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Rate of Convergence for the Modified Szász–Mirakyan Operators on Functions of Bounded Variation
Journal of Mathematical Analysis and Applications 233, 476᎐483 Ž1999. Article ID jmaa.1999.6289, available online at http:rrwww.idealibrary.com on
Rate of Convergence for the Modified Szasz᎐Mirakyan ´ Operators on Functions of Bounded Variation Vijay Gupta Department of Mathematics, Institute of Engineering and Technology, M.J.P. Rohilkhand Uni¨ ersity, Bareilly 243 006, Uttar Pradesh, India
and R. P. Pant Department of Mathematics, Kumaon Uni¨ ersity, D.S.B. Campus, Nainital 263 002, Uttar Pradesh, India Submitted by H. M. Sri¨ asta¨ a Received December 23, 1997
In this paper we obtain an estimate of the rate of convergence of modified Szasz᎐Mirakyan operators on functions of bounded variation. Our result essen´ tially improves the results due to A. Sahai and G. Prasad Ž1993, Publ. Inst. Math. Ž Beograd. Ž N.S.. 53, 73᎐80. and V. Gupta and P. N. Agrawal Ž1991, Publ. Inst. Math. Ž Beograd. Ž N.S.. 49, 97᎐103.. 䊚 1999 Academic Press
1. INTRODUCTION The modified Szasz᎐Mirakyan operator w4x is defined as ´ Mn Ž f , x . s n
⬁
pk Ž nx .
Ý ks0
⬁
H0
pk Ž nt . f Ž t . dt,
x g 0, ⬁ . ,
Ž 1.1.
where k
pk Ž nx . s eyn x Ž nx . rk!. Gupta and Agrawal w3x estimated the rate of convergence for the operator Ž1.1. for functions of bounded variation. Recently, Sahai and Prasad w5x 476 0022-247Xr99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.
477
RATE OF CONVERGENCE
improved and corrected the results of w3x. They have taken the function to be of growth e ␣ t , ␣ ) 0. The improved estimate obtained by Sahai and Prasad w5x is not entirely correct. The aim of this paper is to improve and correct the results of Sahai and Prasad w5x and Gupta and Agrawal w3x. The main result obtained by Sahai and Prasad w5x is THEOREM A. Let f be a function of bounded ¨ ariation on e¨ ery finite subinter¨ al of w0, ⬁. and let f Ž t . s O Ž e ␣ t . for some ␣ ) 0 as t ª ⬁. If x g Ž0, ⬁., then for n sufficiently large Mn Ž f , x . y F
1
f Ž xq. q f Ž xy. 4
2
Ž x 2 q 6 x q 3 . xy2 n
n
'k Ž g . Ý V xx qy xr x xr 'k
ks1
Ž 4 x q 3 x q 1 . xy1r2 2
q
f Ž xq . y f Ž xy .
'n
q O Ž 1 . ey␣ x
Ž 2 x q 1 . xy2 n
,
where Vab Ž g x . is the total ¨ ariation of g x on w a, b x and gx
¡f Ž t . y f Ž x q. , ¢f Ž t . y f Ž xy. ,
' g Ž t . s ~0, x
if x - t - ⬁ if t s x if 0 F t - x.
Ž 1.2 .
2. AUXILIARY RESULTS To prove our main result, we shall need the following lemmas: LEMMA 2.1.
For e¨ ery x g Ž0, ⬁., we ha¨ e
pk Ž nx . F
Ž x.
'n
, where
Ž x. s
32 x 2 q 24 x q 5 2'x
.
Proof. Let i 4 be a sequence of independent and identically distributed random variables all having the same Poisson Ž x . distribution. Let n s Ý nks1 k ; then yn x
P Ž n s k . s e
Ž nx . k!
k
s pk Ž nx . .
478
GUPTA AND PANT
Also,
2s x,  3 s E < 1 y x < 3 F E Ž 13 . q 3 xE Ž 12 . q 3 x 2 E Ž 1 . q x 3 s 8 x 3 q 6 x 2 q x. Next, pk Ž nx . s P Ž k y 1 - n F k . s P
ž
k y 1 y nx
'nx
-
n y nx
'nx
F
k y nx
'nx
/
.
By w1, pp. 104 and 110; 2x, we have pk Ž nx . y
1
Ž kynx .r 'nx
'2
yt 2 r2
HŽ ky1ynx .r'nx e
dt - 2 Ž 0.82. -
8 x2 q 6 x q 1
'nx
16 x 2 q 12 x q 2
'nx
.
