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Bounds for Bernstein Basis Functions and Meyer–König and Zeller Basis Functions
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
219, 364]376 Ž1998.
AY975819
Bounds for Bernstein Basis Functions and Meyer]Konig ¨ and Zeller Basis FunctionsU Zeng Xiaoming† Department of Mathematics, Xiamen Uni¨ ersity, Xiamen, 361005, People’s Republic of China Submitted by Joseph A. Ball Received July 24, 1996
In this paper, the inequality estimates of Bernstein basis functions and Meyer]Konig ¨ and Zeller basis functions are studied. Exact bounds for these two basis functions are obtained. Moreover, some application results of the new estimates in estimating the rate of convergence of Durrmeyer operators and Meyer]Koning and Zeller operators for functions of bounded variation are also ¨ given. Q 1998 Academic Press
1. INTRODUCTION In approximation theory, it is important to estimate some basis functions of operators. There are a number of papers dealing with these estimates Žsee, for instance, w1, Chap. 1; 2]4x.. In the case of Bernstein basis functions Pn k Ž x . s Ž nk . x k Ž1 y x . ny k Ž0 F k F n, x g w0, 1x. and Meyer]Konig and Zeller basis functions Mn k Ž x . s Ž n q kk y 1 . x k Ž1 y x . n ¨ Ž k g N, x g w0, 1x., where N denote all nonnegative integers, for studying the degree of approximation of Durrmeyer operators and Meyer]Konig ¨ and Zeller operators, Guo w3, Lemma 2; 4x proved that
Pn k Ž x . F
5
1
2
'n 'x Ž 1 y x .
,
for 0 F k F n, x g Ž 0, 1 .
Ž 1.
* Research supported by the National Science Foundation and Fujian Provincial Science Foundation of China. † E-mail address: [email protected]. 364 0022-247Xr98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
BOUNDS FOR BASIS FUNCTIONS
365
and Mn k Ž x . F 33r Ž 'n x 3r2 . ,
for k g N, x g Ž 0, 1 . .
Ž 2.
Recently Vijay Gupta w5x improved Ž2. and obtained Mn k Ž x . F 76r Ž 15'n x 3r2 . ,
for k g N, x g Ž 0, 1 . .
Ž 3.
In this paper, we improve these inequalities giving the best coefficients by direct calculation and proving that the order ny1r2 in our estimates is the optimal also. We obtain the following: THEOREM 1. Let fixed j g N and C j s ŽŽ j q 1r2. jq1r2rj!. eyŽ jq1r2.. Then for all k, x such that j F k F n y j, x g Ž0, 1., there holds Pn k Ž x . - C j
1
'nx Ž 1 y x .
.
Ž 4.
Moreo¨ er, the coefficient C j is the best possible Ž that is to say, for arbitrary « ) 0, it cannot be replaced by C j y « ., and the estimate order ny1 r2 is the optimal also. In particular, for 0 F k F n, x g Ž0, 1., we ha¨ e Pn k Ž x . -
1
1
'2 e
'nx Ž 1 y x .
.
Ž 5.
Moreo¨ er, the coefficient 1r '2 e s 0.4288819 ??? is the best possible. We prove also, THEOREM 2.
For e¨ ery k g N, x g Ž0, 1x, we ha¨ e Mn k Ž x . -
1
1
'2 e 'nx
,
Ž 6.
where the coefficient 1r '2 e and the estimate order ny1 r2 in Ž6. are the best possible. In the last part of this paper, as applications of these inequalities, we shall give sharp estimates on the rate of convergence of Durrmeyer operators and Meyer]Konig ¨ and Zeller operators for bounded variation function.
366
ZENG XIAOMING
2. PROOFS OF THE RESULTS We first consider the proof of Theorem 1. For 0 F k F n, and x g w0, 1x, let f n k Ž x . s x Ž 1 y x . Pn k Ž x .. It is easy to verify that
'
fn k Ž x . F fn k
ž
k q 1r2 nq1
/
for x g w 0, 1 x
,
and fn k
ž
k q 1r2 nq1
/
s
n!
k q 1r2
ž
k! Ž n y k . !
nq1
kq 1r2
/ ž
n y k q 1r2 nq1
nykq1r2
/
.
Ž 7. We need to estimate f n k ŽŽ k q 1r2.rŽ n q 1... Noticing that the right hand side of Ž7. is symmetric for k and n y k, we need only deal with the case 0 F k F nr2. The following lemmas are useful for proving Theorem 1 and Theorem 2. For x ) 0, we ha¨ e
LEMMA 1.
2 2xq1
- ln 1 q
ž
1 x
/
-
1
.
'x Ž 1 q x .
Ž 8.
Proof. We first consider the left hand inequality of Ž8.. For t ) 0, X
Ž 1 q t . ln Ž 1 q t . y t s ln Ž 1 q t . ) 0, consequently t
X
Ž 2 q t . ln Ž 1 q t . y 2 t s ln Ž 1 q t . y
1qt
) 0.
Hence, for t ) 0,
Ž 2 q t . ln Ž 1 q t . y 2 t ) 0. The proof of the right hand inequality of Ž8. is similar. LEMMA 2. Ži. For all natural numbers n, we ha¨ e nq1 n Žii.
)
ž
n2 q 4 n q 4 n2 q 4 n q 3
nq 3
/
.
Ž 9.
For 0 F k F nr2, we ha¨ e
n y k q 3r2 nykq1
ž
n y k q 3r2 n y k q 1r2
ny kq1r2
/
)
nq3 nq3
ž
nq2 nq1
Ž nq1 .r2
/
. Ž 10 .
367
BOUNDS FOR BASIS FUNCTIONS
Proof. By Lemma 1, we get ln 1 q
ž
1 n
/
)
2 2n q 1
)
nq3
'Ž n
) Ž n q 3 . ln 1 q
ž
q 4 n q 3 . Ž n2 q 4 n q 4 .
2
1 n2 q 4 n q 3
/
.
It follows that nq1
)
n
ž
n2 q 4 n q 4 n2 q 4 n q 3
nq 3
/
and let hŽ y . s
n y y q 3r2 nyyq1
ž
n y y q 3r2 n y y q 1r2
ny yq1r2
/
.
