AbstractA conjecture of Z. Ditzian on Bernstein polynomials is proved. This yields additional information on the problem of characterizing the rate of convergence for Bernstein polynomials

Zhou, Ding-Xuan

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On a conjecture of Z. Ditzian 

:OURNAL OF APPROXIMATION THEORY 69, 167-172 (1992) 
On a Conjecture of Z. Ditzian 
DING-XUAN ZHOU * 
Deportment of Mathematics, Zhejiang University, and Advanced 
Institute of Mathematics, Hangzhou, Zhejiang, People’s Republic of China 310027 
Communicated bv Zeeo Ditzian 
Received June 5, 1990; revised April 1, 1991 
A conjecture of 2. Ditzian on Bernstein polynomia!s is proved. This yields 
additional information on the problem of characterizing the rate of convergence for 
Bernstein polynomiak. ‘6Z 1992 Academic Press, Inc. 
The Bernstein polynomials on C[O, l] are given by 
B,(f, x) = i f (k/n) Pnk(-xh 
k=O 
where 
P&(x)= ; xk(l -.y)n-k. 0 (2) 
The relation between the rate of the polynomials’ convergence and the 
smoothness of the function they approximate has been investigated in 
many papers. Some of these results can be stated in the following theorem, 
THEOREM A. For f~ C[O, 11, X=,X( 1 -xi, 0 <a < 2, 0 < /3 < 2, :he 
following statements are equivalent : 
(1) XPzl’IB,(f, x)-f(x)/ <MM,n-g~r; 13) 
(2) rx;‘If(x- t)-2f(x)+f(x+ t)l Gif,(t2/X)“? 141 
Here Ml, M, are constants independent of n, x, and t. 
In 1972, H. Berens and G. G. Lorentz [l] proved this theorem for X= p. 
M. Becker [Z] gave an elementary proof of this case. The case z = 0 was 
* Supported by ZPSF and NSF of China. 
167 
ooa-9045!92 $3.00 
Copyright ,ca 1992 by Academic Press. inc. 
All rights of rcprnduction in any form rcser~cti. 
168 DING-XUAN ZHOU 
proved by Z. Ditzian [3] and V. Totik [9]. Z. Ditzian [4] proved this 
theorem for 0 < a < /? < 2 as well in 1980. 
Theorem A had remained open for the case 2 > IX > p > 0 for a few years. 
In 1987, Z. Ditzian [S] gave the proof of this case with the restriction 
c( + /I f 2. In that paper he also gave the following conjecture which allows 
us to complete the proof for the remaining cases. In this paper we give a 
proof of this conjecture, and show that Theorem A holds. 
THEOREM (Conjecture of Z. Ditzian). Suppose for 0 < /? < CY < 2, 
f E C[O, 1 ] satisfies f (0) = f (1) = 0 and 
xY2 If(x-t)-2f(x)+f(x+t)lqt’/X)B’2 (5) 
for any x E (0, l), x-t t E [0, 11; then we nzust have .for the case a + /I > 2 
(1) f(x)=A,x+f,(x), 
where fi(x) = O(X(~+P)/~) usx + O+; 
(2) f(x) = A*(1 -x) +fz(x), 
where f2(x) = O(( 1 - x)@+fl)12) us x + 1 -. 
ProoJ We need only prove (1). 
From (5), we have for XE(O, $) 
) f (x) - 2f (x/2)1 <x(=++)‘2, 
which implies 
1 f(x)/x-2f(x/2)/xl <X(“+fl)‘l-l. (6) 
Define g,(x) = 2”f (x/2”)/x for XE [a, $1, then g,E C[f, $1. For these 
functions we have from (6) for n E N, p E N, x E [i, 41 
I b%(x) - gH+,w = 1 (2kf (x/2Vx -2k- ‘f (x/2kp1)/x) 
k=n+l 
n+P 
< 1 (X/2k--)(~+8)/2-- 
k=n+l 
n-t-p 
< 1 (2’-(“+8W)k-l. 
k=n+l 
ON A CONJECTURE OF DITZIAN 169 
Thus by E + p > 2, we know that (g,, > is convergent uniformly on 
[i, f]. So we can define 
g(s) = lim g,,(x) = lim 2”fc(s/2”)~~. (7) 12 + 22 n--x 
then g E CC;, $1. 
Now we prove that g is constant on [a, $1. 
Let h(x)=,yg(.x). From (5), for x~[$,iJ, t~(O,$j, x&t~[$,i]~ and 
n EN, we have 
and 
1(x-t)g,(x-t)-Zxg,(x)+ (x+t)g,(x+ r)i ~IP~~t-(l-P”22”(1~iZ+8),‘2’~ 
By letting II --+ co, we have from the definition of 12 
h(x-t)-2h(x)+h(x+tj=O, 