Now 1
kynx r 'nx '2 HŽ ky1ynx .r'nx e Ž
.
yt 2 r2
dt -
1
-
'2 nx
1 2'nx
.
Therefore 16 x 2 q 12 x q 2
pk Ž nx . F
'nx
q
1
s
2'nx
Ž x.
'n
.
Remark. We observe that the constant 0.41 taken in w5x holds only for sufficiently large n. For all n, we should take it 0.82 as given in w1, 2x. For n G 2, we have 2 xrn F MnŽŽ t y x . 2 , x . F Ž2 x q 1.rn. If K nŽ x, t . s nÝ⬁ks 0 pk Ž nx . pk Ž nt ., then it is easy to verify that Ži. for 0 F y - x, we have y
H0
K n Ž x, t . dt F
2xq1 nŽ x y y .
Ž 2.1.
2
Žii. for x - z - ⬁, we have ⬁
Hz
K n Ž x, t . dt F
2xq1 nŽ z y x .
2
.
Ž 2.2.
479
RATE OF CONVERGENCE
3. MAIN RESULT THEOREM 3.1 . Let f be a function of bounded ¨ ariation on e¨ ery finite subinter¨ al of w0, ⬁. and let f Ž t . s O Ž e ␣ t . for some ␣ ) 0 as t ª ⬁. If x g Ž0, ⬁. and n G 4␣ , then Mn Ž f , x . y F
1 2
f Ž xq. q f Ž xy. 4
Ž x 2 q 6 x q 3. nx 2
n
'k Ž g . Ý V xx qy xr x xr 'k
ks1
Ž 32 x q 24 x q 5 . 2
q
f Ž xq . y f Ž xy .
2'nx
(
q
2 Ž 2 x q 1. e 2 ␣ x n
x
q
e ␣ x Ž 2 x q 1. nx 2
,
Ž 3.1.
where Vab Ž g x . is the total ¨ ariation of g x on w a, b x as gi¨ en in Ž1.2.. Proof. First we have Mn Ž f , x . y
1 2
f Ž xq. q f Ž xy. 4
F Mn Ž g x , x . q
1 2
f Ž xq . y f Ž xy . ⭈ Mn Ž sign Ž t y x . , x . . Ž 3.2.
Thus, to estimate the left hand side, we need estimates for MnŽ g x , x . and MnŽsignŽ t y x ., x .. We have ⬁
Mn Ž sign Ž t y x . , x . s
H0
s
Hx
⬁
sign Ž t y x . K n Ž x, t . dt K n Ž x, t . dt y
x
H0
K n Ž x, t . dt
s A n Ž x . y Bn Ž x . , say. Proceeding as in the proof of the theorem in w5x, we have A n Ž x . y Bn Ž x . F
Ž x.
'n
.
480
GUPTA AND PANT
To estimate MnŽ g x , x ., we decompose w0, ⬁. into three parts, as follows: Mn Ž g x , x . s s s
⬁
H0
K n Ž x, t . g x Ž t . dt ⬁
xqxr 'n qH H0xyxr'n q Hxyxr xqxr 'n 'n
ž žH
I1
q
HI q HI 2
3
/
/
K n Ž x, t . g x Ž t . dt
K n Ž x, t . g x Ž t . dt s E1 q E2 q E3 , say.
First we estimate E2 . For t g I2 , we have xqxr 'n g x Ž t . s g x Ž t . y g x Ž x . F Vxyx r 'n Ž g x . ,
and so xqxr 'n E2 F Vxyx r 'n Ž g x .
xqxr 'n K Hxyxr 'n
n
xqxr 'n Ž x, t . dt s Vxyx r 'n Ž g x . H
xqxr 'n
xyxr 'n
d t n Ž x, t . ,
where
n Ž x, t . s
t
H0 K
n
Ž x, u . du.
Since Hab d t nŽ x, t . F 1 for all w a, b x : w0, ⬁., we have xqxr 'n E2 F Vxyx r 'n Ž g x . F
n
1 n
xqxr 'k Ý Vxyx r 'k Ž g x . .