After simple calculation and using Lemma 1, we get hX Ž y . s h Ž y .
ž
1 nyyq1
y ln
n y y q 3r2 n y y q 1r2
/
- 0,
for y - n q 1r2. Thus Ž10. holds. The proof is complete. Using Lemma 2 and Stirling’s formula, we prove: LEMMA 3. Ži.
For x g Ž0, 1., we ha¨ e
For e¨ ery k satisfying 0 F k F nr2, there holds Pn k Ž x . -
Žii.
Ž k q 1r2.
kq 1r2
eyŽ kq1r2.
k!
1
'n 'x Ž 1 y x .
.
Ž 11 .
For e¨ ery k satisfying nr2 F k F n, there holds ny kq1r2
Pn k Ž x . -
Ž n y k q 1r2. Ž n y k. !
eyŽ nykq1r2.
1
'n 'x Ž 1 y x .
, Ž 12 .
where the coefficient ŽŽ k q 1r2. kq 1r2rk!. eyŽ kq1r2. and ŽŽ n y k q 1r 2. ny kq1r2r Ž n y k .!. eyŽ nykq1r2. are the best possible. Proof. We first consider Ži.. From Ž7. fn k
ž
k q 1r2 nq1
/
s
Ž k q 1r2. k!
kq 1r2
n!
Ž n y k. !
Ž n y k q 1r2. Ž n q 1.
nykq1r2
nq1
'n 'n
368
ZENG XIAOMING
writing T Ž n, k . s
Ž n y k q 1r2.
n!
Ž n y k. !
Ž n q 1.
ny kq1r2
'n
nq 1
for 0 F k F nr2, by Lemma 2, we get T Ž n q 1, k . T Ž n, k .
s
s
nq1
n
nq2
nq2
( ž / ž ?
G
nq1
n y k q 3r2 n y k q 1r2
ny kq1r2
/
n y k q 3r2 nykq1 nq1
nq1
n
nq2
nq2
nq3
( ž / ž / ( ž / nq1
n2 q 4 n q 3
n
n2 q 4 n q 4
Ž nq1 .r2
?
nq1
nq3 nq2
Ž nq3 .r2
)1
and using Stirling’s formula n!s n n eyn '2p n e u n r12 n ,
0 - un - 1
we get lim T Ž n, k . s eyŽ kq1r2. .
Ž 13 .
nªq`
Hence, for x g Ž0, 1., we have Pn k Ž x . F
s
-
f n k Ž Ž k q 1r2 . r Ž n q 1 . .
'x Ž 1 y x . Ž k q 1r2.
kq 1r2
T Ž n, k .
k!
Ž k q 1r2. k!
1
'n 'x Ž 1 y x .
kq 1r2
eyŽ kq1r2.
1
'n 'x Ž 1 y x .
Ž 14 .
and from Ž13., we deduce that the coefficient ŽŽ k q 1r2. kq 1r2rk!. eyŽ kq1r2. in Ž14. is the best possible. By symmetry of Ž7. for k and n y k, Ž12. is obtained immediately. Proof of Theorem 1. Let H Ž k . s ŽŽ k q 1r2. kq 1r2rk!. eyŽ kq1r2.. It is easy to verify that H Ž k . is monotone decreasing. Therefore for fixed
369
BOUNDS FOR BASIS FUNCTIONS
j g N and C j s ŽŽ j q 1r2. jq1r2rj!. eyŽ jq1r2., when k satisfies j F k F n y j, from Ž11. and Ž12., we get Ž4., and by Lemma 3, we know that the coefficient C j in Ž4. is the best possible. For proving the estimate order ny1 r2 in Ž4. is the optimal, we take k s 0. From Ž14. we have Pn o Ž x . F
f n o Ž Ž 1r2 . Ž n q 1 . .
'x Ž 1 y x .
1
-
1
.
'2 e 'n 'x Ž 1 y x .
On the other hand, we can write fn o
ž
1r2 nq1
/
s
1
'2
ž
nq 1
1 nq1
Ž n q 1r2.
/
nq 1r2
na na
.
Since lim
nªq`
ž
1 nq1
nq 1
Ž n q 1r2.
/
nq 1r2
n a s q`,
as a )
1 2
it is clear that the estimate order ny1 r2 in Ž4. cannot be improved. It is of interest to note that lim H Ž k . s 1r'2p s 0.3989422 . . . .
kªq`
Hence for all j g N, there holds 1r'2p - C j F 1r'2 e .
Ž 15 .
Relation Ž15. gives an estimate to coefficient C j in Theorem 1 uniformly for all j g N. Proof of Theorem 2. For x g w0, 1x, let g n k Ž x . s x 1r2 Mn k Ž x .. It is easy to verify that gnk Ž x . F gnk
ž
k q 1r2 n q k q 1r2
/
,
for x g w 0, 1 x
and gnk
ž
k q 1r2 n q k q 1r2
/
s
Ž k q 1r2.
kq 1r2
k! =
Ž n q k q 1 . !n n Ž n y 1 . ! Ž n q k q 1r2.
nq kq1r2
?
'n 'n
.
370
ZENG XIAOMING
Let GŽ n, k . s Ž n q k y 1.!n nq 1r2rŽ n y 1.!Ž n q k q 1r2. nq kq1r2 . We can prove that GŽ n, k . is monotone increasing for n, and lim G Ž n, k . s eyŽ kq1r2. .
nªq`
Thus for every k g N, x g Ž0, 1., it follows that Mn k Ž x . F xy1 r2 g n k s
ž
k q 1r2 n q k q 1r2
Ž k q 1r2.
/
kq 1r2
1
G Ž n, k .
k!
'nx
1
-
1
'2 e 'n 'x
.
By the same reason as Theorem 1, the coefficient 1r '2 e and the estimate order ny1 r2 in Ž6. are the best possible.
