General Mathematics Vol. 14, No. 4 (2006), 29–36
Rate of convergence on the mixed
summation integral type operators 1
Jyoti Sinha and V. K. Singh
In memoriam of Associate Professor Ph. D. Luciana Lupaş
Abstract
Gupta and Erkus [3] introduced the mixed sequence of summationintegral type operators Sn (f, x) and estimated some direct results in
simultaneous approximation. We extend the study on these operators Sn (f, x) and here we study the rate of convergence for functions
having derivatives of bounded variation.
2000 Mathematics Subject Classification: ....
1
Introduction
Very recently Gupta and Erkus [3] defined a mixed summation-integral type
operators to approximate integrable functions on the interval [0, ∞) .The
operators introduced in [3] are defined as
R∞
(1)
Sn (f, x) = 0 Wn (x, t)f (t)dt
= (n − 1)
∞
X
0
sn,v (x)
Z
∞
bn,v−1 (t)f (t)dt + exp (−nx)f (0),
0
1
Received 30 September, 2006
Accepted for publication (in revised form) 6 December, 2006
29
x ∈ [0, ∞)
30
where
Jyoti Sinha and V. K. Singh
∞
X
sn,v (x)bn,v−1 (t) + exp−nx δ(t)
Wn (x, t) = (n − 1)
v=1
¡
¢v
v
and bn,v (t) = n+v−1
t (1+
δ(t) being Dirac delta function, sn,v (x) = exp (−nx)nx
v!
v
−n−v
t)
are respectively Szasz and Baskakov basis functions. Although these
operators are similar to the generalized summation integral type operators
recently introduced by Srivastava and Gupta [4], but the approximation
properties of the operators (1) are different from those introduced in [4].
Here in summation and integration we are taking different basis functions.
We define
βn (x, t) =
Z
t
Wn (x, s)ds
0
then in particular,
βn (x, ∞) =
Z
∞
Wn (x, s)ds = 1.
0
Let DBγ (0, ∞), γ ≥ 0 be the class of absolutely continuous functions f
defined on (0, ∞) satisfying the growth condition f (t) = O(tγ ), t → ∞ and
having a derivative f ′ on the interval (0, ∞) coinciding a.e. with a function
which is of bounded variation on every finite subinterval of (0, ∞). It can
be observed that all functions f ∈ BDγ (0, ∞) posses for each c > 0 a
representation
Z
x
f (x) = f (c) +
ψ(t)dt,
c
x ≥ c.
We denote the auxiliary function fx by


f (t) − f (x− ),
0 ≤ t < x;





fx (t) =
0,
t = x;






f (t) − f (x+ ), x < t < ∞.
In [3] the authors studied some direct results in simultaneous approximation
for the operators (1).The rates of convergence for functions having derivatives of bounded variation on Bernstein polynomials were studied in [1] and
Rate of convergence on the mixed summation integral type operators
31
[2]. This motivated us to study further on summation integral type operators and here we estimate the rate of convergence for the operator (1) with
functions having derivatives of bounded variation.
2
Auxiliary Results
We shall use the following Lemmas to prove our main theorem.
Lemma 2.1. Let the function µn,m (x), m ∈ ℵ0 , be defined as
Z ∞
∞
X
sn,v (x)
µn,m (x) = (n − 1)
bn,v−1 (t)(t − x)m dt + (−x)m exp (−nx).
0
v=1
Then µn,0 (x) = 1, µn,1 (x) =
recurrence relation:
2x
,
n−2
µn,2 (x) =
nx(x+2)+6(x2 )
,
(n−2)(n−3)
also we have the
(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);
n > m + 2.
Consequently for each x ∈ [0, ∞) we from this recurrence relation that
µn,m (x) = O(n−[(m+1)/2] )
Remark 2.2.In particular given any number λ > 1 and x ∈ (0, ∞),by
lemma 2.1,we have for n sufficiently large
(2)
Sn ((t − x)2 , x) ≡ µn,2 ≤ λx(x+2)
n
Remark 2.3. From equation (2) it follows thatp
1/2
(3)
Sn (| t − x |, x) ≤ [Sn ((t − x)2 , x)] ≤ λx(x + 2)/n
Lemma 2.4. Let x ∈ (0, ∞) and Wn (x, t)are as in (1). then for λ > 1
and for n sufficiently large, we have
Z y
λx(x + 2)
Wn (x, t)dt ≤
(i) βn (x, y) =
, 0≤y<x
n(x − y)2
0
Z ∞
λx(x + 2)
Wn (x, t)dt ≤
(ii) 1 − βn (x, z) =
, x<z<∞
n(z − x)2
z
32
Jyoti Sinha and V. K. Singh
Proof. First we prove (i), by (2), we have
Z y
Z y
(x − t)2
Wn (x, t)dt ≤
Wn (x, t)dt
2
0
0 (x − y)
≤ (x, y)−2 µn,2 (x) ≤
λx(x + 2)
.
x(x − y)2
The proof of (ii) is similar, we omit the details.