Ž 3.3.
ks0
Next, using Ž2.1. and proceeding as in w5x, we have E1 F
2 Ž 2 x q 1. nx 2
n x Ý Vxyx r 'k Ž g x . .
ks1
Finally, we estimate E3 . Setting z s x q xr 'n , we obtain E3 s
⬁
Hz
g x Ž t . K n Ž x, t . dt s
⬁
Hz
g x Ž t . d t Ž n Ž x, t . . .
We define Q nŽ x, t . on w0, 2 x x as Q n Ž x, t . s
½
1 y n Ž x, ty . , 0,
if 0 F t - 2 x if t s 2 x.
Ž 3.4.
481
RATE OF CONVERGENCE
Therefore E3 s
2x
Hz
g x Ž t . d t Ž Q n Ž x, t . . y g x Ž 2 x .
⬁
H2 x K
n
Ž x, u . du
⬁
H2 x g Ž t . d Ž Ž x, t . .
q
x
t
n
s E31 q E32 q E33 , say.
Ž 3.5.
Now using Ž2.2. and proceeding as in w5x, we have < E31 < F
2 Ž 2 x q 1. nx 2
n
Ý Vxxqx r'k Ž g x .
Ž 3.6.
ks1
and < E32 < F
Ž 2 x q 1. nx 2
n
Ý Vxxqx r'k Ž g x . .
Ž 3.7.
ks1
Finally, we estimate E33 as follows: ⬁
< E33 < ' n
pk Ž nx .
Ý
k
x
H2 x p Ž nt . Ž e
␣t
ks0 ⬁
Fn
pk Ž nx .
Ý ks0
s F
⬁
n
pk Ž nx .
Ý
x
ks0
n
⬁
x
q
pk Ž nx .
Ý ks0
e␣ x x
⬁
H2 x p Ž nt . g Ž t . dt ⬁
k
⬁
H2 x p Ž nt . xe k
⬁
H2 x < t y x < e
⬁
n
2
Ý
pk Ž nx .
ks0
␣t
q e ␣ x . dt
␣t
dt q
e␣ x x
2
⬁
n
Ý ks0
pk Ž nx .
⬁
H2 x p Ž nt . x k
2
dt
pk Ž nt . dt
⬁
H2 x p Ž nt . ⭈ Ž t y x .
2
k
dt
Ž because for t G 2 x, t y x G x . F
n x
q F
⬁
1 x
pk Ž nx .
Ý ks0
e␣ x x
2
⬁
H0
⬁
n
Ý
pk Ž nx .
ks0 ⬁
ž
n
< t y x < e ␣ t pk Ž nt . dt
Ý ks0
pk Ž nx .
⬁
H0
⬁
H0
2
pk Ž nt . ⭈ Ž t y x . dt 1r2 2
pk Ž nt . Ž t y x . dt
/
482
GUPTA AND PANT ⬁
= n
ž
F
1 x
q
ž
Ý
pk Ž nx .
ks0
1r2
⬁
pk Ž nt . ⭈ e
H0
2
Mn Ž Ž t y x . , x .
e␣ x
1r2
/
2␣t
dt
q
/
⬁
ž
n
Ý
pk Ž nx .
ks0
⬁
H0
e␣ x x
2
2
Mn Ž Ž t y x . , x . 1r2
pk Ž nt . e 2 ␣ t dt
/
2
Mn Ž Ž t y x . , x . .
x2
Next, we have by partial integration and by the assumption that n ) 2 ␣ ⬁
n
Ý
pk Ž nx .
ks0
sn
⬁
H0
pk Ž nt . e 2 ␣ t dt
⬁
Ý
pk Ž nx .
ks0
sn
⬁
Ý
pk Ž nx .
ks0
s s s
n n y 2␣ n n y 2␣ n n y 2␣
yn x
e
nk
⬁
H k! 0
t k eyŽ ny2 ␣ .t dt
nk
k!
k! Ž n y 2 ␣ . kq1 ⬁
k
n2 x
ž
s
Ý ks0 Ž n y 2 ␣ .
/
n
⬁
n y 2␣
ks0
Ý
ž
k
n n y 2␣
/
pk Ž nx .
1 k!
⭈ eyn x ⭈ e n xŽ1q2 ␣ rŽ ny2 ␣ .. ⭈ e 2 ␣ n x rŽ ny2 ␣ . s
F 2 e 4 ␣ x , for Therefore < E33 < F s
n n y 2␣
⭈ e 2 ␣ x⭈n rŽ ny2 ␣ .
n G 4␣ .
