193 |
P a g e
ON SIMULTANEOUS APPROXIMATION BY BETA
OPERATORS
Manoj Kumar
Department of Mathematics and Statistics, Gurukula Kangri Vishwavidyalaya,
Haridwar, Uttrakhand-249404, (India)
ABSTRACT
For last three decades applications of beta operators in the area of approximation theory is an active area of research. In the present paper, we obtain asymptotic formula for modified beta operators in linear simultaneous approximation. To establish our result, we have used the technique of linear approximating method, namely, Steklov mean.
Keywords- Simultaneous approximation, Linear combinations, Linear positive operators, Steklov
mean.
MATHEMATICAL SUBJECT CLASSIFICATION:
41A25, 41A36.I. INTRODUCTION
In approximation theory, beta operators have been studied for last three decades. Beta operators were introduced and studied by several researchers [1, 3 , 7, 8]. In the present paper we study an asymptotic formula in simultaneous approximation for the linear combinations of the operators introduced by Gupta et al. [2]. The modified beta operators introduced by Gupta et al. [2] are defined as
0)
(
)
,
(
)
,
(
f
x
W
x
t
f
t
dt
B
n n ,x
[
0
,
)
(1.1)
where
(
)
(
)
)
1
(
1
)
,
(
,1
,
x
b
t
b
n
t
x
W
nvv v n
n
, , 1(
1
)
1)
1
,
(
1
)
(
v n vv
n
t
t
n
v
t
b
and
(
v
,
n
1
)
(
v
1
)!
n
!
/(
n
v
)!
the Beta function.It is easily checked that the operators defined by (1.1) are linear positive operators and it is obvious that
1
)
,
1
(
x
B
n . Also it is observed that the order of approximation by operators (1.1) is, at best O(n-1),
194 |
P a g e
have
B
n(
t
x
,
x
)
0
, which is essential property for making iterative combinations. Yet another approach for improving the order of approximation is the technique of linear combinations which was first considered by May [4] to improve the order of approximation for exponential type operators. In the present paper, we use the later approach, which described as:Let d0, d1, d2… dk be (k+1) arbitrary but fixed distinct positive integers. Then the linear
combination
B
n(
f
,
k
,
x
)
ofB
d n(
f
,
x
)
j , j = 0, 1, 2…n is defined as
k k k
k n
d
k n
d
k n
d
n
d
d
d
x
f
B
d
d
d
x
f
B
d
d
d
x
f
B
x
k
f
B
k
....
...
)
,
(
...
....
...
...
...
....
...
...
....
...
...
...
....
...
....
...
)
,
(
....
...
)
,
(
1
)
,
,
(
2 1
1 2
1 1 1
0 2
0 1 0
1 0
(1.2)
where
k k k
k
k k
d
d
d
d
d
d
d
d
d
....
...
1
...
....
...
...
...
...
...
....
...
...
...
...
....
...
1
....
...
1
2 1
1 2
1 1 1
0 2
0 1 0
The above expression (1.2) after simplification may be written as
(
,
,
)
(
,
)
(
,
)
0x
f
B
k
j
C
x
k
f
B
d nk
j
n
j
(1.3)
where
(
,
)
,
0
0
k
d
d
d
k
j
C
k
j i
i j i
j
and C(0,0)=1.
Some basic properties of
b
n,v(
x
)
are as follows(i).
)
1
(
)
1
(
)
(
0, 2
n
n
v
v
dt
t
b
t
nv(1.4)
(ii).
(
)
(
1
)
1
,
n
x
b
v v
n
(1.5)
(iii).
(
)
(
1
)[
1
(
2
)
]
1,
x
n
n
x
b
v
v v
n
195 |
P a g e
(iv).
(
)
(
1
)[
1
3
(
2
)
(
2
)(
3
)
2]
1 , 2
x
n
n
x
n
n
x
b
v
v
v
n
(1.7)
(v).
x
(
1
x
)
b
n
,v(
x
)
[(
v
1
)
(
n
2
)
x
]
b
n,v(
x
)
(1.8)
where
n
N
andx
[
0
,
)
.Throughout this paper, we may assume that
0
a
1
a
3
a
2
b
2
b
3
b
1
and]
,
[
i ii
a
b
I
where i =1, 2, 3.Let
H
[
0
,
)
be the class of all measurable functions defined on[
0
,
)
satisfying
dt
t
t
f
n 0
1
)
1
(
)
(
for some positive integer n.