3. SOME RELATED PROPOSITIONS From the above discussion, we find that the factor x Ž 1 y x . is very important for estimating Pn k Ž x . with all x g Ž0, 1. and 0 F k F n uniformly. Using the factor x Ž1 y x . instead of x Ž 1 y x . , can the result of Theorem 1 be improved? We will study this problem below. Let qn k Ž x . s x Ž1 y x . Pn k Ž x ., x g w0, 1x. It is easy to know that for every x g w0, 1x, there holds
'
'
qn k Ž x . F qn k
ž
kq1 nq2
/
s
n! k! Ž n y k . !
ž
kq1 nq2
kq 1
/ ž
nykq1 nq2
nykq1
/ Ž 16 .
when k s 0, qn0
ž
0q1 nq2
/
s
1
ž
nq1
nq1 nq2
nq 1
/
n n
.
w nrŽ n q 2.x ? wŽ n q 1.rŽ n q 2.x nq 1 is monotone increasing and tends to 1re. Hence for x g Ž0, 1., we have Pn o Ž x . F
qn o Ž 1r Ž n q 2 . . xŽ1 y x.
-
1
where the coefficient 1re is the best possible.
1
e nx Ž 1 y x .
,
371
BOUNDS FOR BASIS FUNCTIONS
Similarly, we can get Pn1 Ž x . -
2
2
ž / e
1 nx Ž 1 y x .
for x g Ž 0, 1 . .
,
For every k satisfying 0 F k F nr2, x g Ž0, 1., there holds Pn k Ž x . -
kq 1
Ž k q 1.
eyŽ kq1.
k!
1 nx Ž 1 y x .
;
Ž 17 .
and for every k satisfying nr2 F k F n, x g Ž0, 1., there holds ny kq1
Pn k Ž x . -
Ž n y k q 1. Ž n y k. !
eyŽ nykq1.
1 nx Ž 1 y x .
.
Ž 18 .
Remark 1. For every k satisfying nr2 - k F n, x g Ž0, 1., there still holds Pn k Ž x . -
Ž k q 1.
kq 1
eyŽ kq1.
k!
1 nx Ž 1 y x .
.
Ž 19 .
However, it is easy to verify that ny kq1
Ž n y k q 1. Ž n y k. !
eyŽ nykq1. -
Ž k q 1.
kq1
eyŽ kq1. ,
k!
for
n 2
- k F n.
Hence, inequality Ž18. is better than inequality Ž19.. In Ž17., we write
Ž k q 1. k! s
kq 1
eyŽ kq1.
Ž k q 1.
1 nx Ž 1 y x .
kq 1r2
k!
eyŽ kq1.
'k q 1 1 'n 'n x Ž 1 y x .
,
ŽŽ k q 1. kq 1r2rk!. eyŽ kq1. is monotone increasing and tends to 1r '2p . Now it is clearly that from Ž17. we have the following PROPOSITION 1.
For fixed 0 - d - 1r2, the inequality Pn k Ž x . -
1
'p
1 1r2q d
n
xŽ1 y x.
holds uniformly for all 0 F k F n1y 2 d and x g Ž0, 1..
Ž 20 .
372
ZENG XIAOMING
Furthermore, we point out that using factor x Ž1 y x . instead of in Theorem 1. By a similar method to proving Theorem 1, we get
'x Ž 1 y x . , we can improve the coefficient 1r '2 e
For all k satisfying 0 F k F n, x g Ž0, 1., there holds
PROPOSITION 2.
Pn k Ž x . -
1
1
,
'8p 'n x Ž 1 y x .
Ž 21 .
where the coefficient 1r '8p s 0.1994711 ??? is the best possible. Proof. For every x g w0, 1x, we have from Ž16.
'n x Ž 1 y x . Pn k Ž x . F s
'n n!
ž
k! Ž n y k . !
Ž k q 1.
kq1
/ ž
nq2
kq 1r2
?
k! ?
kq 1
Ž k q 1.
nykq1 nq2
1r2
nykq1
/
Ž n y k q 1.
1r2
nq2
'n n! Ž n y k q 1. ny kq1r2 Ž n y k . ! Ž n q 2.
nq 1
.
Ž 22 .
It is obvious that
Ž k q 1.
1r2
Ž n y k q 1.
1r2
nq2
F
1 2
for 0 F k F n.
,
Using a similar method to prove Theorem 1, we can prove that for k satisfying 0 F k F n, 'n n!Ž n y k q 1. ny kq1r2rŽ n y k .!Ž n q 2. nq1 is monotone increasing for n and tends to e yŽ kq1.. Hence, We get from Ž22., Pn k Ž x . F
1 Ž k q 1.
kq 1r2
eyŽ kq1.
k!
2
1
'n x Ž 1 y x .
.
Ž 23 .
Since ŽŽ k q 1. kq 1r2rk!. eyŽ kq1. is monotone increasing and tends to 1r '2p , Ž21. now follows from Ž23.. Moreover, by the facts that
Ž k q 1.
1r2
Ž n y k q 1.
1r2
nq2
s
1 2
,
as k s
n 2
and lim kªq`
Ž k q 1. k!
kq 1r2
eyŽ kq1. s
1
'2p
,
we deduce that the coefficient 1r '8p in Ž21. is the best possible.
373
BOUNDS FOR BASIS FUNCTIONS
On the other hand, we point out that the behavior of Mn k Ž x . is different from that of Pn k Ž x .. If we substitute x 3r2 for factor x 1r2 , we cannot improve the coefficient 1r '2 e in the inequality Ž6.. There still holds For all k g N, x g Ž0, 1x, we ha¨ e
PROPOSITION 3.
1
Mn k Ž x . -
1
'2 e 'n x 3r2 ,
Ž 24 .
where the coefficient 1r '2 e is the best possible. Proof. It is obvious that Ž24. holds from Theorem 2. Suppose the estimate coefficient 1r '2 e in Ž24. is not the best possible. Then, there exists an « satisfying 0 - « - 1r '2 e , such that Mn k Ž x . -
ž'
1
2e
y«
/'
1
for all k g N and x g Ž 0, 1 . Ž 25 .
,
n x 3r2
Taking x k s Ž k q 1r2.rŽ n q k q 1r2., k s 0, 1, 2, . . . , from Ž25., we have
ž
1r2
k q 1r2 n q k q 1r2 -
ž'
1
/
Mn k
y«
2e
ž
k q 1r2 n q k q 1r2
/
1
n q k q 1r2
n
k q 1r2
/' ž
/
.