1 2xq1 x
ž
(
n
1r2
/
⭈ Ž 2 e 4␣ x .
2 Ž 2 x q 1. e 2 ␣ x n
x
q
1r2
q
e␣ x 2 x q 1 x2
e␣ x 2 x q 1 x2
n
n .
Ž 3.8.
Using Ž3.5. to Ž3.8., we have, for n G 4␣ , < E3 < F
3 Ž 2 x q 1. nx 2 q
e
␣x
n
Ý Vxxqx r'k Ž g x . q ks1
Ž 2 x q 1. nx 2
.
(
2 Ž 2 x q 1. e 2 ␣ x n
x
Ž 3.9.
RATE OF CONVERGENCE
483
Combining Ž3.2. with Ž3.3., Ž3.4. and Ž3.9., we get the required result. Remark. We may remark that the estimate Ž3.9. in w5x seems to be incorrect. In the estimation of E33 in w5x, the term 1 q ␣rm4 2 kq1 cannot be considered constant.
ACKNOWLEDGMENTS The first author is thankful to U.G.C. for research support under Unassigned Grant, Group D, Minor Research Project 1996᎐1997. The authors are also thankful to the referee for suggesting improvements in the paper.
REFERENCES 1. R. N. Bhattacharya and R. R. Rao, ‘‘Normal Approximation and Asymptotic Expansions,’’ Wiley, New York, 1976. 2. Y. S. Chow and H. Teicher, ‘‘Probability Theory,’’ Springer-Verlag, New York, 1978. 3. V. Gupta and P. N. Agrawal, An estimate of the rate of convergence for modified Szasz-Mirakyan operators of functions of bounded variation, Publ. Inst. Math. Ž Beograd. ´ Ž N.S.. 49 No. 63 Ž1991., 97᎐103. 4. H. S. Kasana, G. Prasad, P. N. Agrawal, and A. Sahai, Modified Szasz ´ operators, in ‘‘Proceedings, International Conference on Mathematical Analysis and Its Applications, Kuwait,’’ pp. 29᎐41, Pergamon, Oxford, 1985. 5. A. Sahai and G. Prasad, On the rate of convergence for modified Szasz᎐Mirakyan ´ operators on functions of bounded variation, Publ. Inst. Math. Ž Beograd. Ž N.S.. 53, No. 67 Ž1993., 73᎐80.
Rate of Convergence for the Modified Szasz᎐Mirakyan ´ Operators on Functions of Bounded Variation Vijay Gupta Department of Mathematics, Institute of Engineering and Technology, M.J.P. Rohilkhand Uni¨ ersity, Bareilly 243 006, Uttar Pradesh, India
and R. P. Pant Department of Mathematics, Kumaon Uni¨ ersity, D.S.B. Campus, Nainital 263 002, Uttar Pradesh, India Submitted by H. M. Sri¨ asta¨ a Received December 23, 1997
In this paper we obtain an estimate of the rate of convergence of modified Szasz᎐Mirakyan operators on functions of bounded variation. Our result essen´ tially improves the results due to A. Sahai and G. Prasad Ž1993, Publ. Inst. Math. Ž Beograd. Ž N.S.. 53, 73᎐80. and V. Gupta and P. N. Agrawal Ž1991, Publ. Inst. Math. Ž Beograd. Ž N.S.. 49, 97᎐103.. 䊚 1999 Academic Press
1. INTRODUCTION The modified Szasz᎐Mirakyan operator w4x is defined as ´ Mn Ž f , x . s n
⬁
pk Ž nx .
Ý ks0
⬁
H0
pk Ž nt . f Ž t . dt,
x g 0, ⬁ . ,
Ž 1.1.
where k
pk Ž nx . s eyn x Ž nx . rk!. Gupta and Agrawal w3x estimated the rate of convergence for the operator Ž1.1. for functions of bounded variation. Recently, Sahai and Prasad w5x 476 0022-247Xr99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.
