SIMULTANEOUS APPROXIMATION
BY BETA-SZASZ OPERATORS
Manoj Kumar
Department of Applied Science,
Krishna Engineering College,
Mohan Nagar, Ghaziabad, Uttar Pradesh-201007,
India
ABSTRACT
The study of direct and inverse theorems remains an active area of research. This research now centers on approximation by members of various non-classical and nonlinear classes such as wavelets, shift-invariant subspaces, radial basis functions, ridge functions, neural nets, multivariate splines and the like. In the present paper, we study inverse approximation property of Beta-Szasz operators in simultaneous approximation.
Keywords-Simultaneous approximation, Summation-integral type operators, Linear positive
operators, Peetre’s K-functional.
MATHEMATICAL SUBJECT CLASSIFICATION:
41A25, 41A36.I. INTRODUCTION
In approximation theory, the results, which determine structural characteristics of functions from their
degree of approximation, are known as inverse theorems. There are many-many generalizations of
direct and inverse results [1, 4, 6, 7, 9, 11]. The excellent textbooks of Timan [10] and of DeVore and
Lorentz [3] contain an abundance of information on direct and inverse theorems for approximation by
algebraic and trigonometric polynomials. In the present paper we study inverse approximation
estimate for these operators, which were defined as
)
,
0
[
)
(
)
(
)
(
1
)
,
(
0 , 1
,
x
dt
t
f
t
s
x
b
n
n
x
f
B
nvv v n
n
(1.1)
where
f
C
[
0
,
)
{
f
C
[
0
,
)
:
f
(
t
)
Mt
for some
M
0
,
0
}
,
1
1 ,
)
1
)(
,
1
(
)
(
v n v vn
x
v
n
B
x
x
b
,
!
)
(
)
(
,v
nx
e
x
s
v nx v
n
and
B
(
n
1
,
v
)
n
!
(
v
1
)!
/(
n
v
)!.
It is easily checked that the operators
B
nare linear positive operators and it is obvious
that
B
n(
1
,
x
)
1
. Alternately the operators (1.1) may be written as
0
)
(
)
,
(
)
,
(
f
x
W
x
t
f
t
dt
where
(
)
(
)
1
)
,
(
,1
,
x
s
t
b
n
n
t
x
W
nvk v n
n
.As far as the rate of approximation is concerned the operators (1.1) are just like exponential type operators [8]. But the operators
B
n are not exponential type operators, since they do not satisfy the following condition:)
)(
,
(
)
(
)
,
(
W
x
t
t
x
x
P
n
t
x
W
x
n
n
,
P
(
x
)
is a function of x. (1.2)The above equation (1.2) is the necessary condition for the operators to be of exponential type. The above condition (1.2) is frequently used in the analysis to prove the inverse theorem for exponential type operators. In the present paper we study an inverse result in simultaneous approximation for the operators (1.1).
By
C
0, we denote the class of continuous functions on the interval(
0
,
)
having a compact support andr
C
0 is r times continuously differentiable functions withC
0C
0r
. Suppose
{
:
0 2,
)
(r
rC
g
g
G
supp]
,
[
a
b
g
, where[
a
,
b
]
(
a
,
b
)}
. For r times continuously differentiable functions f with supp f]
,
[
a
b
, the Peetre’s K-functional are defined as
( 2) [ , ]
] , [ ) ( ]
, [ ) ( ) (
) (
inf
)
,
,
,
(
b a C r b
a C r b
a C r r G g
r
f
a
b
f
g
g
g
K
r
,0
1
.II. PRELIMINARIES
This section consists of the following preliminary results, which will be helpful to prove the inverse approximation theorem in next section.
Lemma 2.1 [5]. For
m
N
{
0
}
, if the m-th order moment be defined asm k
v n m
n
x
n
v
x
b
n
x
U
2
1
)
(
1
1
)
(
1 ,
, , then
U
n,0(
x
)
1
,
U
n,1(
x
)
0
and)].