3
Main Result
In this section, we prove the following main theorem.
Theorem 3.1 Let f ∈ DBγ (0, ∞),γ > 0, and x ∈ (0, ∞).Then for λ > 2
and forn sufficiently large, we have
√
√
[ n] x+x/k
x+x/ n
λ(x + 2) X _
x _
′
((f ′ )x ))
| Sn (f, x) − f (x) | ≤
((f )x ) + √
(
n
n
√
k=1
x−x/k
+
x−x/ n
λ(x + 2)
(| f (2x) − f (x) − xf ′ (x+ ) | + | f (x) |)
n
p
−γ
+ λx(x + 2)/n(M 2γ O(n 2 )+ | f (x+ ) |)
p
+ 1/2 λx(x + 2)/n | f (′ x+ ) | − | f ′ (x− ) |
x
+
| f ′ (x+ ) | + | f ′ (x− ) |,
n−2
W
where ba f (x) denotes the total variation of fx on [a, b].
Proof.We have
Z ∞
Sn (f, x) − f (x) =
Wn (x, t)(f (t) − f (x))dt
0
=
Z
0
∞
Z t
( Wn (x, t)(f ′ (u)du)dt)
x
Using the identity
f ′ (u) = 1/2[f ′ (x+ ) + f ′ (x− )] + (f ′ )x (u) + 1/2[f ′ (x+ ) − f ′ (x− )]sgn(u − x)
Rate of convergence on the mixed summation integral type operators
33
+ [f ′ (x) − 1/2[f ′ (x+ ) + f ′ (x− )]]χx (u),
it is easily verified that
Z ∞ Z t
( f ′ (x) − 1/2[f ′ (x+ ) + f ′ (x− )]χx (u)du))Wn (x, t)dt = 0
0
x
Also
∞
Z
0
Z t
( 1/2[f ′ (x+ ) − f ′ (x− )]sgn(u − x)du)Wn (x, t)dt
x
= 1/2[f ′ (x+ ) − f ′ (x− )]Sn (| t − x |, x)
and
Z
0
∞
Z
(
x
t
1 ′ +
[f (x ) + f ′ (x− )]du)Wn (x, t)dt
2
1
= [f ′ (x+ ) + f ′ (x− )]Sn ((t − x), x).
2
Thus we have
(4)
| Sn (f, x) − f (x) |
Z x Z t
Z ∞ Z t
′
( (f ′ )x (u)du)Wn (x, t)dt |
( (f )x (u)du)Wn (x, t)dt −
≤|
x
0
x
+
x
1
| f ′ (x+ ) − f ′ (x− ) | Sn (| t − x |, x)
2
1
| f ′ (x+ ) + f ′ (x− ) | Sn ((t − x), x)
2
1
=| An (f, x) + Bn (f, x) | + | f ′ (x+ ) − f ′ (x− ) | Sn (| t − x |, x)
2
1
+ | f ′ (x+ ) + f ′ (x− ) | Sn ((t − x), x).
2
To complete the proof of the theorem it is sufficient to estimate the
terms An (f, x) and Bn (f, x). Applying integration by parts, using Lemma
√
2.4 and taking y = x − x/ n,we have
+
| Bn (f, x) |=|
Z
0
x
Z t
( (f ′ )x (u)du)dtβn (x, t)dt |
x
34
Jyoti Sinha and V. K. Singh
x
Z
Z
βn (x, t)(f )x (t)dt ≤ (
′
0
y
+
0
Z
x
)| (f ′ )x (t) || βn (x, t) | dt
y
Z x_
x
λx(x + 2)
1
′
dt +
((f ′ )x )dt
≤
((f )x )
2
n
(x
−
t)
y
0
t
t
Z y_
x
x
λx(x + 2)
1
x _
√
≤
((f ′ )x ).