Hence, as k ) nr« , we get
ž
1r2
k q 1r2 n q k q 1r2
/
ž
Mn k
k q 1r2 n q k q 1r2
-
/ ž'
1
2e
y
« 2
1
/'
n
.
It follows that
x
1r2
Mn k Ž x . F -
ž ž'
k q 1r2
n q k q 1r2 1
2e
y
« 2
1r2
/ 1
/'
n
Mn k ,
ž
k q 1r2 n q k q 1r2
/
for k ) nr« and all x g Ž 0, 1 x .
Ž 26 .
374
ZENG XIAOMING
However, according to Theorem 2, for the coefficient 1r '2 e y «r2, there exist a k 0 satisfying k 0 ) nr« and a x 0 g Ž0, 1., such that x 01r2 Mn k 0Ž x 0 . )
ž'
1
«
y
2e
2
1
/'
n
,
which is a contradiction with Ž26.. The proof is complete.
4. EXAMPLES OF APPLICATIONS In this section we give some applications of inequalities Ž5. and Ž24.. EXAMPLE 1. For Durrmeyer operators Dn w3x n
Dn Ž f , x . s Ž n q 1 .
1
Ý Pn k Ž x . H f Ž t . Pn k Ž t . dt, 0
ks0
where f g L1w0, 1x. By inequality Ž5. and proceeding along the lines of w3x, we give a sharp estimate as follows: THEOREM 4.1. Let f be a function of bounded ¨ ariation on w0, 1x, x g Ž0, 1.. Then for n sufficiently large, we ha¨ e Dn Ž f , x . y F
1 2
f Ž x q. q f Ž x y. 4
5Ž x Ž 1 y x . .
y1
n
n
Ý ks1
Ž q Ž 2 q 1r'8 e .
xq Ž1yx .r k
'
E
Ž gx .
'
xyxr k
xŽ1 y x. .
y1
f Ž x q. y f Ž x y. ,
'n
Ž 27 .
where g x Ž t . and Eba Ž g x . are defined as in the Theorem of w3x. It may be remarked that the estimate in the Theorem of w3x, i.e., Ž13r4.ŽŽ x Ž1 y x ..y1 r 'n .< f Ž x q . y f Ž x y .<, can be improved to Ž2 q 1r '8 e .ŽŽ x Ž1 y x ..y1 r 'n .< f Ž x q . y f Ž x y .<. EXAMPLE 2. For integrated Meyer]Konig ¨ and Zeller operators Mˆn w4x
ˆn Ž f , x . s Ž n q 1 . M
`
Ž kq1 .r Ž nqkq1 .
Ý Mn k Ž x . H
ks0
kr Ž nqk .
f Ž t . dt,
BOUNDS FOR BASIS FUNCTIONS
375
where f g L1 w0, 1x. By inequality Ž24. and proceeding along the lines of w4x, we get THEOREM 4.2. Let f be a function of bounded ¨ ariation on w0, 1x. Then for e¨ ery x g Ž0, 1. and n sufficiently large, we ha¨ e 1
ˆn Ž f , x . y M F
f Ž x q. q f Ž x y. 4 xq Ž1qx .r k
n
5 nx
2
E
Ý ks1
'
Ž gx .
'
xyxr k
q Ž 16 q 1r'2 e .
1
< < 'n x 3r2 f Ž x q. y f Ž x y. ,
Ž 28 .
where g x Ž t . and Eba Ž g x . are defined as in the Theorem of w4x. It may be remarked that the estimate in the Theorem of w4x, i.e., Ž49r 'n x 3r2 .< f Ž x q . y f Ž x y .< can be improved to ŽŽ16 q 1r '2 e .r 'n x 3r2 .< f Ž x q . y f Ž x y .<. EXAMPLE 3. For modified Meyer]Konig ¨ and Zeller operators Mn w5x Mn Ž f , x . s
`
Ý Mn , kq1Ž x . ks1
=
1
H0 M
ny 2, ky1
Ž n q k y 2. Ž n q k y 3. Ž n y 2.
Ž t . f Ž t . dt.
By inequality Ž24. and proceeding along the lines of w5x, the estimate term Ž114r15'n x 3r2 .< f Ž x q . y f Ž x y .< in Theorem 2.1 of w5x can be improved to Ž3r8e'n x 3r2 .< f Ž x q . y f Ž x y .< and the following result holds: THEOREM 4.3. Let f be a function of bounded ¨ ariation on w0, 1x. Then for e¨ ery x g Ž0, 1. and n sufficiently large, we ha¨ e 1
Mn Ž f , x . y F
7 nx
2 n
Ý ks1
f Ž x q. q f Ž x y. 4 xq Ž1yx .r k
E
xyxr 'k
'
Ž gx . q
3
'8 e x 3r2 f Ž x q. y f Ž x y.
ACKNOWLEDGMENT The author is thankful to the referee for his valuable comments.
. Ž 29 .
376
ZENG XIAOMING
REFERENCES 1. G. G. Lorentz, ‘‘Bernstein Polynomials,’’ 2nd ed., Chelsea, New York, 1986. 2. R. Bojanic and F. Cheng, Rate of convergence of Bernstein polynomials for functions with derivative of bounded variation, J. Math. Anal. Appl. 141 Ž1989., 136]151. 3. S. Guo, On the rate of convergence of the Durrmeyer operator for functions of bounded variation, J. Approx. Theory 51 Ž1987., 183]192. 4. S. Guo, On the rate of convergence of the integrated Meyer]Konig ¨ and Zeller operators for functions of bounded variation, J. Approx. Theory 56 Ž1989., 245]255. 5. V. Gupta, A sharp estimate on the degree of approximation to functions of bounded variation by certain operators. Approx. Theory Appl. 11, No. 3 Ž1995., 106]107. 6. A. Piriou and X. M. Zeng, On the rate of convergence of the Bernstein]Bezier operators, ´ C. R. Acad. Sci. Paris Ser. ´ I 321 Ž1995., 575]580.