(
)
(
)[
1
(
)
(
)
2
(
n
U
n,m1x
x
x
U
n(1,m)x
mU
n,m1x
Consequently,U
n,m(
x
)
O
n
[(m1)/2]
.Lemma 2.2. Let the function
n,m(
x
),
m
N
0, be defined as
0 , 1
,
,
(
)
(
)(
)
1
)
(
b
x
s
t
t
x
dt
n
n
x
nv mv v n m
n
.Then
n
x
x
x
nn
)
1
(
2
)
(
,
1
)
(
,1 0,
and2
2 2
,
)
1
(
6
)
2
(
)
(
n
x
n
x
x
x
n
and there holds the recurrence relation
[( 1)/2]
,
(
)
mm
n
x
O
n
.Proof. The values of
n,0(
x
),
n,1(
x
)
easily follow from the definition. We prove the recurrence relation
1 0
, )
1 (
, )
1 (
,
(
1
)
(
)
(
)(
)
1
)
(
)
1
(
v
m v
n k n m
n
x
x
b
x
s
t
t
x
dt
n
n
x
x
x
dt
x
t
t
s
x
b
x
x
n
n
m
mv n v
k n
0
1 ,
1
,
(
)
(
)(
)
)
1
(
1
Now using the identities
x
(
1
x
)
b
n(1,v)(
x
)
((
v
1
)
(
n
2
)
x
)
b
n,v(
x
)
and)
(
]
[(
)
(
,) 1 (
,
t
v
nt
s
t
s
t
nv
nv , we obtain)]
(
)
(
)[
1
(
, 1) 1 (
,
x
m
x
x
x
nm
nm
1 0
,
,
(
)
(
)(
)
)
)
2
(
1
(
1
vm v
n v
n
x
s
t
t
x
dt
b
x
n
v
n
n
dt
x
t
t
s
x
x
t
n
nt
v
x
b
n
n
mv n v
v
n
(
)
[(
)
(
)
(
1
2
)]
(
)
(
)
1
1 0 ,,
b
x
t
s
t
t
x
dt
n
n
mv n v
v
n
(
)
(
)
(
)
1
0) 1 (
, 1
,
+
n
n,m1(
x
)
(
1
2
x
)
n,m(
x
)
b
x
s
t
t
x
dt
n
n
mv n v
v n
1
0 ) 1 (
, 1
,
(
)
(
)
(
)
1
+b
x
s
t
t
x
dt
n
nx
mv n v
v
n
(
)
(
)
(
)
1
0) 1 (
, 1
,
+
n
n,m1(
x
)
(
1
2
x
)
n,m(
x
)
.
(
m
1
)
n,m(
x
)
n
n,m1(
x
)
mx
n,m1(
x
)
(
1
2
x
)
n,m(
x
).
This completes the proof of recurrence relation. The values of
n,2(
x
)
,
n,m(
x
)
follow from therecurrence relation.
Lemma 2.3 [5]. There exist the polynomials
Q
i,j,r(
x
)
independent of n and k such that
{
(
1
)}
[
(
)]
(
2
)
(
1
(
2
)
)
, ,(
)
,(
)
0 , 2
,
x
n
v
n
x
Q
x
b
x
b
D
x
x
j i jr nvj i
r j i
i v
n r
r
, where
dx
d
D
.Lemma 2.4. Let 0<
<2 and0
a
a
a
b
b
b
. Iff
C
0 with suppf
[
a
,
b
]
and
/2
] , [ ) ( )
(
)
,
(
f
f
O
n
B
b a C r r
n , then
K
r(
,
f
)
M
5
n
/2n
K
r(
n
1,
f
)
Consequently
K
r(
,
f
)
M
6
/2,
M
2
0
.
Proof. It is sufficient to prove
(
,
)
)
,
(
f
M
5n
/2n
K
n
1f
K
r r
, for sufficiently large n.
