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ON APPROXIMATION BY MODIFIED SZASZ OPERATORS
Suresh P. Singh
(received 23 January 1984, revised 20 February 1986)
1.
Introduction
In 1950, 0. Szasz introduced a sequence of positive linear operators
{S }
n
defined as
(S/)(x)
=
*'”* I
fc=o
\ )
•
J”€ C-([0 ,-))
which is a generalization of the Bernstein polynomials
(B/)(x) =
to the infinite interval
lQ
’
f
{B^} ,
(1 .1 )
namely
€ C([0, 1 ]) ,
(1 .2)
[0 ,<=°) .
To obtain approximation results by positive linear operators for
integrable functions, Kantorovic modified the polynomials
(1 .2 )
by
substituting
f(fe+l)/(n+l)
f(t)dt
(n+ 1 )
*k/ (n+i)
for
f{k/n ) .
Another polynomial
=
(n+1)
I
{B*}
defined as
[jj] xk (l-x ) ” ^ ! 1 | " J tk { l - t f - kftt)dt
(1.3)
appear, due to Derriennic [l] .
Here we define a sequence of positive linear operators of order
Math. Chronicle 15(1986), 39-48 .
39
n
as
(1.4)
cV ) (x)
where
This sequence of operators
(1.2)
(1.4)
modifies the operators
(1.1)
interval
[0 ,“) .
In the present paper we study some approximation theorems on the
derivative of the function by the corresponding
operators
(1.4) .
and polynomials
2.
and
and approximates the Lebesgue integrable functions defined over the
r-th
r-th
derivative of the
We also compare to the results of Szasz operators (1.1)
(1.3) .
Preliminary Lemmas
In this section we give some lemmas which are useful in proving the
main results.
Lemma 2.1.
n > r , r = 0 , 1,
For
2,
....
one obtains
( 2 . 1)
Proof.
n > 1 ,
We note that for
differentiable function of
respect to
x
the expression
(L^f)(x) ,
=
71
=
71
n
I *
k=o
(1.4)
On differentiating
and setting
— - (L /)(*)
(L^mx)
i .
we get
(*)
’
40
is an infinitely
(1.4) ,
r-times
with
Now using
( nr ( n t )k+r +
I
r nr { n t £ \ e -nt
k + r!
k I
=
J
dtr
n >k+r
we get
(L^f)(x)
n
=
n
I * Ax) f
k=0 n,K
Jo dt
U
Again on integrating the right hand side of
(t))t-l)r f(t)dt .
(2.2)
(2.2)
by parts and using the
limit,
lim (-1 )k [nt)k e'nt
=
0 ,
(fe>0) ,
(2.3)
we get
( ^ r)/)(x)
=
n
l * k( x ) \ « t t ) £ ^ d t .
k=o
Lemma 2.2.
1o
’
dt
’
Let
rn > m .r (x)
-
(m-r)! K=o
’
1o
*
(2.4)
Then we have the following relation,
m-r+l
777+ 1 J
n,m-r +1
(2.5)
(m+1) T
(x) + 2mx 21
(x) + x T
(x)
n,m-rK
n,m-r-\y J
n,m-rK J
with
41
Tn t0W
=
r• •
In particular,
Tn
I
Proof.
fx)
n,2v '
On differentiating
=
-
(r+ 1 ) (r+ 1 ) !/n ,
(2 .6 )
tr* 2)! ( 2* > ( ^ 1 ) ^ - 2 ) 1
2n
(2.4)
n
I
with respect to
j
x ,
we get
xT'
Or)
n,m-r
n — m-' -
=
Now using
x f
=
' 7,(x) f «f> t,
(t) (t-x)m r<it - mx T
n-m-r-1
J0
x
^ix ) (.^~n x ) >
n,m-r
£,(x )
=
we again get that
(x)
n — —— ) iji 7 (x) [ { [k-r) -nt}$ ,
( t ) (t-x)m ~rdt
( m - r ) ! fc^ 0 "» fc
Jq
n,k-r
T
(x) - r r
{x) - m x T
(x) ,
j n,m-r+i
n,m-r
n,m-r- 1
+ w I
1
=
I
( m - r ) ! k t 0 n >k
m!
r .