219, 364]376 Ž1998.
AY975819
Bounds for Bernstein Basis Functions and Meyer]Konig ¨ and Zeller Basis FunctionsU Zeng Xiaoming† Department of Mathematics, Xiamen Uni¨ ersity, Xiamen, 361005, People’s Republic of China Submitted by Joseph A. Ball Received July 24, 1996
In this paper, the inequality estimates of Bernstein basis functions and Meyer]Konig ¨ and Zeller basis functions are studied. Exact bounds for these two basis functions are obtained. Moreover, some application results of the new estimates in estimating the rate of convergence of Durrmeyer operators and Meyer]Koning and Zeller operators for functions of bounded variation are also ¨ given. Q 1998 Academic Press
1. INTRODUCTION In approximation theory, it is important to estimate some basis functions of operators. There are a number of papers dealing with these estimates Žsee, for instance, w1, Chap. 1; 2]4x.. In the case of Bernstein basis functions Pn k Ž x . s Ž nk . x k Ž1 y x . ny k Ž0 F k F n, x g w0, 1x. and Meyer]Konig and Zeller basis functions Mn k Ž x . s Ž n q kk y 1 . x k Ž1 y x . n ¨ Ž k g N, x g w0, 1x., where N denote all nonnegative integers, for studying the degree of approximation of Durrmeyer operators and Meyer]Konig ¨ and Zeller operators, Guo w3, Lemma 2; 4x proved that
Pn k Ž x . F
5
1
2
'n 'x Ž 1 y x .
,
for 0 F k F n, x g Ž 0, 1 .
Ž 1.
* Research supported by the National Science Foundation and Fujian Provincial Science Foundation of China. † E-mail address: [email protected]. 364 0022-247Xr98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
BOUNDS FOR BASIS FUNCTIONS
365
and Mn k Ž x . F 33r Ž 'n x 3r2 . ,
for k g N, x g Ž 0, 1 . .
Ž 2.
Recently Vijay Gupta w5x improved Ž2. and obtained Mn k Ž x . F 76r Ž 15'n x 3r2 . ,
for k g N, x g Ž 0, 1 . .
Ž 3.
In this paper, we improve these inequalities giving the best coefficients by direct calculation and proving that the order ny1r2 in our estimates is the optimal also. We obtain the following: THEOREM 1. Let fixed j g N and C j s ŽŽ j q 1r2. jq1r2rj!. eyŽ jq1r2.. Then for all k, x such that j F k F n y j, x g Ž0, 1., there holds Pn k Ž x . - C j
1
'nx Ž 1 y x .
.
Ž 4.
Moreo¨ er, the coefficient C j is the best possible Ž that is to say, for arbitrary « ) 0, it cannot be replaced by C j y « ., and the estimate order ny1 r2 is the optimal also. In particular, for 0 F k F n, x g Ž0, 1., we ha¨ e Pn k Ž x . -
1
1
'2 e
'nx Ž 1 y x .
.
Ž 5.
Moreo¨ er, the coefficient 1r '2 e s 0.4288819 ??? is the best possible. We prove also, THEOREM 2.
For e¨ ery k g N, x g Ž0, 1x, we ha¨ e Mn k Ž x . -
1
1
'2 e 'nx
,
Ž 6.
where the coefficient 1r '2 e and the estimate order ny1 r2 in Ž6. are the best possible. In the last part of this paper, as applications of these inequalities, we shall give sharp estimates on the rate of convergence of Durrmeyer operators and Meyer]Konig ¨ and Zeller operators for bounded variation function.
366
ZENG XIAOMING
2. PROOFS OF THE RESULTS We first consider the proof of Theorem 1. For 0 F k F n, and x g w0, 1x, let f n k Ž x . s x Ž 1 y x . Pn k Ž x .. It is easy to verify that
'
fn k Ž x . F fn k
ž
k q 1r2 nq1
/
for x g w 0, 1 x
,
and fn k
ž
k q 1r2 nq1
/
s
n!
k q 1r2
ž
k! Ž n y k . !
nq1
kq 1r2
/ ž
n y k q 1r2 nq1
nykq1r2
/
.
Ž 7. We need to estimate f n k ŽŽ k q 1r2.rŽ n q 1... Noticing that the right hand side of Ž7. is symmetric for k and n y k, we need only deal with the case 0 F k F nr2. The following lemmas are useful for proving Theorem 1 and Theorem 2. For x ) 0, we ha¨ e
LEMMA 1.
2 2xq1
- ln 1 q
ž
1 x
/
-
1
.
'x Ž 1 q x .
Ž 8.
Proof. We first consider the left hand inequality of Ž8.. For t ) 0, X
Ž 1 q t . ln Ž 1 q t . y t s ln Ž 1 q t . ) 0, consequently t
X
Ž 2 q t . ln Ž 1 q t . y 2 t s ln Ž 1 q t . y
1qt
) 0.
Hence, for t ) 0,
Ž 2 q t . ln Ž 1 q t . y 2 t ) 0. The proof of the right hand inequality of Ž8. is similar. LEMMA 2. Ži. For all natural numbers n, we ha¨ e nq1 n Žii.
)
ž
n2 q 4 n q 4 n2 q 4 n q 3
nq 3
/
.
Ž 9.
For 0 F k F nr2, we ha¨ e
n y k q 3r2 nykq1
ž
n y k q 3r2 n y k q 1r2
ny kq1r2
/
)
nq3 nq3
ž
nq2 nq1
Ž nq1 .r2
/
. Ž 10 .
367
BOUNDS FOR BASIS FUNCTIONS
Proof. By Lemma 1, we get ln 1 q
ž
1 n
/
)
2 2n q 1
)
nq3
'Ž n
) Ž n q 3 . ln 1 q
ž
q 4 n q 3 . Ž n2 q 4 n q 4 .
2
1 n2 q 4 n q 3
/
.
It follows that nq1
)
n
ž
n2 q 4 n q 4 n2 q 4 n q 3
nq 3
/
and let hŽ y . s
n y y q 3r2 nyyq1
ž
n y y q 3r2 n y y q 1r2
ny yq1r2
/
.
