Journal of Inequalities in Pure and
Applied Mathematics
SIMULTANEOUS APPROXIMATION FOR THE PHILLIPS-BÉZIER
OPERATORS
volume 7, issue 4, article 141,
2006.
VIJAY GUPTA, P.N. AGRAWAL AND HARUN KARSLI
School of Applied Sciences
Netaji Subhas Institute of Technology
Sector 3 Dwarka
New Delhi 110075, India.
EMail: vijay@nsit.ac.in
Department of Mathematics
Indian Institute of Technology
Roorkee 247667, India.
EMail: pnappfma@iitr.ernet.in
Ankara University
Faculty of Sciences Department of Mathematics
06100 Tandogan-Ankara-Turkey.
EMail: karsli@science.ankara.edu.tr
Received 28 July, 2006;
accepted 17 November, 2006.
Communicated by: F. Qi
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
204-06
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Abstract
We study the simultaneous approximation properties of the Bézier variant of
the well known Phillips operators and estimate the rate of convergence of the
Phillips-Bézier operators in simultaneous approximation, for functions of bounded
variation.
2000 Mathematics Subject Classification: 41A25, 41A30.
Key words: Rate of convergence, Bounded variation, Total variation, Simultaneous
approximation.
Vijay Gupta, P.N. Agrawal and
Harun Karsli
Vijay Gupta, P.N. Agrawal and
Harun Karsli
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
3
5
8
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1.
Introduction
For α ≥ 1, the Phillips-Bézier operator is defined by
Z ∞
∞
X
(α)
(α)
Qn,k (x)
pn,k−1 (t)f (t)dt + Qn,0 (x) f (0),
(1.1)
Pn,α (f, x) = n
k=1
0
where n ∈ N, x ∈ [0, ∞),
(α)
Qn,k
α
α
(x) = [Jn,k (x)] − [Jn,k+1 (x)] ,
Jn,k (x) =
∞
X
pn,j (x)
and
j=k
(nx)k
.
k!
For α = 1, the operator (1.1) reduces to the Phillips operator [1]. Some approximation properties of the Phillips operators were recently studied by Finta and
Gupta [2]. The rates of convergence in ordinary and simultaneous approximations on functions of bounded variation for the Phillips operators were estimated
in [3], [4] and [5]. In the present paper we extend the earlier study and here we
investigate and estimate the rate of convergence for the Bézier variant of the
Phillips operators in simultaneous approximations by means of the decomposition technique of functions of bounded variation. We denote the class Br,β
by
n
(r)
Br,β = f : f (r−1) ∈ C[0, ∞), f± (x) exist everywhere
pn,k (x) = e−nx
and are bounded on every finite subinterval of
o
(r)
[0, ∞) and f± (x) = O(eβt ) (t → ∞), for some β > 0 ,
Vijay Gupta, P.N. Agrawal and
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Harun Karsli
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(0)
r = 1, 2, . . . . By f± (x) we mean f (x±). Our main theorem is stated as:
Theorem 1.1. Let f ∈ Br,β , r = 1, 2, ... and β > 0. Then for every x ∈ (0, ∞)
and n ≥ max {r2 + r, 4β} , we have
n
o r + α − 1 (r)
(r)
(r)
(r)
(r)
Pn,α (f, x) − 1
f+ (x) + α f− (x) ≤ √
f+ (x) − f− (x)
α+1
2enx
√
√
n x+x/ k
2x + 1
1
2α(1 + 2x) X _
(gr,x ) + α √ 2r/2 e2βx ,
+
1+
2
n
x
x n
√
k=1
x−x/ k
where gr,x is the auxiliary function defined by

(r)

 f (r) (t) − f+ (x), x < t < ∞


0,
t=x
,
gr,x (t) =



 f (r) (t) − f (r) (x), 0 ≤ t < x
−
Wb
is the total variation of gr,x (t) on [a, b]. In particular g0,x (t) ≡
gx (t), defined in [4].
a (gr,x (t))
Vijay Gupta, P.N. Agrawal and
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2.
Auxiliary Results
In this section we give certain lemmas, which are necessary for proving the
main theorem.