((f ′ )x )
dt
+
2
n
(x
−
t)
n
0
√x
t
Z
x
y_
x−
x
.Then
x−t
Letu =
λx(x + 2)
n
Z
n
we have
0
x
y_
t
1
λx(x + 2)
((f )x )
dt
=
(x − t2 )
n
′
Z
√
1
n
x
_
((f ′ )x )du
x
x− u
√
[ n] x
λx(x + 2) X _
≤
((f ′ )x ).
n
x
k=1
x− u
Thus
(5)
| βn (f, x) | ≤
√
λx(x+2) P[ n]
k=1
n
Wx
x ((f
x− u
′
)x ) +
√x
n
Wx
x− √xn ((f
′
)x ).
On the other hand, we have
R∞ Rt
(6)
| An (f, x) |=| x ( x (f ′ )x (u)du)Wn (x, t)dt |
=|
Z
∞
2x
≤|
+|
Z
Z t
Z
′
( (f )x (u)du)Wn (x, t)dt +
x
Z
x
∞
2x
2x
Z t
( (f ′ )x (u)du)dt(1 − βn (x, t)) | dt
′
′
(f )x (u)du) || (1 − βn (x, 2x) | +
M
≤
x
+
(f (t) − f (x))Wn (x, t)dt | +| f (x ) ||
2x
x
x
Z
∞
2x
Z
x
∞
2x
(t − x)Wn (x, t)dt |
2x
| (f ′ )x (t) | | (1 − βn (x, t) | dt
| f (x) |
Wn (x, t)t | t − x |dt +
x2
γ
Z
Z
∞
2x
Wn (x, t)(t − x)2 dt
Rate of convergence on the mixed summation integral type operators
′
+
+| f (x ) |
Z
∞
2x
Wn (x, t)| (t − x) |dt +
√
+
n
λ(x + 2)
(| f (2x) − f (x) − xf ′ (x+ ) |
nx
x+ √xn
x
[ n] x+
λ(x + 2) X _k
k=1
x
35
x _
((f ′ )x ).
((f ′ )x ) + √
n x
Next applying Holderş inequality, and Lemma 2.1, we proceed as follows
for the estimation of the first two terms in the right hand side of (6):
(7)
M
x
Z
∞
2x
| f (x) |
Wn (x, t)t | t − x |dt +
x2
γ
Z
∞
2x
Wn (x, t)(t − x)2 dt
1
Z ∞
Z ∞
2
1
M
2
2γ
Wn (x, t)(t − x) dt)
≤
(
Wn (x, t)t dt) 2 + (
x 2x
0
Z
| f (x) | ∞
(
Wn (x, t)(t − x)2 dt)
+
x2
2x
p
λx(x + 2)
λ(x + 2)
√
+ | f (x) |
≤ M 2γ O(n−γ/2 )
nx
n
Also the third term of the right side of (6) is estimated as
Z ∞
′ +
| f (x ) |
Wn (x, t)| t − x |dt
2x
Z
≤ | f (x ) |
′
+
0
Z
≤ | f (x ) |(
′
+
0
∞
∞
Wn (x, t)| t − x |dt
1
2
Z
Wn (x, t)(t − x) dt) (
2
∞
Wn (x, t)dt)
0
p
λx(x + 2)
√
=| f ′ (x+ ) |
n
Combining the estimates (4)-(7), we get the desired result.
This completes the proof of Theorem 3.1.
1
2
36
Jyoti Sinha and V. K. Singh
References
[1] R. Bojanic and F. Cheng, Rate of convergence of Bernstein polynomials
for functions with derivatives of bounded variation, J. Math. Anal.
Appl. 141 (1989), no. 1, 136-151.
[2] R. Bojanic and F. Cheng, Rate of convergence of Hermite-Fejér polynomials for functions with derivatives of bounded variation, Acta Math.
Hungar. 59 (1992), no. 1-2, 91-102.
[3] V. Gupta and E. Erkus, On a hybrid family of summation integral type
operators, J. Inequal. Pure and Appl. Math. 7(1)(2006), Art 23.
[4] H. M. Srivastava and V. Gupta, A certain family of summation integral
type operators, Math. Comput Modelling 37 (2003), 1307-1315.
Inderprastha Engineering College
Sahibabad, Ghaziabad (U.,P.) India
E-mail address: romjyoti@yahoo.co.in
E-mail address: vinai singh2004@yahoo.com