After simple calculation and using Lemma 1, we get hX Ž y . s h Ž y .
ž
1 nyyq1
y ln
n y y q 3r2 n y y q 1r2
/
- 0,
for y - n q 1r2. Thus Ž10. holds. The proof is complete. Using Lemma 2 and Stirling’s formula, we prove: LEMMA 3. Ži.
For x g Ž0, 1., we ha¨ e
For e¨ ery k satisfying 0 F k F nr2, there holds Pn k Ž x . -
Žii.
Ž k q 1r2.
kq 1r2
eyŽ kq1r2.
k!
1
'n 'x Ž 1 y x .
.
Ž 11 .
For e¨ ery k satisfying nr2 F k F n, there holds ny kq1r2
Pn k Ž x . -
Ž n y k q 1r2. Ž n y k. !
eyŽ nykq1r2.
1
'n 'x Ž 1 y x .
, Ž 12 .
where the coefficient ŽŽ k q 1r2. kq 1r2rk!. eyŽ kq1r2. and ŽŽ n y k q 1r 2. ny kq1r2r Ž n y k .!. eyŽ nykq1r2. are the best possible. Proof. We first consider Ži.. From Ž7. fn k
ž
k q 1r2 nq1
/
s
Ž k q 1r2. k!
kq 1r2
n!
Ž n y k. !
Ž n y k q 1r2. Ž n q 1.
nykq1r2
nq1
'n 'n
368
ZENG XIAOMING
writing T Ž n, k . s
Ž n y k q 1r2.
n!
Ž n y k. !
Ž n q 1.
ny kq1r2
'n
nq 1
for 0 F k F nr2, by Lemma 2, we get T Ž n q 1, k . T Ž n, k .
s
s
nq1
n
nq2
nq2
( ž / ž ?
G
nq1
n y k q 3r2 n y k q 1r2
ny kq1r2
/
n y k q 3r2 nykq1 nq1
nq1
n
nq2
nq2
nq3
( ž / ž / ( ž / nq1
n2 q 4 n q 3
n
n2 q 4 n q 4
Ž nq1 .r2
?
nq1
nq3 nq2
Ž nq3 .r2
)1
and using Stirling’s formula n!s n n eyn '2p n e u n r12 n ,
0 - un - 1
we get lim T Ž n, k . s eyŽ kq1r2. .
Ž 13 .
nªq`
Hence, for x g Ž0, 1., we have Pn k Ž x . F
s
-
f n k Ž Ž k q 1r2 . r Ž n q 1 . .
'x Ž 1 y x . Ž k q 1r2.
kq 1r2
T Ž n, k .
k!
Ž k q 1r2. k!
1
'n 'x Ž 1 y x .
kq 1r2
eyŽ kq1r2.
1
'n 'x Ž 1 y x .
Ž 14 .
and from Ž13., we deduce that the coefficient ŽŽ k q 1r2. kq 1r2rk!. eyŽ kq1r2. in Ž14. is the best possible. By symmetry of Ž7. for k and n y k, Ž12. is obtained immediately. Proof of Theorem 1. Let H Ž k . s ŽŽ k q 1r2. kq 1r2rk!. eyŽ kq1r2.. It is easy to verify that H Ž k . is monotone decreasing. Therefore for fixed
369
BOUNDS FOR BASIS FUNCTIONS
j g N and C j s ŽŽ j q 1r2. jq1r2rj!. eyŽ jq1r2., when k satisfies j F k F n y j, from Ž11. and Ž12., we get Ž4., and by Lemma 3, we know that the coefficient C j in Ž4. is the best possible. For proving the estimate order ny1 r2 in Ž4. is the optimal, we take k s 0. From Ž14. we have Pn o Ž x . F
f n o Ž Ž 1r2 . Ž n q 1 . .
'x Ž 1 y x .
1
-
1
.
'2 e 'n 'x Ž 1 y x .
On the other hand, we can write fn o
ž
1r2 nq1
/
s
1
'2
ž
nq 1
1 nq1
Ž n q 1r2.
/
nq 1r2
na na
.
Since lim
nªq`
ž
1 nq1
nq 1
Ž n q 1r2.
/
nq 1r2
n a s q`,
as a )
1 2
it is clear that the estimate order ny1 r2 in Ž4. cannot be improved. It is of interest to note that lim H Ž k . s 1r'2p s 0.3989422 . . . .
kªq`
Hence for all j g N, there holds 1r'2p - C j F 1r'2 e .
Ž 15 .
Relation Ž15. gives an estimate to coefficient C j in Theorem 1 uniformly for all j g N. Proof of Theorem 2. For x g w0, 1x, let g n k Ž x . s x 1r2 Mn k Ž x .. It is easy to verify that gnk Ž x . F gnk
ž
k q 1r2 n q k q 1r2
/
,
for x g w 0, 1 x
and gnk
ž
k q 1r2 n q k q 1r2
/
s
Ž k q 1r2.
kq 1r2
k! =
Ž n q k q 1 . !n n Ž n y 1 . ! Ž n q k q 1r2.
nq kq1r2
?
'n 'n
.
370
ZENG XIAOMING
Let GŽ n, k . s Ž n q k y 1.!n nq 1r2rŽ n y 1.!Ž n q k q 1r2. nq kq1r2 . We can prove that GŽ n, k . is monotone increasing for n, and lim G Ž n, k . s eyŽ kq1r2. .
nªq`
Thus for every k g N, x g Ž0, 1., it follows that Mn k Ž x . F xy1 r2 g n k s
ž
k q 1r2 n q k q 1r2
Ž k q 1r2.
/
kq 1r2
1
G Ž n, k .
k!
'nx
1
-
1
'2 e 'n 'x
.
By the same reason as Theorem 1, the coefficient 1r '2 e and the estimate order ny1 r2 in Ž6. are the best possible.