Obviously the class
H
[
0
,
)
is bigger than the class of all lebesgue integrable functions on[
0
,
)
. Therefore the operators (1.1) may be applicable for studying a larger class.The main object of the present paper is to study a Voronovskaja-type asymptotic formula in simultaneous approximation for linear combinations of the operators (1.1).
II. AUXILIARY RESULTS
To prove our main results, we shall require the following preliminary results:
Lemma 2.1. For
m
N
0(the set of non-negative integers) andn
m
, let the function
n,m(
x
)
be defined as
0 , 1
,
,
(
)
(
)(
)
)
1
(
1
,
)
(
)
(
b
x
b
t
t
x
dt
n
x
x
t
B
x
nv mv v n m
n m
n
.Then
n
x
x
x
nn
1
2
)
(
,
1
)
(
,10 ,
, and there holds the recurrence relation.
)
(
)
2
1
)(
1
(
)]
(
2
)
(
)[
1
(
)
(
)
(
n
m
n,m1x
x
x
n/,mx
m
n,m1x
m
x
n,mx
Moreover, we have the following consequences about
n,m(
x
)
:(i).
n,m(
x
)
is a polynomial in x of degree m, (ii). for everyx
[
0
,
)
,
n,m(
x
)
O
n
[(m1)/2]
where [
] denotes the integral part of
.Consequently, on using Holder’s inequality, we have from this recurrence relation that
r
r/2,
x
x
O
n
t
B
n for each r >0 and for every fixedx
[
0
,
)
.196 |
P a g e
(
)
,
(
1
,
)
1
)
(
00
,
x
B
nt
x
x
B
nx
n
and
n
x
x
xB
x
t
B
x
x
t
B
x
n n nn
2
1
)
,
1
(
)
,
(
),
(
)
(
1 ,
To prove the recurrence relation we shall make the use of the following identity
)
(
]
)
2
(
)
1
[(
)
(
)
1
(
x
b
/,x
v
n
x
b
,x
x
nv
nvNow, we have
1 0
, /
, /
,
(
1
)
(
)
(
)(
)
)
1
(
1
)
(
)
1
(
v
m v
n v n m
n
x
x
b
x
b
t
t
x
dt
n
x
x
x
x
x
b
x
b
t
t
x
dt
n
m
nv mv
v
n
0
1 ,
1
,
(
)
(
)(
)
)
1
(
)
1
(
1
or
1 0
, ,
1 , /
,
[(
1
)
(
2
)
]
(
)
(
)(
)
)
1
(
1
)]
(
)
(
)[
1
(
v
m v
n v n m
n m
n
v
n
x
b
x
b
t
t
x
dt
n
x
m
x
x
x
dt
x
t
t
b
x
t
n
t
n
v
x
b
n
m v
n v
v
n
(
)
[(
1
)
(
2
)
(
2
)(
)]
(
)
(
)
)
1
(
1
0
, 1
,
)
(
)
2
(
)
(
)
(
)
1
(
)
(
)
1
(
1
1 , 0
/ , 1
,
x
t
t
b
t
t
x
dt
n
x
b
n
nmm v
n v
v
n
)
(
)
2
(
)
(
)
(
)]
1
(
)
)(
2
1
(
)
[(
)
(
)
1
(
1
1 , 0
/ , 2
1
,
x
t
x
x
t
x
x
x
b
t
t
x
dt
n
x
b
n
nmm v
n v
v
n
1 0
2 ,
/
,
(
)
(
)(
)
)
1
(
1
v
m v
n v
n
x
b
t
t
x
dt
b
n
+
1 0
1 ,
/
,
(
)
(
)(
)
)
1
(
)
2
1
(
v
m v
n v
n
x
b
t
t
x
dt
b
n
x
+
1 0
, /
,
(
)
(
)(
)
)
1
(
)
1
(
v
m v
n v
n
x
b
t
t
x
dt
b
n
x
x
+
(
n
2
)
n,m1(
x
)
(
m
2
)
n,m1(
x
)
(
m
1
)(
1
2
x
)
n,m(
x
)
mx
(
1
x
)
n,m1(
x
)
(
n
2
)
n,m1(
x
)
Thus, we get the required recurrence relation
)]
(
2
)
(
)[
1
(
)
(
)
2
1
)(
1
(
)
(
)
(
n
m
n,m1x
m
x
n,mx
x
x
n/,mx
m
n,m1x
The other consequences follow easily from the above recurrence relation.