Lemma 2.1. For all x ∈ (0, ∞), α ≥ 1 and k ∈ N ∪ {0} , we have
pn,k (x) ≤ √
1
2enx
and
(α)
Qn,k (x) ≤ √
Vijay Gupta, P.N. Agrawal and
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α
,
2enx
√
where the constant 1/ 2e and the estimation order n−1/2 (for n → ∞) are the
best possible.
Lemma 2.2 ([3]). If f ∈ L1 [0, ∞), f
L1 [0, ∞), then
Pn(r) (f, x) = n
∞
X
(r−1)
Z
pn,k (x)
∈ A.C.loc , r ∈ N and f
(r)
∈
∞
pn,k+r−1 (t) f (r) (t)dt.
0
k=0
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Lemma 2.3 ([3]). For m ∈ N ∪ {0} , r ∈ N, if we define the m-th order moment
by
Z ∞
∞
X
pn,k (x)
pn,k+r−1 (t) (t − x)m dt
µr,n,m (x) = n
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0
k=0
then µr,n,0 (x) = 1, µr,n,1 (x) =
Vijay Gupta, P.N. Agrawal and
Harun Karsli
r
n
and µr,n,2 (x) =
2nx+r(r+1)
.
n2
J. Ineq. Pure and Appl. Math. 7(4) Art. 141, 2006
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Also there holds the following recurrence relation
nµr,n,m+1 (x) = x[µ(1)
r,n,m (x) + 2mµr,n,m−1 (x)] + (m + r)µr,n,m (x).
Consequently by the recurrence relation, for all x ∈ [0, ∞), we have
µr,n,m (x) = O n−[(m+1)/2] .
Remark 1. In particular, by Lemma 2.3, for given any number n ≥ r2 + r and
0 < x < ∞, we have
(2.1)
µr,n,2 (x) ≤
2x + 1
.
n
Vijay Gupta, P.N. Agrawal and
Harun Karsli
Remark 2. We can observe from Lemma 2.2 and Lemma 2.3 that for r = 0, the
summation over k starts from 1. For r = 0, Lemma 2.3 may be defined as [5,
Lemma 2], with c = 0.
Lemma 2.4. Suppose x ∈ (0, ∞), r ∈ N ∪ {0} , α ≥ 1 and
Kr,n,α (x, t) = n
∞
X
(α)
Qn,k (x)pn,k+r−1 (t).
k=0
Then for n ≥ r2 + r, there hold
Z y
α(2x + 1)
(2.2)
Kr,n,α (x, t)dt ≤
,
n(x − y)2
0
Z ∞
α(2x + 1)
(2.3)
Kr,n,α (x, t)dt ≤
,
n(z − x)2
z
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x < z < ∞.
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Proof. We first prove (2.2) as follows:
Z y
Z
Kr,n,α (x, t)dt ≤
0
y
(x − t)2
K
(x, t)dt
2 r,n,α
0 (x − y)
α
≤
Pn ((t − x)2 , x)
(x − y)2
α(2x + 1)
αµr,n,2 (x)
≤
≤
,
(x − y)2
n(x − y)2
by using (2.1). The proof of (2.3) follows along similar lines.
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3.
Proof
Proof of Theorem 1.1. Clearly
(3.1)
n
o
(r)
(r)
(r)
Pn,α (f, x) − 1
f+ (x) + α f− (x) α+1
Z ∞
∞
X
(α)
≤n
Qn,k (x)
pn+r−1,k (x) |gr,x (t)| dt
0
k=0
1
+
α+1
(r)
(r)
(r)
(sgn α (t − x), x) .
f+ (x) − f− (x) Pn,α
Vijay Gupta, P.N. Agrawal and
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(r)
We first estimate Pn,α (sgnα (t − x), x) as follows:
(r)
Pn,α
(sgn α (t − x), x)
Z ∞
Z
∞
X
(α)
=n
Qn,k (x)
αpn,k+r−1 (t)dt − (1 + α)
0
k=0
= α − (1 + α)n
= α − (1 + α)n
∞
X
k=0
∞
X
0
Z
(α)
Qn,k (x)
= (1 + α)n
k=0
(α)
pn,k+r−1 (t)dt
0
(α)
Qn,k (x)
k+r−1
X
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x
pn,k+r−1 (t)dt
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k+r−1
X
!
pn,j (x)
j=0
k=0
∞
X
∞
Qn,k (x) 1 −
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pn,j (x) − 1
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j=0
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!
r−1
= (1 + α)n
pn,j (x) + √
−1
2enx
j=0
k=0
#
"∞
∞
X
X
r
−
1
(α)
= (1 + α)
Qn,k (x) + √
−1
pn,j (x)
2enx
j=0
k=j
#
"∞
∞
X
X
r−1
(α)
α
−
Qn,j (x).
