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• 76 • Lin Xudong: Conjecture of Haason
ON A CONJECTURE OF HASSON
Lin Xudong (~)
(Hangzhou University, China)
Received Feb. I, 1990
Abstract
In this paper, we consider the situation o f l i m ~ , where f e C [ - 1,1] and 1" does not
xist at some interior point o f f - l , I], and solve a modified conjecture o f Hasson.
Let fe C[a,b] and denote by It the class of all algebraic polynomials of degree at
~ost n. Define E ( ] ) = m i n l l f - P[I and E~(]) = m i n { l l f - pll: p e r t and p(*)(0)-- 0},
#here Ilhll -- max Ih(x)l for heC[a,b]. Hasson in [1] made a conjecture t ha t if f e C [ m ~ , , b
- 1,1] and )* does no t exist at some interior point o f [-1,1], then
lim-----=-~ _ < ~ , k ~ 1 integer, (1) , . . E if)
In paper [2], Chengrnin Yang gave a counterexample to the con~c tu re of Hasson , in
ae meantime, he proposed how a b o u t the situation if the liminf in place o f the limsup in
~uotient (1).
In this paper, this quest ion will be answered. The result is: Fo r k >~ 2, it is ture; and
;= 1, it is not . We express it as following
Theorem. There exist a function f 0 e C [ - 1,I], fro does not exist a t some interior
aoint off-I ,1] , such that
lira ~ -- oo. (2) .-.e,CY0)
For all k >~ 2, i f f eC[ - 1,1] and]' does not exist at some interior point of f - I , 1 ] , then
Approx. Theory & its Appl., 7:1, Mar., 1991 • 77 •
E~(¢) l i m ~ - ~ < co.
,(D
Lemma 1. [Bernstein's inequality] ([3] Lel P ~II,, then
lip ~k)11 [~Jl ~< Nn~ l i P , It ta~l '
where a < = < fl < b, constant N only depends on ~',B and k.
Lemma 2. ([1]) Let k be an integer >I 1, There exist positive constants N k
such dmt, /'or every integer n >I I,
M k N~ < t~(x~, [ - 1,1]) ~ k
n n
To prove the first part of the theorem, we define The proof o f the lheroem.
= 1 n :oCx)= .7cos(5 arccos,O, x~[- 1,]]
and M
(3)
It is clearly tha t f o e C [ - I , I ] and its best approximation polynomial of degree at most n (see [4])
Do=sx]
.~ (x)= ~ kcos(S'arco~) I = o 5 i
and
® 1 1 . 1 1
1- ~ss ' l+ 151 (4)
First, we show f 0 does not exist at x = 0. Let h = = ~ • 5 - " ~ 0(m ~ 0o)
f°~h=)-f°(O)h = h-~-.~05 -~[c°s(5~arcc°sh-)-c°s(Sjarcc°sO)]
[h__~._,=~ 1 cos(SiarccosO)] 4 5 . ® I (5)
By differential mean value theorem,
cos(5 Jareccosk,, ) - cos(5 JarccosO)
= s i n ( 5 / a r c c o s ~ ) • 5 / . 1
1-¢ : h=, Cje(O,h ). ]
And
• 78 •
Hence
Lin Xudong: Conjecture ofHfUson
sin(Siarccos~.) ~ sin(Siarccosh )/ ~
~sin(Si(~-h »=sin(~-Sih )~.J2 j=O,I,"',m2 '" 2 ~ 2'
1
m -. 00.
This shows f does not exist at x = Oe( - 1,1). But
- -1 1 (~) -where liP ~ - f o II = E ~ (fo)' liP ~ - f o II = E ,.fo)' and b 1 is the coefficient of x in P ~
Y ~ wBy (cos(2m + l)arccosx ... -0 = (- l) (2m + 1). we have b 1 = Dog
5n] + 1 . By lemma
1 N2, E (x) ~ -, N = const. So '
~ n
E 1 (f. ) (Dog 5 ~] + I)N
lim E ~ (1.0
) ~ lim 1 n - I = 00 .• -c ,,0 11-" _
4n
This shows that (2) is proved.