477
RATE OF CONVERGENCE
improved and corrected the results of w3x. They have taken the function to be of growth e ␣ t , ␣ ) 0. The improved estimate obtained by Sahai and Prasad w5x is not entirely correct. The aim of this paper is to improve and correct the results of Sahai and Prasad w5x and Gupta and Agrawal w3x. The main result obtained by Sahai and Prasad w5x is THEOREM A. Let f be a function of bounded ¨ ariation on e¨ ery finite subinter¨ al of w0, ⬁. and let f Ž t . s O Ž e ␣ t . for some ␣ ) 0 as t ª ⬁. If x g Ž0, ⬁., then for n sufficiently large Mn Ž f , x . y F
1
f Ž xq. q f Ž xy. 4
2
Ž x 2 q 6 x q 3 . xy2 n
n
'k Ž g . Ý V xx qy xr x xr 'k
ks1
Ž 4 x q 3 x q 1 . xy1r2 2
q
f Ž xq . y f Ž xy .
'n
q O Ž 1 . ey␣ x
Ž 2 x q 1 . xy2 n
,
where Vab Ž g x . is the total ¨ ariation of g x on w a, b x and gx
¡f Ž t . y f Ž x q. , ¢f Ž t . y f Ž xy. ,
' g Ž t . s ~0, x
if x - t - ⬁ if t s x if 0 F t - x.
Ž 1.2 .
2. AUXILIARY RESULTS To prove our main result, we shall need the following lemmas: LEMMA 2.1.
For e¨ ery x g Ž0, ⬁., we ha¨ e
pk Ž nx . F
Ž x.
'n
, where
Ž x. s
32 x 2 q 24 x q 5 2'x
.
Proof. Let i 4 be a sequence of independent and identically distributed random variables all having the same Poisson Ž x . distribution. Let n s Ý nks1 k ; then yn x
P Ž n s k . s e
Ž nx . k!
k
s pk Ž nx . .
478
GUPTA AND PANT
Also,
2s x,  3 s E < 1 y x < 3 F E Ž 13 . q 3 xE Ž 12 . q 3 x 2 E Ž 1 . q x 3 s 8 x 3 q 6 x 2 q x. Next, pk Ž nx . s P Ž k y 1 - n F k . s P
ž
k y 1 y nx
'nx
-
n y nx
'nx
F
k y nx
'nx
/
.
By w1, pp. 104 and 110; 2x, we have pk Ž nx . y
1
Ž kynx .r 'nx
'2
yt 2 r2
HŽ ky1ynx .r'nx e
dt - 2 Ž 0.82. -
8 x2 q 6 x q 1
'nx
16 x 2 q 12 x q 2
'nx
.
Now 1
kynx r 'nx '2 HŽ ky1ynx .r'nx e Ž
.
yt 2 r2
dt -
1
-
'2 nx
1 2'nx
.
Therefore 16 x 2 q 12 x q 2
pk Ž nx . F
'nx
q
1
s
2'nx
Ž x.
'n
.
Remark. We observe that the constant 0.41 taken in w5x holds only for sufficiently large n. For all n, we should take it 0.82 as given in w1, 2x. For n G 2, we have 2 xrn F MnŽŽ t y x . 2 , x . F Ž2 x q 1.rn. If K nŽ x, t . s nÝ⬁ks 0 pk Ž nx . pk Ž nt ., then it is easy to verify that Ži. for 0 F y - x, we have y
H0
K n Ž x, t . dt F
2xq1 nŽ x y y .
Ž 2.1.
2
Žii. for x - z - ⬁, we have ⬁
Hz
K n Ž x, t . dt F
2xq1 nŽ z y x .
2
.
Ž 2.2.
479
RATE OF CONVERGENCE
3. MAIN RESULT THEOREM 3.1 . Let f be a function of bounded ¨ ariation on e¨ ery finite subinter¨ al of w0, ⬁. and let f Ž t . s O Ž e ␣ t . for some ␣ ) 0 as t ª ⬁. If x g Ž0, ⬁. and n G 4␣ , then Mn Ž f , x . y F
1 2
f Ž xq. q f Ž xy. 4
Ž x 2 q 6 x q 3. nx 2
n
'k Ž g . Ý V xx qy xr x xr 'k
ks1
Ž 32 x q 24 x q 5 . 2
q
f Ž xq . y f Ž xy .
2'nx
(
q
2 Ž 2 x q 1. e 2 ␣ x n
x
q
e ␣ x Ž 2 x q 1. nx 2
,
Ž 3.1.
where Vab Ž g x . is the total ¨ ariation of g x on w a, b x as gi¨ en in Ž1.2.. Proof. First we have Mn Ž f , x . y
1 2
f Ž xq. q f Ž xy. 4
F Mn Ž g x , x . q
1 2
f Ž xq . y f Ž xy . ⭈ Mn Ž sign Ž t y x . , x . . Ž 3.2.