= (1 + α)
pn,j (x) [Jn,j (x)] + √
2enx
j=0
j=0
∞
X
(α)
Qn,k (x)
k
X
Vijay Gupta, P.N. Agrawal and
Harun Karsli
By the mean value theorem, we find that
(α+1)
Qn,j (x)
α+1
= [Jn,j (x)]
α+1
− [Jn,j+1 (x)]
α
= (α + 1)pn,j (x) [γn,j (x)] ,
where
Jn,j+1 (x) < γn,j (x) < Jn,j (x).
Hence by Lemma 2.1, we have
(r)
Pn,α (sgn α (t − x), x)
"∞
#
(1 + α)(r − 1)
X
√
≤ (1 + α)
pn,j (x) ([Jn,j (x)]α − [γn,j (x)]α ) +
2enx
j=0
"∞
#
X
r−1
(α)
≤ (1 + α)
pn,j (x)Qn,k (x) + √
2enx
j=0
α
r−1
(1 + α)(r + α − 1)
√
(3.2) ≤ (1 + α) √
+√
=
.
2enx
2enx
2enx
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(r)
Next we estimate Pn (gr,x , x). By the Lebesgue-Stieltjes integral representation, we have
Z ∞
(r)
Pn,α (gr,x , x) =
gr,x (t)Kr,n,α (x, t)dt
0
Z
Z
Z
Z =
+
+
+
gr,x (t)Kr,n,α (x, t)dt
I1
I2
I3
I4
= R1 + R2 + R3 + R4,
√
√
√
√
say, where I1 = [0, x−x/ n], I2 = [x−x/ n, x+x/ n], I3 = [x+x/ n, 2x]
and I4 = [2x, ∞). Let us define
Z t
ηr,n,α (x, t) =
Kr,n,α (x, u)du.
(3.3)
0
√
We first estimate R1 . Writing y = x − x/ n and using integration by parts, we
have
Z y
R1 =
gr,x (t)dt (ηr,n,α (x, t))
0
Z y
= gr,x (y)ηr,n,α (x, y) −
ηr,n,α (x, t)dt (gr,x (t)).
0
|R1 | ≤
y
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By Remark 1, it follows that
x
_
Vijay Gupta, P.N. Agrawal and
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Z
y
ηr,n,α (x, t)dt −
(gr,x )ηr,n,α (x, y) +
0
x
_
t
!
(gr,x )
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≤
x
_
y
α(2x + 1) λx
+
(gr,x )
n(x − y)2
n
Z
0
y
!
x
_
1
dt − (gr,x ) .
(x − t)2
t
Integrating by parts the last term, we have after simple computation,
Wx
Z y Wx
α(2x + 1)
0 (gr,x )
t (gr,x )
|R1 | ≤
+2
dt .
3
n
x2
0 (x − t)
√
Now replacing the variable y in the last integral by x − x/ t, we get
n
x
2α(2x + 1) X _
|R1 | ≤
(gr,x ).
nx2
√
k=1
(3.4)
Vijay Gupta, P.N. Agrawal and
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Vijay Gupta, P.N. Agrawal and
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x−x/ k
√
√
Next we estimate R2 . For t ∈ [x − x/ n, x + x/ n], we have
√
x+x/ n
|gr,x (t)| = |gr,x (t) − gr,x (x)| ≤
_
√
(gr,x ).