Because supp f
[
a
,
b
]
, therefore by [8, Th.3.2], there exists a functionh
(i)
G
(r),
i
r
,
r
2
such that 1 6 ] , [ ) ( ) ()
,
(
f
h
M
n
B
b a C i i n Therefore ] , [ ) ( ) ( 17
(
,
)
3
)
,
(
b a C r r nr
f
M
n
B
f
f
K
( 2) [ , ]
] , [ ) (
)
,
(
)
,
(
b a C r n b a C rn
f
B
f
B
Next, it is sufficient to show that there exists a constant
M
6 such that for eachg
G
(r)
1 ( 2) [ , ]
] , [ ) ( ) ( 8 ] , [ ) 2 (
)
,
(
b a C r b a C r r b a C rn
f
M
n
f
g
n
g
B
(2.1)Also using linearity property, we have
] , [ ) 2 ( ] , [ ) 2 ( ] , [ ) 2 (
)
,
(
)
,
(
)
,
(
b a C r n b a C r n b a C rn
f
B
f
g
B
g
B
(2.2)
Applying Lemma 2.3, we get
2 0 1 , , 2 2 , , 0 , 2 1 0 2 2
)
1
)(
1
)...(
1
)(
(
)
(
)
(
)}
1
(
{
)
(
)
,
(
r n k n k n r r j i j j i r j i k i n r rx
r
n
n
n
dt
t
b
x
p
x
x
x
Q
nx
k
n
dt
t
x
W
x
Therefore by Schwarz inequality and Lemma 2.1, we obtain ] , [ ) ( ) ( 9 ] , [ ) 2 (
)
,
(
b a C r r b a C rn
f
g
M
n
f
g
B
(2.3)
where the constant
M
9 is independent of f and g. Next by Taylor’s expansion, we have,
)
(
)!
2
(
)
(
)
(
!
)
(
)
(
2 ) 2 ( 1 0 ) (
rr i r i i
x
t
r
g
x
t
i
x
g
t
g
where
lies between t and x. Using above expansion we get] , [ 2 2 2 ] , [ ) 2 ( ] , [ ) 2 (
)
)(
,
(
)!
2
(
1
)
,
(
b a C r n r r b a C r b a C rn
W
x
t
t
x
dt
x
g
r
g
B
(2.4)] , [ ) 2 ( 8 ] , [ )
2 (
)
,
(
b a C r b
a C r
n
g
M
g
B
(2.5)
Combining the estimates of (2.2)-(2.5), we get (2.1). The other consequence follows from [2].
This completes the proof of the lemma.
Lemma 2.5. Let
a
a
a
b
b
b
andf
(r)
C
0 with suppf
[
a
,
b
]
then if),
,
,
1
,
(
0
a
b
C
f
r
we have
f
(r)
Liz
(
,
1
,
a
,
b
).
Proof. Let
h
andg
G
(r), then forf
C
0(
,
1
,
a
,
b
)
r
)
(
)
(
)
(
2 ( ) ( ) 2 () )( 2
x
g
g
f
x
f
r r r r
2 109 ]
, [ ) 2 ( 2 ] , [ ) ( ) ( 2
)
,
(
4
2
f
g
g
M
K
rf
M
b a C r b
a C r
r
It follows that
f
Liz
(
,
1
,
a
,
b
)
i.e.f
Lip
(
,
a
,
b
).
III.MAIN RESULT
In this section, we shall prove our main result, namely, inverse approximation theorem, which is stated as Theorem: Let
0
2
,
0
a
1
a
2
b
2
b
1
and supposef
C
[
0
,
)
. Then in the following statements(
i
)
(
ii
)
(i)
/2
] , [ ) ( )
(
1 1
,.)
(
f
f
O
n
B
b a C r r
n
(ii)
f
(r)
Lip
(
,
a
2,
b
2).
Proof. Let us choose
a
,
a
,
b
,
b
in such a way thata
1
a
a
a
2
b
2
b
b
b
1. Also supposeg
C
0 with suppg
[
a
,
b
]
andg
(
x
)
1
on[
a
2,
b
2]
. Forx
[
a
,
b
]
withdx
d
D
, we have))
),
)(
(
)
)(
((
(
)
(
)
(
)
,
(
()) (
x
x
fg
t
fg
B
D
x
fg
x
fg
B
nr r
r
n
))
)),
(
)
(
)(
(
(
(
))
)),
(
)
(
)(
(
(
(
B
f
t
g
t
g
x
x
D
B
g
x
f
t
f
x
x
D
nr n
r
2 1
J
J
( say)By Leibnitz theorem, we have
0
1
W
(
x
,
t
)
f
(
t
)(
g
(
t
)
g
(
x
))
dt
x
J
r nr
0 ) ( 0
))]
(
)
(
)(
(
[
)
,
(
f
t
g
t
g
x
dt
x
t
x
W
i
r
i r
i r i
n r
0 ) ( )
( ) ( 1
0
))
(
)
(
)(
(
)
,
(
)
,
(
)
(
x
B
f
x
W
x
t
f
t
g
t
g
x
dt
g
i
r
rn i
n i r r
i
J
3
J
4, say. Following [8, Th. 3.3], we obtain)
(
)
(
)
(
() /2 )( 1
0 3
g
x
f
x
o
n
i
r
J
r i ir i
, uniformly in
x
[
a
,
b
]
Next following [8, Th. 3.2], Schwarz inequality, Taylor’s expansion of f and g and Lemma 2.2, we get
)
(
!