Lemma 2.2. For
m
N
and sufficiently largen
N
, there holds the following recurrence relation
(
t
x
)
,
k
,
x
n
( 1)
Q
(
m
,
k
,
x
)
o
(
1
)
B
n
m
k
197 |
P a g e
where
Q
(
m
,
k
,
x
)
is a certain polynomial in x of degree m andx
[
0
,
)
is arbitrary but fixed.Proof. Using Lemma 2.1, for sufficiently large n, we can write
(
)
,
(
)
[( 1(
1)/)
2]1...
[ /2](
)
...
]2 / ) 1 [(
0
m mm m
m n
n
x
q
n
x
q
n
x
q
x
x
t
B
where
q
i(
x
)
,i
0
, 1, 2, 3, … are certain polynomials in x of degree at most m.
Now, we have
k k k
k m
k m
k
k m
m
k m
m
m n
d
d
d
n
d
x
q
n
d
x
q
d
d
d
n
d
x
q
n
d
x
q
d
d
d
n
d
x
q
n
d
x
q
x
k
x
t
B
...
...
)
(
)
(
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
)
(
)
(
...
...
)
(
)
(
1
,
,
)
(
2 1 1
] 2 / ) 1 [(
1 ]
2 / ) 1 [( 0
1 2
1 1 1 1
] 2 / ) 1 [( 1
1 ]
2 / ) 1 [( 1
0
0 2
0 1 0 1
] 2 / ) 1 [( 0
1 ]
2 / ) 1 [( 0
0
=
n
(k1)
Q
(
m
,
k
,
x
)
o
(
1
)
,for each fixedx
[
0
,
)
. This completes the proof of the Lemma 2.2.Lemma 2.3. For
m
N
0, if the mth order moment for the operators (1.1) be defined asm
v v n m
n
x
n
v
x
b
n
x
U
2
1
)
(
)
1
(
1
)
(
1 ,
, ,
then, we have
U
n,0(
x
)
1
,U
n,1(
x
)
0
and there holds the recurrence relation]
)
(
)
(
)[
1
(
)
(
)
2
(
n
U
n,m1x
x
x
U
n/,mx
mU
n,m1x
. Consequently, for eachx
[
0
,
)
, we have from this relation that
[( 1)/2]
,
(
)
mm
n
x
O
n
U
,where [
] denotes the integral part of
.Proof. Using the definition of
U
n,m(
x
)
and basic properties ofb
n,v(
x
)
, we obtain1
)
x
(
U
n,0
andU
n,1(
x
)
0
.Now, we have
m
v
v n m
n
x
n
v
x
b
x
x
n
x
U
x
x
2
1
)
(
)
1
(
)
1
(
1
)
(
)
1
(
1
198 |
P a g e
11 ,
2
1
)
(
)
1
(
1
)
1
(
m
v v
n
x
n
v
m
x
b
n
x
x
Thus using basic properties of
b
n,v(
x
)
, we get
m
v
v n m
n m
n
x
n
v
x
b
x
n
v
n
x
mU
x
U
x
x
2
1
)
(
)
2
(
)
1
(
)
1
(
1
)
(
)
(
)
1
(
1
, 1
, ,
1
1 ,
2
1
)
(
)
1
(
)
2
(
m
v v
n
x
n
v
x
b
n
n
(
n
2
)
U
n,m1(
x
)
This completes the proof of the recurrence relation.
The other consequence follows easily from the recurrence relation.
Lemma 2.4([6]). There exist the polynomials
Q
i,j,r(
x
)
independent of n and v such that)
x
(
b
)
x
(
Q
]
x
)
2
n
(
)
1
v
[(
)
2
n
(
)
x
(
b
D
)]
x
1
(
x
[
j i,j,r n,v0 j , i
r j i 2
i v
, n r
r
,
where
D
r is the rth order differentiation operator.Lemma 2.5. If C( j, k ) , j = 0, 1, 2, …, k are defined as in (1.3), then we have
k
m
m
d
k
j
C
k
j
m
j
0
for
1
,
2
,
...,
0
for
1
)
,
(
0
.
Proof. From relations (1.2) and (1.3) we get
k k k
k
k k
k k k
k m k
k m
k m
k
j
m j
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
k
j
C
....
...
1
...
....
...
...
...
...
...
....
...
...
...
...
....
...
1
....
...
1
....
...
...
....
...
...
...
...
...
....
...
...
...