3. SOME RELATED PROPOSITIONS From the above discussion, we find that the factor x Ž 1 y x . is very important for estimating Pn k Ž x . with all x g Ž0, 1. and 0 F k F n uniformly. Using the factor x Ž1 y x . instead of x Ž 1 y x . , can the result of Theorem 1 be improved? We will study this problem below. Let qn k Ž x . s x Ž1 y x . Pn k Ž x ., x g w0, 1x. It is easy to know that for every x g w0, 1x, there holds
'
'
qn k Ž x . F qn k
ž
kq1 nq2
/
s
n! k! Ž n y k . !
ž
kq1 nq2
kq 1
/ ž
nykq1 nq2
nykq1
/ Ž 16 .
when k s 0, qn0
ž
0q1 nq2
/
s
1
ž
nq1
nq1 nq2
nq 1
/
n n
.
w nrŽ n q 2.x ? wŽ n q 1.rŽ n q 2.x nq 1 is monotone increasing and tends to 1re. Hence for x g Ž0, 1., we have Pn o Ž x . F
qn o Ž 1r Ž n q 2 . . xŽ1 y x.
-
1
where the coefficient 1re is the best possible.
1
e nx Ž 1 y x .
,
371
BOUNDS FOR BASIS FUNCTIONS
Similarly, we can get Pn1 Ž x . -
2
2
ž / e
1 nx Ž 1 y x .
for x g Ž 0, 1 . .
,
For every k satisfying 0 F k F nr2, x g Ž0, 1., there holds Pn k Ž x . -
kq 1
Ž k q 1.
eyŽ kq1.
k!
1 nx Ž 1 y x .
;
Ž 17 .
and for every k satisfying nr2 F k F n, x g Ž0, 1., there holds ny kq1
Pn k Ž x . -
Ž n y k q 1. Ž n y k. !
eyŽ nykq1.
1 nx Ž 1 y x .
.
Ž 18 .
Remark 1. For every k satisfying nr2 - k F n, x g Ž0, 1., there still holds Pn k Ž x . -
Ž k q 1.
kq 1
eyŽ kq1.
k!
1 nx Ž 1 y x .
.
Ž 19 .
However, it is easy to verify that ny kq1
Ž n y k q 1. Ž n y k. !
eyŽ nykq1. -
Ž k q 1.
kq1
eyŽ kq1. ,
k!
for
n 2
- k F n.
Hence, inequality Ž18. is better than inequality Ž19.. In Ž17., we write
Ž k q 1. k! s
kq 1
eyŽ kq1.
Ž k q 1.
1 nx Ž 1 y x .
kq 1r2
k!
eyŽ kq1.
'k q 1 1 'n 'n x Ž 1 y x .
,
ŽŽ k q 1. kq 1r2rk!. eyŽ kq1. is monotone increasing and tends to 1r '2p . Now it is clearly that from Ž17. we have the following PROPOSITION 1.
For fixed 0 - d - 1r2, the inequality Pn k Ž x . -
1
'p
1 1r2q d
n
xŽ1 y x.
holds uniformly for all 0 F k F n1y 2 d and x g Ž0, 1..
Ž 20 .
372
ZENG XIAOMING
Furthermore, we point out that using factor x Ž1 y x . instead of in Theorem 1. By a similar method to proving Theorem 1, we get
'x Ž 1 y x . , we can improve the coefficient 1r '2 e
For all k satisfying 0 F k F n, x g Ž0, 1., there holds
PROPOSITION 2.
Pn k Ž x . -
1
1
,
'8p 'n x Ž 1 y x .
Ž 21 .
where the coefficient 1r '8p s 0.1994711 ??? is the best possible. Proof. For every x g w0, 1x, we have from Ž16.
'n x Ž 1 y x . Pn k Ž x . F s
'n n!
ž
k! Ž n y k . !
Ž k q 1.
kq1
/ ž
nq2
kq 1r2
?
k! ?
kq 1
Ž k q 1.
nykq1 nq2
1r2
nykq1
/
Ž n y k q 1.
1r2
nq2
'n n! Ž n y k q 1. ny kq1r2 Ž n y k . ! Ž n q 2.
nq 1
.
Ž 22 .
It is obvious that
Ž k q 1.
1r2
Ž n y k q 1.
1r2
nq2
F
1 2
for 0 F k F n.
,
Using a similar method to prove Theorem 1, we can prove that for k satisfying 0 F k F n, 'n n!Ž n y k q 1. ny kq1r2rŽ n y k .!Ž n q 2. nq1 is monotone increasing for n and tends to e yŽ kq1.. Hence, We get from Ž22., Pn k Ž x . F
1 Ž k q 1.
kq 1r2
eyŽ kq1.
k!
2
1
'n x Ž 1 y x .
.
Ž 23 .
Since ŽŽ k q 1. kq 1r2rk!. eyŽ kq1. is monotone increasing and tends to 1r '2p , Ž21. now follows from Ž23.. Moreover, by the facts that
Ž k q 1.
1r2
Ž n y k q 1.
1r2
nq2
s
1 2
,
as k s
n 2
and lim kªq`
Ž k q 1. k!
kq 1r2
eyŽ kq1. s
1
'2p
,
we deduce that the coefficient 1r '8p in Ž21. is the best possible.
373
BOUNDS FOR BASIS FUNCTIONS
On the other hand, we point out that the behavior of Mn k Ž x . is different from that of Pn k Ž x .. If we substitute x 3r2 for factor x 1r2 , we cannot improve the coefficient 1r '2 e in the inequality Ž6.. There still holds For all k g N, x g Ž0, 1x, we ha¨ e
PROPOSITION 3.
1
Mn k Ž x . -
1
'2 e 'n x 3r2 ,
Ž 24 .
where the coefficient 1r '2 e is the best possible. Proof. It is obvious that Ž24. holds from Theorem 2. Suppose the estimate coefficient 1r '2 e in Ž24. is not the best possible. Then, there exists an « satisfying 0 - « - 1r '2 e , such that Mn k Ž x . -
ž'
1
2e
y«
/'
1
for all k g N and x g Ž 0, 1 . Ž 25 .
,
n x 3r2
Taking x k s Ž k q 1r2.rŽ n q k q 1r2., k s 0, 1, 2, . . . , from Ž25., we have
ž
1r2
k q 1r2 n q k q 1r2 -
ž'
1
/
Mn k
y«
2e
ž
k q 1r2 n q k q 1r2
/
1
n q k q 1r2
n
k q 1r2
/' ž
/
.