Thus, to estimate the left hand side, we need estimates for MnŽ g x , x . and MnŽsignŽ t y x ., x .. We have ⬁
Mn Ž sign Ž t y x . , x . s
H0
s
Hx
⬁
sign Ž t y x . K n Ž x, t . dt K n Ž x, t . dt y
x
H0
K n Ž x, t . dt
s A n Ž x . y Bn Ž x . , say. Proceeding as in the proof of the theorem in w5x, we have A n Ž x . y Bn Ž x . F
Ž x.
'n
.
480
GUPTA AND PANT
To estimate MnŽ g x , x ., we decompose w0, ⬁. into three parts, as follows: Mn Ž g x , x . s s s
⬁
H0
K n Ž x, t . g x Ž t . dt ⬁
xqxr 'n qH H0xyxr'n q Hxyxr xqxr 'n 'n
ž žH
I1
q
HI q HI 2
3
/
/
K n Ž x, t . g x Ž t . dt
K n Ž x, t . g x Ž t . dt s E1 q E2 q E3 , say.
First we estimate E2 . For t g I2 , we have xqxr 'n g x Ž t . s g x Ž t . y g x Ž x . F Vxyx r 'n Ž g x . ,
and so xqxr 'n E2 F Vxyx r 'n Ž g x .
xqxr 'n K Hxyxr 'n
n
xqxr 'n Ž x, t . dt s Vxyx r 'n Ž g x . H
xqxr 'n
xyxr 'n
d t n Ž x, t . ,
where
n Ž x, t . s
t
H0 K
n
Ž x, u . du.
Since Hab d t nŽ x, t . F 1 for all w a, b x : w0, ⬁., we have xqxr 'n E2 F Vxyx r 'n Ž g x . F
n
1 n
xqxr 'k Ý Vxyx r 'k Ž g x . .
Ž 3.3.
ks0
Next, using Ž2.1. and proceeding as in w5x, we have E1 F
2 Ž 2 x q 1. nx 2
n x Ý Vxyx r 'k Ž g x . .
ks1
Finally, we estimate E3 . Setting z s x q xr 'n , we obtain E3 s
⬁
Hz
g x Ž t . K n Ž x, t . dt s
⬁
Hz
g x Ž t . d t Ž n Ž x, t . . .
We define Q nŽ x, t . on w0, 2 x x as Q n Ž x, t . s
½
1 y n Ž x, ty . , 0,
if 0 F t - 2 x if t s 2 x.
Ž 3.4.
481
RATE OF CONVERGENCE
Therefore E3 s
2x
Hz
g x Ž t . d t Ž Q n Ž x, t . . y g x Ž 2 x .
⬁
H2 x K
n
Ž x, u . du
⬁
H2 x g Ž t . d Ž Ž x, t . .
q
x
t
n
s E31 q E32 q E33 , say.
Ž 3.5.
Now using Ž2.2. and proceeding as in w5x, we have < E31 < F
2 Ž 2 x q 1. nx 2
n
Ý Vxxqx r'k Ž g x .
Ž 3.6.
ks1
and < E32 < F
Ž 2 x q 1. nx 2
n
Ý Vxxqx r'k Ž g x . .
Ž 3.7.
ks1
Finally, we estimate E33 as follows: ⬁
< E33 < ' n
pk Ž nx .
Ý
k
x
H2 x p Ž nt . Ž e
␣t
ks0 ⬁
Fn
pk Ž nx .
Ý ks0
s F
⬁
n
pk Ž nx .
Ý
x
ks0
n
⬁
x
q
pk Ž nx .
Ý ks0
e␣ x x
⬁
H2 x p Ž nt . g Ž t . dt ⬁
k
⬁
H2 x p Ž nt . xe k
⬁
H2 x < t y x < e
⬁
n
2
Ý
pk Ž nx .
ks0
␣t
q e ␣ x . dt
␣t
dt q
e␣ x x
2
⬁
n
Ý ks0
pk Ž nx .
⬁
H2 x p Ž nt . x k
2
dt
pk Ž nt . dt
⬁
H2 x p Ž nt . ⭈ Ž t y x .
2
k
dt
Ž because for t G 2 x, t y x G x . F
n x
q F
⬁
1 x
pk Ž nx .
