• 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.

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Hanszhou University

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PRC .