0 k=i
Z ∞
×
Sn,k (y)f (y)dy
0
∞
X
(n + k + r − 1)!
Z ∞X
r
xk
r−i r
=n
·
(−1)
Sn,k+i (y)f (y)dy
n+k+r
(n
−
1)!k!
(1
+
x)
i
0
i=0
k=0
Z ∞X
∞
r
r
X
(n + r − 1)!
Pn+r,k (x)
(−1)r−i
Sn,k+i (y)f (y)dy.
=n
(n − 1)! k=0
i
0
i=0
Again using Leibniz’s theorem,
(r)
Sn,k+r (y) =
i=0
i
r
X
i=0
−ny
(−1)i nr
(−1)i
e
r
i
k+i
(ny)
(k + i)!
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Sn,k+i (y).
∞
(n + r − 1)! X
= r−1
Pn+r,k (x)
n (n − 1)! k=0
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Hence
(Bn(r) f )(x)
Naokant Deo
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r X
r
= nr
Simultaneous Approximation by
Lupaş Modified Operators with
Weighted Function of Szasz
Operators
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(r)
r
Sn,k+r (y)(−1) f (y)dy
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and integrating by parts r times, we get the required result.
J. Ineq. Pure and Appl. Math. 5(4) Art. 113, 2004
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3.
Main Results
Theorem 3.1. Let f be integrable in [0, ∞), admitting a derivative of order
(r + 2) at a point x ∈ [0, ∞). Also suppose f (r) (x) = o(eαx ) as x → ∞, then
lim n[(Bn(r) f )(x) − f (r) (x)] = [1 + r(1 + x)]f (r+1) (x) + x(2 + x)f (r+2) (x).
n→∞
Proof. By Taylor’s formula, we get
(r)
(3.1) f (y) − f (r) (x)
= (y − x)f (r+1) (x) +
(y − x)2 (r+2)
(y − x)2
f
(x) +
η(y, x),
2
2
Simultaneous Approximation by
Lupaş Modified Operators with
Weighted Function of Szasz
Operators
Naokant Deo
where
(r)
η(y, x) =
f (y) − f (r) (x) − (y − x)f (r+1) (x) −
= 0 if
(y−x)2 (r+2)
f
(x)
2
(y−x)2
2
x = y.
Now, for arbitrary ε > 0, A > 0∃ aδ > 0 s. t.
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if
x 6= y
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(3.2)
|η(y, x)| ≤ ε
for
|y − x| < δ, x ≤ A.
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Using (2.2) in (3.1)
nr (n − 1)!
(B (r) f )(x) − f (r) (x)
(n + r − 1)! n
Z ∞
∞
X
=n
Pn+r,k (x)
Sn,k+r (y)f (r) (y)dy − f (r) (x)
=n
k=0
∞
X
0
Z
∞
Sn,k+r (y) f (r) (y) − f (r) (x) dy
Pn+r,k (x)
0
k=0
= Tn,1 f (r+1) (x) + Tn,2 f (r+2) (x) + En,r (x),
where
Simultaneous Approximation by
Lupaş Modified Operators with
Weighted Function of Szasz
Operators
Naokant Deo
∞
En,r (x) =
nX
Pn+r,k (x)
2 k=0
Z
∞
Sn,k+r (y)(y − x)2 η(y, x)dy.
0
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In order to completely prove the theorem it is sufficient to show that
nEn,r (x) → 0 as
n → ∞.
Now
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nEn,r (x) = Rn,r,1 (x) + Rn,r,2 (x),
where
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Rn,r,1 (x) =
∞
2 X
n
2
k=0
Z
Pn+r,k (x)
|y−x|<δ
Sn,k+r (y)(y − x)2 η(x, y)dy
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and
Z
∞
n2 X
Rn,r,2 (x) =
Pn+r,k (x)
2 k=0
Sn,k+r (y)(y − x)2 η(y, x)dy
|y−x|>δ
By (3.2) and (2.1)

(3.3)
|Rn,r,1 (x)| <
∞
X

nε 
Pn+r,k (x)
n
2
k=0
Z

Sn,k+r (y)(y − x)2 dy 
|y−x|≤δ
≤ εx(2 + x)
as n → ∞.
Finally we estimate Rn,r,2 (x). Using Corollary 2.2 we have
Z
∞
n2 X
Pn+r,k (x)
(3.4)
Rn,r,2 (x) =
Sn,k+r (y)eαy dy
2 k=0
|y−x|>δ
n
= Mm,x n−m = 0
2
as n → ∞.
Theorem 3.2. Let f ∈ C (r+1) [0, a] and let w(f (r+1) ; ·) be the modulus of continuity of f (r+1) , then r = 0, 1, 2, . . .
(r)
(Bn f )(x) − f (r) (x) ≤ [1 + r(1 + a)] f (r+1) (x)
n
q
Tn,2 (a)
1
Tn,2 (a) +
w f (r+1) ; n−2
+ 2
2
n
Simultaneous Approximation by
Lupaş Modified Operators with
Weighted Function of Szasz
Operators
Naokant Deo
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where k·k is the sup norm [0, a].
Proof. We have by Taylor’s expansion
f (r) (y) − f (r) (x)
= (y − x)f
(r+1)
Z
y
(x) +
[f (r+1) (t) − f (r+1) (x)]dt
x
nr (n − 1)!
×
(B (r) f )(x) − f (r) (x)
(n + r − 1)! n
Z ∞
∞
X
=n
Pn+r,k (x)
Sn,k+r (y) f (r) (y) − f (r) (x) dy
=n
k=0
∞
X
0
Z
∞
Sn,k+r (y) (y − x)f (r+1) (x)
[f
(r+1)
Pn+r,k (x)
Simultaneous Approximation by
Lupaş Modified Operators with
Weighted Function of Szasz
Operators
Naokant Deo
0
k=0
Z
+
y
(t) − f
(r+1)
(x)]dt dy.
x
Also
(r+1)
f
(t) − f (r+1) (x) ≤
1+
|t − x|
δ
w(f (r+1) ; δ)
Hence
r
n (n − 1)!
(r)
(r)
(n + r − 1)! (Bn f )(x) − f (x)
|T | p
(r+1) n,2
≤ |Tn,1 | · f
(x) + Tn,2 +
· w(f (r+1) ; δ).
2δ
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By Schwarz’s inequality. Choosing δ =
required result.
1
n2
and using (i) and (2.1) we obtain the
Simultaneous Approximation by
Lupaş Modified Operators with
Weighted Function of Szasz
Operators
Naokant Deo
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References
[1] M.M. DERRIENNIC, Sur l’approximation de fonctions integrables sur
[0, 1] par des polynomes de Bernstein modifies, J. Approx, Theory, 31
(1981), 325–343.
[2] V. GUPTA, A note on modified Szasz operators, Bull. Inst. Math. Academia
Sinica, 21(3) (1993), 275–278.
[3] A. SAHAI AND G. PRASAD, On simultaneous approximation by modified
Lupaş operators, J. Approx., Theory, 45 (1985), 122–128.
Simultaneous Approximation by
Lupaş Modified Operators with
Weighted Function of Szasz
Operators
Naokant Deo
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