x−x/ n
Also by the fact that
Rb
a
dt (ηr,n (x, t)) ≤ 1 for (a, b) ⊂ [0, ∞), therefore
√
x+x/ n
(3.5)
|R2 | ≤
_
(gr,x ) ≤
√
x−x/ n
1
n
√
n x+x/
X
_k
√
k=1 x−x/ k
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√
To estimate R3 , we take z = x + x/ n, thus
Z 2x
R3 =
Kr,n,α (x, t)gr,x (t)dt
z
Z 2x
=−
gr,x (t)dt (1 − ηr,n,α (x, t))
z
= −gr,x (2x)(1 − ηr,n,α (x, 2x)) + gr,x (z)(1 − ηr,n,α (x, z))
Z 2x
+
(1 − ηr,n,α (x, t))dt (gr,x (t)).
z
Thus arguing similarly as in the estimate of R1 , we obtain
√
|R3 | ≤
(3.6)
n x+x/
2α(2x + 1) X _
nx2
k=1
(gr,x ).
x
2x
≤ nα
nα
≤
x
k=0
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∞
Z
pn,k+r−1 (t) eβt dt
pn,k (x)
k=0
∞
X
Vijay Gupta, P.N. Agrawal and
Harun Karsli
k
Finally we estimate R4 as follows
Z ∞
|R4 | = Kr,n,α (x, t)gr,x (t)dt
∞
X
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2x
Z
pn,k (x)
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∞
pn,k+r−1 (t) eβt |t − x| dt
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0
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α
≤
x
n
∞
X
Z
pn,k (x)
pn,k+r−1 (t) (t − x)2 dt
0
k=0
×
! 21
∞
n
∞
X
Z
pn,k (x)
! 12
∞
pn,k+r−1 (t) e2βt dt
.
0
k=0
For the first expression above we use Remark 1. To evaluate the second expression, we proceed as follows:
n
∞
X
Z
pn,k (x)
∞
pn,k+r−1 (t) e2βt dt
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Harun Karsli
0
k=0
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Harun Karsli
∞
X
nk+r−1
=n
pn,k (x)
(k + r − 1)!
k=0
Z
∞
tk+r−1 e−(n−2β)t dt,
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0
∞
X
nk+r−1
Γ(k + r)
n
pn,k (x)
(k + r − 1)! (n − 2β)k+r
k=0
k
∞ X
nr
n
=
pn,k (x) ,
(n − 2β)r k=0 n − 2β
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k
∞ X
n
n2 x
1
nr
−nx
e
=
e2nxβ/(n−2β) ≤ 2r e4βx
r
(n − 2β)r
n
−
2β
k!
(n
−
2β)
k=0
r
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for n > 4β. Thus
α
|R4 | ≤
x
n
∞
X
√
≤
pn,k (x)
! 12
∞
pn,k+r−1 (t) (t − x)2 dt
0
k=0
×
(3.7)
Z
n
∞
X
Z
pn,k (x)
k=0
∞
! 12
pn,k+r−1 (t) e2βt dt
0
α 2x + 1 r/2 2βx
√
2 e .
x n
Combining the estimates of (3.1)-(3.7), we get the required result.
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References
[1] R.S. PHILLIPS, An inversion formula for semi-groups of linear operators,
Ann. Math. (Ser. 2), 59 (1954), 352–356.
[2] Z. FINTA AND V. GUPTA, Direct and inverse estimates for Phillips type
operators, J. Math Anal Appl., 303 (2005), 627–642.
[3] N.K. GOVIL, V. GUPTA AND M.A. NOOR, Simultaneous approximation
for the Phillips operators, International Journal of Math and Math Sci., Vol.
2006 Art Id. 49094 92006, 1-9.
[4] V. GUPTA AND G.S. SRIVASTAVA, On the rate of convergence of Phillips
operators for functions of bounded variation, Ann. Soc. Math. Polon. Ser. I
Commentat. Math., 36 (1996), 123–130.
Vijay Gupta, P.N. Agrawal and
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Vijay Gupta, P.N. Agrawal and
Harun Karsli
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[5] H.M. SRIVASTAVA AND V. GUPTA, A certain family of summationintegral type operators, Math. Comput. Modelling, 37 (2003), 1307–1315.
[6] X.M. ZENG AND J.N. ZHAO, Exact bounds for some basis functions of
approximation operators, J. Inequal. Appl., 6 (2001), 563–575.
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