...
....
...
....
...
)
,
(
2 1
1 2
1 1 1
0 2
0 1 0
2 1
1 2
1 1 1 1
0 2
0 1 0 0
0
which implies that
k
m
m
d
k
j
C
k
j
m
j
0
for
1
,
2
,
...,
0
for
1
)
,
(
0
.
III. MAIN RESULT
199 |
P a g e
Theorem: Let
r
N
. If the functionf
H
[
0
,
)
is bounded on every finite subinterval of[
0
,
)
admitting a derivative of order (2k+r+2) at a fixed pointx
(
0
,
)
, satisfyingf
(
t
)
O
(
t
)
ast
for some
0
, then we have
lim
[
(
,
,
)
(
)
]
(
)
(
,
,
,
)
2 r 2
1 r
) ( )
r ( )
r ( 1
x
r
k
i
Q
x
f
x
f
x
k
f
B
n
n
k
i i n
k
(3.1)
and
lim
1[
(r)(
,
1
,
)
(r)(
)
]
0
x
f
x
k
f
B
n
n
nk
,
(3.2)
where
Q
(
i
,
k
,
r
,
x
)
are certain polynomials in x.Moreover, if
f
(2kr2)exists and is continuous on(
a
,
b
)
(
0
,
)
where
0
, then (3.1) and (3.2) hold uniformly on[
a
,
b
]
.Proof. Since
f
(2kr2)exists atx
(
0
,
)
, therefore by Taylor’s expansion of f (t) , we have2 r 2 2
r 2
0 ) (
)
(
)
,
(
)
(
!
)
(
)
(
i kk
i i
x
t
x
t
x
t
i
x
f
t
f
where
(
t
,
x
)
0
ast
x
and
(
t
,
x
)
O
(
t
x
)
ast
.Using linearity of
B
n(r)(
,
k
,
x
)
, relation (1.3) and the above expansion of f (t), we get
)
(
,
,
)
(
!
)
(
)]
(
,
),
(
[
(r) (r)2 r 2
0 ) ( 1
) r ( )
r ( 1
x
f
x
k
x
t
B
i
x
f
n
x
f
x
k
t
f
B
n
n ik
i i k
n k
0
2 r 2 ,
1 ) r (
, 0
1
)
(
)
,
(
)
(
)
(
)
1
(
1
)
,
(
b
x
b
t
t
x
t
x
dt
n
d
k
j
C
n
dnv kv v n d j
k
j k
j
j
=J1+J2 (say)
Using Lemma 2.2, we obtain
)
(
,
,
)
(
!
)
(
(r) (r)2 r 2
0 ) ( 1
1
B
t
x
k
x
f
x
i
x
f
n
J
n ik
i i k
)
(
)
1
(
)
,
,
(
!
)
(
)
(
r ( 1) (r)2 r 2
1 r
) ( )
r ( 1
x
f
o
x
k
i
Q
n
D
i
x
f
x
f
n
kk
i i k
(
)
(
,
,
r,
)
(
1
)
2r 2
1 r
) (
o
x
k
i
Q
x
f
k
i
i
200 |
P a g e
Let
0 2 r 2 , 1 ) r ( , 121
(
)
(
)
(
,
)(
)
)
1
(
1
dt
x
t
x
t
t
b
x
b
n
n
J
nv kv v n
k
.