Now we prove the inquality (3).
Let P be the best approximation algebraic polynomial of degree at most n of fee [~
-1,1] and 21~ n < 2
1+
1(lbe nonnegative integer).
,P ~ = P ~ - P 2/ + L(P 2
'- P 2 / - I ) + PI'
I-I ,pel) = pel) _ p(~) + "(P I _ P,-I )(l)
~ " 2 I.J 2 2I-I
By lemma 1,
for k ~ 2.
where constant N only depends on k. And 'for I < m
liP", - P III ~ liP", - III + liP I - III ~ 2E I(/)'
Consequently, we have
Ao~ox Theory & its Appi.. 7:1, Mar, 1991 • 79 •
I
i ~ ( t ) k " Ill_],J] ~NtnkE. ( / )+ NtZ2#'E,,(I '), I=. l l
where constant Nk only depends on k. t~) k .
Denote b t is the coefficient of x m P . , then
l
Ib',"l = , - . .~. N~n E (J) + Nt ~21kE=, (j) (6)
Furthermore
E : ( ] ) ~ E ( J ) + {b ~) {E(x k), (7)
in fact,
_ k , ,., ~ . ( . ) k. _ _ l ~ , _ (s) 1~,, E ~ ( f ) ~ U(x)--o t x ) + E (0 k x )
and
By (6), (7) and lemma 2, for k ~ 2,
E~(]) C.) k k Ib t IE (x ) ~ 1 + E ( ] ) E ( [ )
R
l k <~ 1 +M~JVk(n E,(J)+ ~2~E~(J) ) /nkE (J)
t
I + Mk,V k + MkN k ~221Ed (]')/n2E(]) (s)
Because f does not exist at some interior point of [-1,1], for 0 < 6 < 1, we have
lirnnl+~E(]) = oo, otherwise, m
[- l+e, 1-8](0<~< 1), f c L i p 6
some interior point of [-1,1].
1 Without loss of generality, choosing 6 -- ~ and
for n ~n<n,+1, (n---l)
3
nS/2E (t'). sup{m2E (])}--- , ",
/ 1 \
we have E(J)--O~n-~+~) , hence by [5], we know, in
• This is a contradiction to the fact that f does not exist at
a subsequence n such that,
So by (8), for k >~ 2
• 80 • L/n Xmtong: Con/ecture of Ha.vaon
lira--~ I + M,N k + MkNtlim ~2~E2,(/)/ n2E (/) ,-~,E (D ,-.® i.,
~oiza ,]
1 +Mklg~, +M,Nkli__m_m ~ 2~E2,(t)/n~E 09 ,-*so I ' Q
~oSaa ,]
~I+M,N, +MtNtlim ~ 2J/a/nl/2 ., ~oD.
The proof is completed.
Aekmwledl~ent I am grateful of my advisor Professor T.F. Xie for his enthusiastic
guidance.
References
[1] Hmson., ~ . Comparison Between the Degree of Approximation by Lacunary and Ordinary"
Algebraic Folynomials. J. Approx, Theory 29(1980), 103-115.
[2] Yang, Chengmin, A Counterexample to it Conjecture of Hasson. J. Approx. Theory 56(1989),
330--332.
[3] Natanmn, L P. °Cons~uctive Function Theory" Vol.l Unger New York 1964.
[4] Lorentx, G. G. Approximation by Incomplete polynomials (problem and results) in "Parle and Ra-
tional Approximation: Theory and Application" pp.259-302. Academic Press, New York 1977.
[5] Lorentz, G.G. "Approximation of Function" Holt Rinehart and Winston. Inc. 1966.
Institute of Policy Decision and Optimization Research
Hanszhou University
Hanszhou, 310028
PRC .
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