Ý ks0
e␣ x x
2
⬁
H0
⬁
n
Ý
pk Ž nx .
ks0 ⬁
ž
n
< t y x < e ␣ t pk Ž nt . dt
Ý ks0
pk Ž nx .
⬁
H0
⬁
H0
2
pk Ž nt . ⭈ Ž t y x . dt 1r2 2
pk Ž nt . Ž t y x . dt
/
482
GUPTA AND PANT ⬁
= n
ž
F
1 x
q
ž
Ý
pk Ž nx .
ks0
1r2
⬁
pk Ž nt . ⭈ e
H0
2
Mn Ž Ž t y x . , x .
e␣ x
1r2
/
2␣t
dt
q
/
⬁
ž
n
Ý
pk Ž nx .
ks0
⬁
H0
e␣ x x
2
2
Mn Ž Ž t y x . , x . 1r2
pk Ž nt . e 2 ␣ t dt
/
2
Mn Ž Ž t y x . , x . .
x2
Next, we have by partial integration and by the assumption that n ) 2 ␣ ⬁
n
Ý
pk Ž nx .
ks0
sn
⬁
H0
pk Ž nt . e 2 ␣ t dt
⬁
Ý
pk Ž nx .
ks0
sn
⬁
Ý
pk Ž nx .
ks0
s s s
n n y 2␣ n n y 2␣ n n y 2␣
yn x
e
nk
⬁
H k! 0
t k eyŽ ny2 ␣ .t dt
nk
k!
k! Ž n y 2 ␣ . kq1 ⬁
k
n2 x
ž
s
Ý ks0 Ž n y 2 ␣ .
/
n
⬁
n y 2␣
ks0
Ý
ž
k
n n y 2␣
/
pk Ž nx .
1 k!
⭈ eyn x ⭈ e n xŽ1q2 ␣ rŽ ny2 ␣ .. ⭈ e 2 ␣ n x rŽ ny2 ␣ . s
F 2 e 4 ␣ x , for Therefore < E33 < F s
n n y 2␣
⭈ e 2 ␣ x⭈n rŽ ny2 ␣ .
n G 4␣ .
1 2xq1 x
ž
(
n
1r2
/
⭈ Ž 2 e 4␣ x .
2 Ž 2 x q 1. e 2 ␣ x n
x
q
1r2
q
e␣ x 2 x q 1 x2
e␣ x 2 x q 1 x2
n
n .
Ž 3.8.
Using Ž3.5. to Ž3.8., we have, for n G 4␣ , < E3 < F
3 Ž 2 x q 1. nx 2 q
e
␣x
n
Ý Vxxqx r'k Ž g x . q ks1
Ž 2 x q 1. nx 2
.
(
2 Ž 2 x q 1. e 2 ␣ x n
x
Ž 3.9.
RATE OF CONVERGENCE
483
Combining Ž3.2. with Ž3.3., Ž3.4. and Ž3.9., we get the required result. Remark. We may remark that the estimate Ž3.9. in w5x seems to be incorrect. In the estimation of E33 in w5x, the term 1 q ␣rm4 2 kq1 cannot be considered constant.
ACKNOWLEDGMENTS The first author is thankful to U.G.C. for research support under Unassigned Grant, Group D, Minor Research Project 1996᎐1997. The authors are also thankful to the referee for suggesting improvements in the paper.
REFERENCES 1. R. N. Bhattacharya and R. R. Rao, ‘‘Normal Approximation and Asymptotic Expansions,’’ Wiley, New York, 1976. 2. Y. S. Chow and H. Teicher, ‘‘Probability Theory,’’ Springer-Verlag, New York, 1978. 3. V. Gupta and P. N. Agrawal, An estimate of the rate of convergence for modified Szasz-Mirakyan operators of functions of bounded variation, Publ. Inst. Math. Ž Beograd. ´ Ž N.S.. 49 No. 63 Ž1991., 97᎐103. 4. H. S. Kasana, G. Prasad, P. N. Agrawal, and A. Sahai, Modified Szasz ´ operators, in ‘‘Proceedings, International Conference on Mathematical Analysis and Its Applications, Kuwait,’’ pp. 29᎐41, Pergamon, Oxford, 1985. 5. A. Sahai and G. Prasad, On the rate of convergence for modified Szasz᎐Mirakyan ´ operators on functions of bounded variation, Publ. Inst. Math. Ž Beograd. Ž N.S.. 53, No. 67 Ž1993., 73᎐80.