MATEMATIQKI VESNIK
originalni nauqni rad
research paper
63, 1 (2011), 27–32
March 2011
A NOTE ON RATE OF APPROXIMATION
FOR CERTAIN BÉZIER-DURRMEYER OPERATORS
Vijay Gupta and Naokant Deo
Abstract. The present paper deals with certain Bézier-Durrmeyer type sequence of linear
positive operators Mn,α (f, x), having different basis functions in summation and integration. We
estimate the rate of convergence of these operators Mn,α (f, x), for functions having derivatives of
bounded variation.
1. Introduction
To approximate Lebesgue integrable functions on [0, ∞), a new sequence of
linear positive operators Mn introduced and studied in [2] is defined as
∞
P
Mn (f, x) =
Φn,v (f )pn,v (x)
(1)
v=0
where
½
Φn,v (f ) =
¡n+v−1¢
n
R∞
0
qn,v−1 (t)f (t) dt, v > 0;
f (0),
v=0
v
are respectively
and pn,v (x) =
x (1 + x)
and qn,v (t) = e−nt (nt)
v!
v
Baskakov and Szász basis functions.
Very recently Abel et al. [1] introduced a more general sequence of summationintegral type operators, which for parameters a ∈ R and b ∈ Z is defined as
Z ∞
∞
P
Mna,b (f, x) = n
pn+a,v (x)
qn,v+b (t)f (t) dt
(2)
v
−n−v
v=max{0,−b}
0
We may remark here that for the case a = 0, b = −1 the operators (2) are very
similar to the operators (1). The only difference is that the definition (1) contains
an additional term (1 + x)−n f (0) to ensure that the operators preserve constant
functions. For the similar type of operators, we refer the readers to [7–9] where,
some approximation properties of summation-integral type operators have been
studied.
2010 AMS Subject Classification: 41A25, 41A30.
Keywords and phrases: Linear positive operators; Baskakov and Szász basis functions;
summation-integral type operators; Bézier variant; rate of convergence.
28
V. Gupta, N. Deo
It is observed that Bézier curves play an important role in Computer aided
geometric design. Several researchers have considered the Bézier variants of well
known operators and studied their approximation properties. We now consider the
Bézier variant of the operators (1), which for α ≥ 1 or 0 < α < 1 are defined as
Z ∞
∞
P
(α)
(α)
qn,v−1 (t)f (t) dt + Qn,0 (x)f (0)
Mn,α (f, x) = n
Qn,v (x)
v=1
0
Z ∞
=
Wn,α (x, t)f (t) dt
(3)
0
(α)
Qn,v (x)
where
is given by
= (Jn,v (x))α − (Jn,v+1 (x))α , Jn,v (x) =
Wn,α (x, t) = n
∞
P
v=1
P∞
(α)
j=v
pn,j (x) and the kernel
(α)
Qn,v (x)qn,v−1 (t) + Qn,0 (x)δ(t),
δ(t) being Dirac delta function. In case α = 1 the operators defined by (3) reduce
to the operators (1).
We denote by DBr (0, ∞), r ≥ 0 the class of absolutely continuous functions f
defined on (0, ∞) satisfying:
(i) f (t) = O(tr ), t → ∞,
(ii) having a derivative f 0 on the interval (0, ∞) coinciding a.e. with a function
ψ which is of bounded variation on every finite subinterval of (0, ∞). It can
be observed that all functions f ∈ BDr (0, ∞) possess for each c > 0 the
representation
Z
x
f (x) = f (c) +
ψ(t) dt.
c
The rate of convergence for functions of bounded variations for Bernstein and
Bernstein-Durrmeyer operators were established in [4] and [10]. Bojanić and Cheng
[3] obtained interesting results on the rate of convergence of Bernstein polynomials
for functions with derivatives of bounded variation. We now extend the study and
here we estimate the rate of convergence of the operators Mn,α (f, x), defined by
(3) for functions having derivatives of bounded variation.
2. Auxiliary results
In the sequel, we shall need the following lemmas:
S
Lemma 1 [2] Let the m-th order moment µn,m (x), m ∈ N {0} be defined as
∞
R∞
P
µn,m (x) ≡ Mn ((t−x)m , x) = n
pn,v (x) 0 qn,v−1 (t)(t − x)m dt+(−x)m (1+x)−n .
v=1
Then
µn,0 (x) = 1,
µn,1 (x) = 0,
µn,2 (x) =
x(x + 2)
n
and we have the recurrence relation:
nµn,m+1 (x) = x(1 + x)µ0n,m (x) + mµn,m (x) + mx(x + 2)µn,m−1 (x),
m ≥ 1,
Rate of approximation for Bézier-Durrmeyer operators
where µ0n,m (x) =
d
dx µn,m (x).
29
Consequently, for each x ∈ [0, ∞), we get
µn,m (x) = O(n−[(m+1)/2] ),
where [α] denotes the integral part of α.
Remark 1. From Lemma 1, using Cauchy-Schwarz inequality, it follows that
p
1/2
Mn (| t − x |, x) ≤ (µn,2 (x))
= x(x + 2)/n.