)!
(
!
)
(
)
(
1/21
) ( ) ( 4
r
o
n
i
r
i
x
f
x
g
J
r i
i r i
)
(
)
(
)
(
( ) /2) ( 1
g
x
f
x
o
n
i
r
i r ir i
, uniformly in
x
[
a
,
b
]
Finally applying Leibnitz theorem, we obtaindt
x
f
t
f
t
g
x
t
x
W
i
r
J
r ii r r
i
i
n
(
,
)
[
(
)(
(
)
(
))]
0 0 ) (
2
ri
r i
n i r
x
fg
x
f
B
x
g
i
r
0
) ( )
( ) (
)
(
)
(
)
,
(
)
(
)
(
)
(
)
(
)
(
)
(
( ) /20
) ( )
(
g
x
f
x
fg
x
o
n
i
r
rr i
i i r
)
(
/2
O
n
, uniformly inx
[
a
,
b
]
.Combining the estimates of
J
1,J
2 ,J
3andJ
4 , we get
/2
] , [ ) ( )
(
)
(
,.)
(
fg
O
n
fg
B
b a C r r
n .
Thus by Lemma 2.4 and Lemma 2.5, we have
(
fg
)
(r)
Lip
(
,
a
,
b
)
, sinceg
(
x
)
1
on[
a
2,
b
2]
, it follows thatf
(r)
Lip
(
,
a
2,
b
2)
. This proves implication(
i
)
(
ii
)
for the case0
1
.Now to prove the implication for
1
2
, for any interval[
a
1,
b
1]
(
a
1,
b
1)
and leta
2,
b
2 be such that(
2,
2)
(
2,
2)
a
b
b
a
and(
2,
2)
(
1,
1)
b
a
b
a
. Let
0
we shall prove the assertion for
2
. From the previous case it implies thatf
(r) exists and belongs toLip
(
1
,
a
1,
b
1)
.Let
g
C
0 be such thatg
(
x
)
1
on[
a
2,
b
2]
and suppg
(
a
2,
b
2)
. Then for characteristic function
2(
t
)
of the interval[
1,
1]
b
a
, we have] , [ ]
, [ ) ( )
(
2 2 2
2
(.),.)
)
(
(.)(
(
[
)
(
,.)
(
b a C n
r b
a C r r
n
fg
fg
D
B
g
f
t
f
B
] , [ 2 2
(.)),.)
)
(
)(
(
(
[
b a C n
r
2 1
I
I
, say.Following [5, Th. 3.3], we have
] , [ ) ( 1 2 2
)
(
),.)]
(
)
(
(
[
b a C r n mfg
t
f
x
g
B
D
I
] , [ ) ( ) ( ) ( 0 2 2)
(
,.)
(
b a C r i n i r ifg
f
B
g
i
r
)
(
)
(
/2] , [ ) ( ) ( ) ( 0 2 2
g
f
fg
O
n
i
r
b a C r i i r r i)
(
/2
O
n
.Next following [8, Th. 3.2] and Leibnitz theorem, we have
] , [ 2 ) ( ) ( ) ( 1 0 2 2 2
),.)
(
(.))
)
(
)(
(
(
,.)