Hence, as k ) nr« , we get
ž
1r2
k q 1r2 n q k q 1r2
/
ž
Mn k
k q 1r2 n q k q 1r2
-
/ ž'
1
2e
y
« 2
1
/'
n
.
It follows that
x
1r2
Mn k Ž x . F -
ž ž'
k q 1r2
n q k q 1r2 1
2e
y
« 2
1r2
/ 1
/'
n
Mn k ,
ž
k q 1r2 n q k q 1r2
/
for k ) nr« and all x g Ž 0, 1 x .
Ž 26 .
374
ZENG XIAOMING
However, according to Theorem 2, for the coefficient 1r '2 e y «r2, there exist a k 0 satisfying k 0 ) nr« and a x 0 g Ž0, 1., such that x 01r2 Mn k 0Ž x 0 . )
ž'
1
«
y
2e
2
1
/'
n
,
which is a contradiction with Ž26.. The proof is complete.
4. EXAMPLES OF APPLICATIONS In this section we give some applications of inequalities Ž5. and Ž24.. EXAMPLE 1. For Durrmeyer operators Dn w3x n
Dn Ž f , x . s Ž n q 1 .
1
Ý Pn k Ž x . H f Ž t . Pn k Ž t . dt, 0
ks0
where f g L1w0, 1x. By inequality Ž5. and proceeding along the lines of w3x, we give a sharp estimate as follows: THEOREM 4.1. Let f be a function of bounded ¨ ariation on w0, 1x, x g Ž0, 1.. Then for n sufficiently large, we ha¨ e Dn Ž f , x . y F
1 2
f Ž x q. q f Ž x y. 4
5Ž x Ž 1 y x . .
y1
n
n
Ý ks1
Ž q Ž 2 q 1r'8 e .
xq Ž1yx .r k
'
E
Ž gx .
'
xyxr k
xŽ1 y x. .
y1
f Ž x q. y f Ž x y. ,
'n
Ž 27 .
where g x Ž t . and Eba Ž g x . are defined as in the Theorem of w3x. It may be remarked that the estimate in the Theorem of w3x, i.e., Ž13r4.ŽŽ x Ž1 y x ..y1 r 'n .< f Ž x q . y f Ž x y .<, can be improved to Ž2 q 1r '8 e .ŽŽ x Ž1 y x ..y1 r 'n .< f Ž x q . y f Ž x y .<. EXAMPLE 2. For integrated Meyer]Konig ¨ and Zeller operators Mˆn w4x
ˆn Ž f , x . s Ž n q 1 . M
`
Ž kq1 .r Ž nqkq1 .
Ý Mn k Ž x . H
ks0
kr Ž nqk .
f Ž t . dt,
BOUNDS FOR BASIS FUNCTIONS
375
where f g L1 w0, 1x. By inequality Ž24. and proceeding along the lines of w4x, we get THEOREM 4.2. Let f be a function of bounded ¨ ariation on w0, 1x. Then for e¨ ery x g Ž0, 1. and n sufficiently large, we ha¨ e 1
ˆn Ž f , x . y M F
f Ž x q. q f Ž x y. 4 xq Ž1qx .r k
n
5 nx
2
E
Ý ks1
'
Ž gx .
'
xyxr k
q Ž 16 q 1r'2 e .
1
< < 'n x 3r2 f Ž x q. y f Ž x y. ,
Ž 28 .
where g x Ž t . and Eba Ž g x . are defined as in the Theorem of w4x. It may be remarked that the estimate in the Theorem of w4x, i.e., Ž49r 'n x 3r2 .< f Ž x q . y f Ž x y .< can be improved to ŽŽ16 q 1r '2 e .r 'n x 3r2 .< f Ž x q . y f Ž x y .<. EXAMPLE 3. For modified Meyer]Konig ¨ and Zeller operators Mn w5x Mn Ž f , x . s
`
Ý Mn , kq1Ž x . ks1
=
1
H0 M
ny 2, ky1
Ž n q k y 2. Ž n q k y 3. Ž n y 2.
Ž t . f Ž t . dt.
By inequality Ž24. and proceeding along the lines of w5x, the estimate term Ž114r15'n x 3r2 .< f Ž x q . y f Ž x y .< in Theorem 2.1 of w5x can be improved to Ž3r8e'n x 3r2 .< f Ž x q . y f Ž x y .< and the following result holds: THEOREM 4.3. Let f be a function of bounded ¨ ariation on w0, 1x. Then for e¨ ery x g Ž0, 1. and n sufficiently large, we ha¨ e 1
Mn Ž f , x . y F
7 nx
2 n
Ý ks1
f Ž x q. q f Ž x y. 4 xq Ž1yx .r k
E
xyxr 'k
'
Ž gx . q
3
'8 e x 3r2 f Ž x q. y f Ž x y.
ACKNOWLEDGMENT The author is thankful to the referee for his valuable comments.
. Ž 29 .
376
ZENG XIAOMING
REFERENCES 1. G. G. Lorentz, ‘‘Bernstein Polynomials,’’ 2nd ed., Chelsea, New York, 1986. 2. R. Bojanic and F. Cheng, Rate of convergence of Bernstein polynomials for functions with derivative of bounded variation, J. Math. Anal. Appl. 141 Ž1989., 136]151. 3. S. Guo, On the rate of convergence of the Durrmeyer operator for functions of bounded variation, J. Approx. Theory 51 Ž1987., 183]192. 4. S. Guo, On the rate of convergence of the integrated Meyer]Konig ¨ and Zeller operators for functions of bounded variation, J. Approx. Theory 56 Ž1989., 245]255. 5. V. Gupta, A sharp estimate on the degree of approximation to functions of bounded variation by certain operators. Approx. Theory Appl. 11, No. 3 Ž1995., 106]107. 6. A. Piriou and X. M. Zeng, On the rate of convergence of the Bernstein]Bezier operators, ´ C. R. Acad. Sci. Paris Ser. ´ I 321 Ž1995., 575]580.