(α)
Remark 2. It is observed that for α ≥ 1 we have Qn,v (x) ≤ αpn,v (x), x ∈
Rt
[0, ∞). We define βn,α (x, t) = 0 Wn,α (x, s) ds. In particular, we have
Z ∞
βn,α (x, ∞) =
Wn,α (x, s) ds = 1.
0
Lemma 2. For x ∈ (0, ∞) and α ≥ 1, we have
(i) βn,α (x, y) ≤
αx(x+2)
n(x−y)2 ,
(ii) 1 − βn,α (x, z) ≤
0 ≤ y < x,
αx(x+2)
n(z−x)2 ,
x < z < ∞.
Proof. First we prove (i). By using Lemma 1 and Remark 2, we have
Z y
Z y
(x − t)2
βn,α (x, y) =
Wn,α (x, t)dt ≤
Wn,α (x, t) dt
2
0
0 (x − y)
αx(x + 2)
≤ (x − y)−2 αµn,2 (x) =
.
n(x − y)2
The proof of (ii) is similar, we omit the details.
3. Rate of convergence
Our main result is stated as follows:
Theorem 1. Let f ∈ DBr (0, ∞), r ∈ N ; also suppose α ≥ 1, and x ∈ (0, ∞).
Then for n sufficiently large, we have
√
√
x+x/v
x+x/ n
_
α(x + 2) [ Pn] _
x
0
|Mn,α (f, x) − f (x)| ≤
((f )x ) + √
((f 0 )x )
n
n
√
v=1
x−x/v
x−x/ n
α(x + 2)
(|f (2x) − f (x) − xf 0 (x+ )| + |f (x)| + x|f 0 (x+ )|)
+
nx
r
r
α(x + 2) r
α
αx(x + 2) 0 +
−r/2
+
2 O(n
)+
|f (x ) − f 0 (x− )|,
nx
α+1
n
where the auxiliary function fx is given by

−

 f (t) − f (x ), 0 ≤ t < x
fx (t) =
0,
t=x


+
f (t) − f (x ), x < t < ∞.
30
V. Gupta, N. Deo
Wb
f (x) denotes the total variation of f on [a, b]. Also by f (x− ) and f (x+ ), we
mean the left and right-hand limits respectively.
a
Proof. By the Mean value theorem, we have
Z ∞
Mn,α (f, x) − f (x) =
Wn,α (x, t)(f (t) − f (x)) dt
0
Z ∞Z t
=
Wn,α (x, t)(f 0 (u) du) dt.
0
(4)
x
Using the following identity
1
1
f 0 (u) =
[f 0 (x+ ) + αf 0 (x− )] + (f 0 )x (u) + [f 0 (x+ ) − f 0 (x− )]×
α+1
2
³
α − 1´
1
× sgn(u − x) +
+ [f 0 (x) − [f 0 (x+ ) + f 0 (x− )]]χx (u)
α+1
2
½
1, u = x
where χx (u) =
Obviously, Mn,α (χx , x) = 0. Also
0, u 6= x.
Z ∞ ³Z t
´
1 0 +
[f (x ) − f 0 (x− )] sgn(u − x) du Wn,α (x, t) dt
0
x 2
1
= [f 0 (x+ ) − f 0 (x− )]Mn,α (|t − x|, x)
2
and
Z ∞³Z t
´
1
[f 0 (x+ ) + αf 0 (x− )] du Wn,α (x, t) dt
0
x α+1
α
≤
[f 0 (x+ ) + αf 0 (x− )]Mn ((t − x), x) = 0.
α+1
Thus using the methods as given in [6], Lemma 1, Remark 1 and above values, we
have
|Mn,α (f, x) − f (x)|
¯Z ∞ ³ Z t
¯
Z x ³Z t
´
´
¯
¯
0
0
¯
≤¯
(f )x (u) du Wn,α (x, t) dt −
(f )x (u) du Wn,α (x, t) dt¯¯
x
x
0
x
α
|f 0 (x+ ) − f 0 (x− )|Mn,α (|t − x|, x)
+
α+1
α
= |An,α (f, x) − Bn,α (f, x)| +
|f 0 (x+ ) − f 0 (x− )|Mn,α (|t − x|, x)
α+1
r
αx(x + 2)
α
0 +
0 −
|f (x ) − f (x )|
. (5)
≤ |An,α (f, x)| + |Bn,α (f, x)| +
α+1
n
Thus it suffice to estimate the terms An,α (f,
√ x) and Bn,α (f, x). Integrating by
parts, using Lemma 2 and taking y = x − x/ n, we have
¯Z x ³ Z t
¯
´
¯
¯
0
¯
|Bn,α (f, x)| = ¯
(f )x (u) du dt βn,α (x, t)¯¯
0
x
¯Z x
¯ µZ y Z x ¶
x
_
¯
¯
= ¯¯
βn,α (x, t)(f 0 )x (t) dt¯¯ ≤
+
|βn,α (x, t)| ((f 0 )x ) dt
0
0
y
t
31
Rate of approximation for Bézier-Durrmeyer operators
≤
αx(x + 2)
n
Z
x
y_
0
((f 0 )x )
t
αx(x + 2)
≤
n
Let u =
Z
1
dt +
(x − t)2
t
y
x
x_
((f 0 )x ) dt
t
x
x _
1
√
((f )x )
dt +
((f 0 )x ).