(
b a C r n i n i r r it
g
t
g
t
f
B
f
B
g
i
r
I
3 4 [ , ]
(
1)
2 2
I
I
CabO
n
(say).Again following [8, Th. 3.3], we get
)
(
)
(
)
(
() /2 ) ( 1 0 3
g
x
f
x
O
n
i
r
I
r i ir i
, uniformly in
[
2,
2]
a
b
x
.Applying Taylor’s expansion of f, we have
I
W
nr(
x
,
t
)[
f
(
t
)(
g
(
t
)
g
(
x
))
2(
t
)]
dt
0 ) (
4
W
x
t
t
x
g
t
g
x
t
dt
i
x
f
r in r p i i
)
(
))
(
)
(
(
)
)(
,
(
!
)
(
2 0 ) ( 1 0 ) (
dt
t
x
g
t
g
x
t
r
x
f
f
t
x
W
r r r rn
(
)
(
(
)
(
))
(
)
!
)
(
)
(
(
)
,
(
2 ) ( ) ( 0 ) (
(
lying between t and x )
I
5
I
6, say.Next following [5, Th. 3.2], we get
)
(
))
(
)
(
(
)
)(
,
(
!
)
(
1 0 ) ( 0 ) ( 5
W
x
t
t
x
g
t
g
x
dt
O
n
i
x
f
I
nr ir i
i
( uniformly in
[
2,
2]
a
b
x
)
I
7
O
(
n
1)
(say) Sinceg
C
0, therefore we can writedt
x
t
t
x
W
m
x
g
i
x
f
I
nr i mdt
x
t
x
t
t
x
W
i
x
f
r i rn r
i i
2
0 ) ( 0
) (
)
)(
,
(
)
,
(
!
)
(
( where
(
t
,
x
)
0
ast
x
)
I
8
I
9 ( say)Following [5, Th. 3.2], we get
)
(
!
)!
(
)
(
!
)
(
( ) 11 ) ( 8
r
O
n
m
r
x
f
m
x
g
I
m r r
m m
(
)
(
)
(
1)
0
) ( )
(
g
x
f
x
O
n
m
r
r m
m r m
. Also
I
9
O
(
n
/2)
uniformly inx
[
a
2,
b
2]
.Finally using mean value theorem and Lemma 2.3, we obtain
0
1 ,
,
0 , 2 ] , [
6
(
,
)
)}
1
(
{
)
(
2 2
r n
r r s m s
m r s m
s m b
a
C
W
x
t
t
x
x
x
x
Q
n
I
] , [ 2
) ( )
(
2 2
)
(
)
(
!
)
(
)
(
b a C r
r
dt
t
g
r
x
f
f
O
(
n
/2)
where
is chosen in such a way that0
2
. Finally combining the above estimates we get
/2
] , [ ) ( )
(
2 2
)
(
,.)
(
fg
fg
O
n
B
b a C r r
n .
Since supp
fg
(
a
2,
b
2)
, it follows from Lemma 2.4 and Lemma 2.5 that)
,
,
(
)
(
2 2)
(
b
a
Lip
fg
r
. Furthermore, sinceg
(
x
)
1
on[
a
2,
b
2]
, we have(
,
2,
2)
)(
b
a
Lip
f
r
.This completes the proof of the theorem.
REFERENCES
[1] P. N. Agrawal and V. Gupta, “ Inverse theorem for linear combinations of modified Bernstein polynomials, ” Pure and Applied Math Sci. 40, 29-38, 1995.
[2] H. Berens and G. G. Lorentz, “ Inverse theorems for Bernstein polynomials, ” Indiana University Mathematics Journal, 21, 693-708, 1972.
[3] R. A. De-Vore and G. G. Lorentz, “ Constructive Approximation, ” Spriger-Verlag, Berlin, 1993.
[4] Z. Finta and V. Gupta, “ Direct and inverse estimates for Phillips type operators, ” J. Math Anal Appl. 303(2), 627-642, 2005.
[6] V. Gupta, P. Maheshwari and V. K. Jain, “ Lp-inverse theorem for modified beta operators, ” Int. J. Math.
Sci., 33(20), 1395-1403, 2003.
[7] H. S. Kasana, P. N. Agrawal and V. Gupta, “ Inverse and saturation theorems for linear combination of modified Baskakov operators, ” Approx Theory and its Appl. 7 (2), 65-81, 1991.
[8] C. P. May, “ Saturation and inverse theorems for combinations of a class of exponential type operators, ” Canad. J. Math. 28, 1224-1250, 1976.