-v I . .
/■*•»/■* .'"-i’ jx
m-r+1
£ *
(x)
* ♦
(t)(t-x)
dt ♦ n
(m -r) ! ^ fJ n >K
J0
rz,fc+r
I m+1
n ---- _
...
T
n,m-r +l
(x) - r T
n,m-r
(x) - m x T
Now on integrating and using the limit
42
(2.3)
n,m-r +l
we get
(x) .
(x)
(x)
x T’
n,m-r
T
r T
n,m-r
(x) - m x T
n,m-r -l
n,m-r+l
(x) ...
(x) .
This completes the proof.
Lemma 2.3.
f € C([0,°°)) ;
Let
ax + 6n
,
0 < 6 < 1 ,
/(x) = Oie**) , a > 0 , x
«
and
then
(2 . 8)
0 (e'6”) , B > 0
where
B
Proof.
3.
depends on
f , x , 6
and
a .
The proof is similar to the proof of lemma
2
of Hermann [3] .
Main Results
In this section we prove some approximation theorems.
Theorem 3.1.
If
f
is an integrable function with the condition that
f{x) = 0(ea X ) , a > 0 , x -+ °
° and has an
point
x 6 [0 ,<=°) ,
order derivative at a
then
lim n { ( ^ / H * )
Proof.
(r+2)-th
- / (r)(x) |
=
r / (r+l)(x) + (x/(r+ 0 (x))' .
(3.1)
By the Taylor formula,
/ (r)(x) - f ir) (x)
(3.2)
43
T\(f,t,x)
where
0
t — *■x and
as
r)
is integrable and is of order
<*> .
0 (eot) , a > 0 , t — *■
Again for
e > 0
|r)(/,t,a:) | < e
Now applying
(2.1)
A > 0 ,
arbitrary,
on
if
(3.2)
there is
|*-r| - X ,
and using
(2.6)
X > 0
such that
x < A .
to
(2.7)
we get
( ^ r)/)(x) - f {T\ x )
=
( (r+l)/^r+1-) (x) )/n + {x/n + (r+ 1 ) (r+2 )/n2 }/(-r+2-) (x) + Rn p (x)
where
f°
°
00
Rn
=
*
We have to show that
^ *n
k=o
’
fe+r. ^ H t - x ) 2Ti(/,t,x)dt .
Jo
»
lim ni!
(x) = 0 .
n-><»
n,r
Clearly
=
and
e(x+(r+l)(r+ 2 )/n)
n i? r (^) = 0(e
),
for certain
nR
but
e
0 > 0 .
(x) S
n,rv
On combining we get
ex,
is arbitrary so we get
lim n R
(x) =
n-yoo
n,r
0
This completes the proof.
Remarks.
We mention the following results.
(i)
The following estimate on polynomials
{£*}
is due to Derriennic
[ 1] I f a function
Theorem.
f
is bounded and integrable in
second order derivative at a point
lim n[(fl*/)(x)-/(x)]
(ii)
and has
then
(l-2x)/'(x) + (l-x)x/"(x) .
The estimate on the variant of Voronoskaya type theorem [4]
n > r , r = 0 , 1 , 2 , ... ,
y(^+ 2 ) € C([ 0 ,=°)) ,
(1 .1 )
=
x € [0 ,1 ] ,
[0 ,1 ]
for
on the polynomials
is
| ( ^ r)/)(~) - / (r)(x)|
£
(r/n)|/(r+i:)(x)|
♦ (*n ^( x ) /n) |/(r+2) (x) | ,
where
K
(x)
Theorem 3.2
If
=
[(x/2) + (r/2/rT) + (r2 /4n) + (x+r 2 /4n )^2 (l+r/2^)] .
f
is an integrable function with the condition that
/(x) = 0 (ea x ) , a > 0
a point
x € [0 ,“
>) t
as
x — *■°
o and has an
r-th
order derivative at
then
lim ( L ^ f ) (x)
45
=
<ff{x)/dxr .