(x − t)2
n √x
x
y_
0
Z
0
x−
n
x
x−t .
Then we have
Z x
Z √n _
x
αx(x + 2) y _ 0
1
αx(x + 2) −1
((f )x )
dt
=
x
((f 0 )x ) du
2
n
(x
−
t)
n
0
1
x
t
α(x + 2)
≤
n
Thus,
√ Z
k+1
[P
n]
k=1
k
x− u
x
_
√
α(x + 2) [ Pn] Wx
((f )x ) du ≤
((f 0 )x ).
x− x
k
n
k=1
x
0
x− u
√
x
x
α(x + 2) [ Pn] _
x _
((f 0 )x ) + √
((f 0 )x ).
|Bn,α (f, x)| ≤
n
n √x
k=1
x
x− k
x−
(6)
n
On the other hand, we have
¯Z ∞³Z t
¯
´
¯
¯
0
¯
|An,α (f, x)| = ¯
(f )x (u) du Wn,α (x, t)dt¯¯
x
x
¯Z ∞³Z t
¯ ¯Z 2x ³Z t
¯
´
´
¯
¯ ¯
¯
≤ ¯¯
(f 0 )x (u) du Wn,α (x, t) dt¯¯ + ¯¯
(f 0 )x (u) du dt (1 − βn,α (x, t)) dt¯¯
2x
x
¯Zx ∞
¯ x
¯Z ∞
¯
¯
¯
¯
¯
0
≤ ¯¯
(f (t) − f (x))Wn,α (x, t) dt¯¯ + |f (x+ )| ¯¯
(t − x)Wn,α (x, t) dt¯¯
2x
2x
¯Z 2x
¯
Z 2x
¯
¯
+ ¯¯
(f 0 )x (u)du)¯¯ |1 − βn,α (x, 2x)| +
|(f 0 )x (t)| |1 − βn,α (x, t)| dt
x
x
Z
Z
|f (x)| ∞
C ∞
r
Wn,α (x, t)t |t − x| dt +
Wn,α (x, t)(t − x)2 dt
≤
x 2x
x2 2x
Z ∞
α(x + 2)
+ |f 0 (x+ )|
Wn,α (x, t)|t − x| dt +
|f (2x) − f (x) − xf 0 (x+ )|
nx
2x
√
x
x+ √x
[ n] x+
n
α(x + 2) X _k
x _
+
((f 0 )x ) + √
((f 0 )x ). (7)
n
n
x
x
k=1
For the estimation of the first two terms in the right-hand side of (7), we proceed
as follows:
Applying Holder’s inequality, Remark 1 and Lemma 1, we have
Z
Z
|f (x)| ∞
C ∞
Wn,α (x, t)tr |t − x| dt +
Wn,α (x, t)(t − x)2 dt
x 2x
x2
2x
µZ ∞
¶ 12 µZ ∞
¶ 12
C
≤
Wn,α (x, t)t2r dt
Wn,α (x, t)(t − x)2 dt
x
2x
0
32
V. Gupta, N. Deo
p
Z
α(x + 2)
|f (x)| ∞
α(x + 2)
2
r
−r/2
+
Wn,α (x, t)(t − x) dt ≤ C2 O(n
) √
+ |f (x)|
.
x2
nx
nx
2x
Also, by Remark 1, the third term of the right-hand side of (6) is given by
Z ∞
Z ∞
|f 0 (x+ )|
Wn,α (x, t)|t − x| dt ≤ |f 0 (x+ )|x−1
Wn,α (x, t)(t − x)2 dt
2x
0
α(x + 2)
.
n
Combining the estimates (4)–(7), we get the desired result. This completes the
proof of the theorem.
Remark 3. We may note here that for a = b = 0, the operators reduce to the
standard Baskakov-Szász operators studied in [5] and [8] which are defined by
Z ∞
∞
P
Mn0,0 (f, x) = n
qn,v (t)f (t) dt, x ∈ [0, ∞)
pn,v (x)
≤ |f 0 (x+ )|
v=0
0
We may consider the Bézier variant of such type of operators and the analogous
results can be obtained along the lines similar to our main result.
Acknowledgements. The authors are thankful to the referees for their
valuable comments leading to the overall improvements in the paper.
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(received 23.09.2009; in revised form 03.06.2010)
School of Applied Sciences, Netaji Subhas Institute of Technology, Sector 3 Dwarka, New Delhi
110078, India
E-mail: vijaygupta2001@hotmail.com
Department of Applied Mathematics, Delhi Technological University (Formerly Delhi College of
Engineering), Bawana Road, Delhi 110042, India
E-mail: dr naokant deo@yahoo.com