Then, using Lemma 2.4 and Schwarz inequality for summation, we obtain
1 v v , n r r , j , i j 0 j , i r j i 2 i 1 k21
b
(
x
)
)]
x
1
(
x
[
)
x
(
Q
x
)
2
n
(
)
1
v
(
)
1
n
(
)
2
n
(
n
J
0 2 r k 2 v ,n
(
t
)
(
t
,
x
)
t
x
dt
b
0 2 r 2 , 1 , 0 j i, r 2 1)
,
(
)
(
)
(
)
2
(
)
1
(
)
1
(
)
2
(
)
(
v
n
x
b
x
b
t
t
x
t
x
dt
n
n
x
C
n
nv kv v n j j i i k
2 / 1 1 , 2 0 j i, r 2 1)
(
)
2
(
)
1
(
)
1
(
1
)
2
(
)
(
x
b
x
n
v
n
n
x
C
n
v v n j j i i k 2 / 1 1 2 0 2 r 2 , ,(
)
(
)
(
,
)
)
1
(
1
v k v n vn
x
b
t
t
x
t
x
dt
b
n
,where r r , j , i
)]
x
1
(
x
[
)
x
(
Q
]
b
,
a
[
x
sup
0
j
,
i
r
j
i
2
sup
)
x
(
C
.Since
(
t
,
x
)
0
ast
x
, therefore , for a given
0
there exists a
0
such that
(
t
,
x
)
whenever0
t
x
.Also, since
(
t
,
x
)
O
(
t
x
)
, therefore there exists a positive constant C1 such that
(
t
,
x
)
C
1t
x
for allt
x
.Now applying Schwarz inequality for integration and Lemma 2.1, we have
1 2 0 2 r 2 , ,211
(
)
(
)
(
,
)
)
1
(
1
v k v n vn
x
b
t
t
x
t
x
dt
b
n
J
0 4 r 2 4 2 , 1 0 , ,(
)
(
)
(
)
(
,
)
(
)
)
1
(
1
dt
x
t
x
t
t
b
dt
t
b
x
b
n
k v n v v n v n
b
x
b
t
t
x
t
x
dt
n
k v n x t x t v v n 4 r 2 4 2 , 1 ,(
)
(
)
(
,
)
(
)
)
1
(
1
201 |
P a g e
x t
k v
n x
t
k v
n v
v
n
x
b
t
t
x
dt
C
b
t
t
x
dt
b
n
4 2 r 2 4 ,
2 1 4 r 2 4 ,
2 1
,
(
)
(
)
(
)
(
)
(
)
)
1
(
1
0
2 ,
) 4 2 r 2 4 ( 2
2 1 0
4 r 2 4 ,
2 1
,
(
)
(
)
(
)
(
)
(
)
)
1
(
1
dt
x
t
t
b
C
dt
x
t
t
b
x
b
n
q v
n k
q k
v n v
v
n
2(
)
) 2 r 2 (
2 k q
n
O
C
n
O
,where q is a positive constant and the constant
C
2is dependent onq
,
k
,
r
,
and
. Next using Lemma 2.3, we get
1/22 ) 2 r 2 ( 2 2 /
0 j i,
r 2 1
21
(
)
(
2
)
(
)
(
)
q k
j
j i
j i k
n
O
C
n
O
n
O
n
x
C
n
J
2 (2 r 2)
2
1/20 j i, r 2
2 / ) 2 2 2 (
)
(
)
(
k qj i
k j i
n
O
C
n
O
n
O
x
C
C
(
x
)
O
n
(2kr2)/2
2O
n
(2kr2)
C
2O
(
n
q)
1/2
C
(
x
)
2O
1
C
2O
(
n
2kr2q)
1/2
C
(
x
)
O
1
, where q is chosen to be greater than 2k + r + 2.Thus, due to arbitrariness of
0
and by Lemma 2.5 it follows thatJ
2
0
asn
. Finally collecting the estimates of J1 and J2, assertion (3.1) follows.The assertion (3.2) can be proved easily in a similar manner noticing the fact that
( 2)
,
1
,
)
(
i
kn
t
x
k
x
O
n
B
fori
1
,
2
,
...
The last uniformity assertion follows from the fact that
(
)
in the proof of the assertion (3.1) can chosen to be independent ofx
[
a
,
b
]
and rest of the estimates hold uniformly on[
a
,
b
]
.This completes the proof of the theorem.
REFERENCES
[1] Gupta, V., “ Convergence of Szasz- Beta operators, ” Ann. Univ. Ferrara, 59, 311-318, 2013.
[2] Gupta, M. K., Gupta, V. and Kumar, M., “ Direct and inverse estimates for a new family of linear positive operators, ” J. Math. Anal. Appl., 330, 799-816, 2007.
[3] Gupta, V. and Yadav, R., “ Some approximation results on q-Beta-Szasz operators, ” Southeast Asian Bulletin of Mathematics, 36 (3), 343-352, 2012.
202 |
P a g e
[5] Micchelli, C. A., “ The saturation class and iterates of the Bernstein polynomials, ” J. Approx. Theory, 8, 1-18, 1973.
[6] Timan, A. F., “ Theory of Approximation of Functions of a Real Variable (English Translation), ” Dover Publications Inc., N.Y., 1994.
[7] Upreti, R., “ Approximation properties of Beta operators, ” J. Approx. Theory, 45, 85-89, 1985. [8] Zhou, D. X., “ Lp-Inverese theorems for Beta operators, ” J. Approx. Theory, 66, 279-287, 1991.