where x, x 2 t E [b, i] are arbitrary. From this fact, we know [8] that A 
is linear on [a, 41, say, 
h(x) = ux $ b. 
Hence g(,u) = a + b/.x. 
Now from (7), we know that 
g($,=g(+)=J~mX 2nf(2--n); 
therefore, we must have b = 0 and 
g(x) = a. (8) 
On the other hand, as for every SE (0, $), there exists n, such that 
.Y~“OE [i, 41; in view of (7) and (8) we get 
Thus, by (6), 
where the constant C is independent of n and x by LX +/I > 2. 
170 DING-XUAN ZHOU 
By letting n -+ 1y3, we have for any x E (0, b) 
1 f(x) - nxl d Cx’” + fl)12. 
The proof is complete. 
Proof of Theorem A (For 2>a>b>O and ol+/-I>2). By [l], we need 
only prove that (4) implies (3). 
It is sufficient for us to prove the theorem on [0, $1, and to prove from 
(4) that 
IB,(f, X)-f(X)\ a!f,x~~*n-~‘* (9) 
with a constant M, independent of n and XE [0, 41. 
In this case we can assume that f 1 13,1, i, = 0 and 1 f(x)\ $ Mx(” + P)D with 
the constant A4 independent of XE [0, 11. We divide the proof into two 
parts. 
For 0 < x < 2/n, the proof is easy: 
IBJf, x)-f(x)1 GM i (k/n)‘“+8)!2-11P,k(x)/n + Mx’@+PJ12 
k=O 
dMx i (k/lz)(a+P)‘*-lP,_L,k--(X)+Mx(a+p)lZ 
k=l 
l-(~+P)/*+Mx c 
( 
’ $$P,-1,&-l(x) 
1 
(LI+p)/2-1 
6 Mxn 
k-2 
+ 4/fx(“‘+B)12 
$ 8Mxa12n-P’2. (10) 
For f 2 x > 2/n, n > 8, let d = (x/n)“’ <x and define a Steklov function 
as 
.fJt) = (2/d)’ j-Joq2 (2f(t + u + u) -f(t + 224 + 2~)) du du, (11) 
where f has been extended to y E [ 1,2] as f( v) = 0. 
Then we have 
If(t)-&(t)/ 4M~(2/d,‘jjo” (u+u)~ (t+u+u)‘a-~“2dudu 
6 M,dB(t+ d)‘“-B”2 
and 
If:(t)1 <9M2dB-‘(r+d) (C(-8)/2~gM2dB~*(di*-8)/*+t(a--8)/2) t12) 
ON A CONJECTURE OF DITZIAN 171 
where we have used the estimate 
B,, 
> 
=B,((t3-ff.?i2)/2-(f3-X3j!3,.~) 
= (B,,( t3. x) - x3)/6 
<.xn-2+3x2.ci 
< 4x “/h. 
Combining (10) and (13), we have proved (9) for n 2 8. 
For y1< 8, it is trivial since lB,(f, x) -f(x)1 Q B,(.Mt, xj + M.x < IM.x d 
14Mx”“n mmp’2. Our proof is therefore complete. 
ACKNOWLEDGMENT 
The author thanks Professor Z. R. Guo and Professor H. Berem for their continnous and 
kind help. Thanks are also due to Professor Z. Ditzian for his comments. 
REFERENCES 
1. H. BERENS AND G. G. LORENTZ, Inverse theorems for Bernstein po!ynomials, Indiana l/m. 
Math. J. 21 (1972), 693-708. 
2. M. BECKER, An elementary proof of the inverse theorem for Bernstein polynomials, 
Aequarioms Math. 19 (1979), 145-150. 
3. Z. DITZIAN, A global inverse theorem for combinations of Bernstein polynomials. 
J. Approx. Theory 26 (1979), 271-292. 
4. Z. DITZIAN, Interpolation theorems and the rate of convergence of Bernstein polyno- 
mials, in “Approximation Theory III” (E. W. Cheney, Ed.), pp. 341-347, Gcademic Press. 
New York. 1980. 
172 DING-XUAN ZHOU 
5. Z. DITZIAN, Rate of convergence for Bernstein polynomials revisited, J. Approx. Theory 50 
(1987), 40-48. 
6. Z. DITZIAN, Derivatives of Bernstein polynomials and smoothness, Proc. Amer. Math. Sot. 
93 (1985), 25-31. 
7. R. A. DEVORE, “The Approximation of Continuous Functions by Positive Linear 
Operators,” Lecture Notes in Mathematics, Vol. 293, Springer-Verlag, Berlin/New York, 
1972. 
8. G. G. LORENTZ, “Approximation of Functions,” Holt, Rinehart & Winston, New York, 
1966. 
9. V. TOTIK, An interpolation theorem and its applications to positive operators, Pacific J. 
Math. 111 (1984), 447481. 


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On a conjecture of Z. Ditzian

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