(3.3)
Proof.
By the Taylor formula,
fit)
=
f{x) + it-x)f'ix) + ( it-x)Z/2)f"ix) + ...
+ ... + ( it-x)r/rl )(cf‘
fix)/dxr ) +
where
lq(t-x) — *■0
T)(t) = 0(e
rtf'
) .
as
t — *• x
and
,
i] is an integrable function with
Now using lemma 2.3 , the proof follows as in Theorem 3.1
Remarks.
(i)
The following result on operators
I f the
Theorem.
f^ix)
point
r-th
= 0 (e<xX) ,
x = x0 ,
(ii)
is due to J.Grof [2] .
derivative of a function exists with condition that
a > 0 , x — ► 00 and i f
then
(1.1)
(S^f)ix)
f^
uniformly at
(x)
is continuous at the
x = xQ .
The following result on the polynomials
{B*}
is due to
Derriennic [l] .
Theorem.
If
f
is bounded and integrable in
order derivative at a point
x € [0 ,l] ,
n X Z (Bn ^ ft ^
Theorem 3.3.
Let
o f continuity o f
f € £7^r+1^ ( [0,°°))
.
Then for
[0, 1 ]
and has an
r-th
then
=
•
and let
; •)
be the modulus
n > 1 ,
\ ( ^ r)f)ix) - / (r)(x)|
(3.4)
<
where
Proof.
((r+ l)/n)|/(r+l)(x)| + (l/v/r T ) { / r W + (l/ 2 ) ^ ( x ) } a ) | / (r+ ) ;J _ j ,
K^ix) = 2x + (r+1 )(r+ 2 ) .
Following [5] , we write that
On applying
(2.1)
to
(3.5)
and using the inequality
l/’
-Joo - / " ‘
’m
£
i
{l+|i/-x|/6 }<i)(/^+1^ ;6 ) ,
we get that
| ( ^ r)/)(a0 - / (r)(x)|
<
|/(r+ 1 ) (x)| • |Z,Cr)(t-x)(x)
+
<
L
| [ U + . . .+ \ y - x \ /6 } d y
(*) ,
Jx
|/(r+ 0 (x)| • |L^r ) (t-x)(x)|
♦ (u(/(r+l); 5 ) ^ / L ^ ) (t-x) 2 (x) + L ^ \ t - x ) 2 (x)/ 25
With the help of
(2.6) , (2.7)
and choosing
5 = l/Sn ,
we get the
required result.
This complets the proof.
Remarks.
(i)
We mention the following estimate [4] on the Sz&sz operators,
| ( ^ r)/)(x) - / (r)(x)|
2
(ii)
polynomials
For
J |/(7,+ 1 ) (x)| ♦ ( K ^ ^ i x U / n )
/ € C ([0,1)) ,
lMT)
.
we find the following estimate on
{B*} , n > 3 ,
I (*#)(*) -/(x)|
s
(i/'wi/h.j,.
(2 * i i ] A „ ( r
47
.-I.) .
Acknowledgement:
The author is thankful to Professor O.P. Varshney,
University of Roorkee, U.P., India for his valuable suggestions.
REFERENCES
1.
M.M. Derriennic, Sur I’approximation de fonctions integrables
sur
[0,l]
par des polyndmes de Bernstein modifies, J. Approx.
Theory 31 (1981), 325-343 .
2.
J. Grof, Otto file operator approximaos tulajdonsagairol , MTA 111 ,
Oszt. Kazl. 20 (1971), 35-44 .
3.
T. Hermann, On the Szasz Mirakian operators, Acta. Math. Acad. Sci
Hungar. 32 (1978), 163-173 .
4.
S.P. Singh, On the degree o f approximation by Szasz operators,
Bull. Austral. Math. Soc. 24 (1981), 221-225 .
5.
S.P. Singh and O.P. Varshney, A note on convergence of linear
positive operators, J. Approx, Theory 39 (1983), 386-388 .
University of Garhwal
Srinagar (U.P.)
INDIA 246174
48
1/--pages
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