A NOTE A.
ON A T H E O R E M V.
OF
SUNOUCHI UDC 517.5
Efimov
We show that f o r n e g a t i v e o~ S u n o u e h i ' s f o r m u l a H n (f, ~z, ~,
x)~ ~
y~z' 13-1 I l (x)--%a (f, x) I, k~_te k=O
a > - - ~ - , ,~>
,
b e c o m e s f a l s e , w h e r e c r y ( f , x) is the (C, a) m e a n of the F o u r i e r s e r i e s f o r the f u n e t i o n f ( x ) E Lip T , 0 < 3/< 1. A b o u n d i s given f o r H n ( f , ~ , f l , x) f o r a l l o t > - l , t 3 > - l , w h i e h f o r ~ + 3 > 0, ce -> 0, 3 -> 0, c o i n c i d e s with the Sunouchi bound. The p r o o f is by a m e t h o d d i f f e r e n t f r o m that of Sunouchi.
Let f ( x ) be an i n t e g r a b l e function of p e r i o d 2v co (/) = ao -~ + ~, ~=~(a~cos~x + b ~ s i n v x )
its F o u r i e r s e r i e s , S n ( f , • t h e p a r t i a l s u m s of the s e r i e s , and I
~
a-i
z~ (/, x) = ~ ~'~=o A~-k S~ (/, x) its (C, o~) m e a n .
Here A:~
(v+ a ) ~ v ~'~ r ( a ~ t ) '
a>--l.
(1)
Let HT(0 < T -< 1) denote t h e c l a s s of continuous functions of p e r i o d 27r s a t i s f y i n g a L i p s c h i t z condition of order y. In [1] Sunouchi f o r m u l a t e s the following a s s e r t i o n ( T h e o r e m 2). I f f ( x ) E HT, 0 < T < 1, and oe> - 1/'2, fl > 1 / 2 , then --
A~
v=0
A~-v I ] (x) -- z= (], x) ] = 0 (n-~).
B u t , f o r - - i / 2 < ~ < 0,(2), in g e n e r a l , is not t r u e . a m p l e . Let
r
(2)
It is e a s y to s e e this by m e a n s o f the following e•
=~ ~ r k=~ k~+~' 0 < T < I .
(3)
C l e a r l y , ~o(x) ~ HT (see, e.g., [2], p. 119) and f o r it we h a v e
( x ) - :~ (/, ~) = ~ ~ A~
Since
~ n/- ~0
for-l
~-o
A~-~-o ~
0,1, ...,
~ Y~
t . ( x ) - s~ (r x)l = A--~ ~=0
A~:i E~ co~ ~=~+~ ~+~
n) , then
/Jk=0 I A n - k ! ~o=k~-J_ ; l+y - -
= [..~] [ n-/~ J
vl+y
M o s c o w Institute of E l e c t r o n i c T e c h n o l o g y . T r a n s l a t e d f r o m M a t e m a t i c h e s k i e Z a m e t k i , Vol. 12, No. 6, pp. 665-670, D e c e m b e r , 1972. Original a r t i c l e s u b m i t t e d J a n u a r y 5, 1972.
9 1973 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. 10011. All rights reserved. This article cannot be reproduced for,any purpose whatsoever without permission of the publisher. A copy of this article is available from the publisher for $15.00.
839
> ~-
But when-l<
(n - k +
-- > -
Y',-o
(~ + t)~-~"
oz < 0 (v + t)~-1 = const ~ 1,
~lv~ 0 and so
C1 I~p(o) - o~ (~, o) 1> ..+, ,
(4)
~, > o.
N o t i n g (4), f o r a n y / ~ > 0, w h e n - 1 < o~ < 0, w e h a v e
(~-~+t)~-~.
_ CI _
> w h e r e c 4 > 0.
Thus, clearly, when-
1 < a < O,
i . e . , (2) d o e s n o t h o l d w h e n - 1 < o~ < 0. We give a more precise THEOREM.
version of this relation.
I f f ( x ) E H'Y, 0 < ,/ < 1, t h e n f o r
A~_I f t(x) -- ~ (f, x) l
i y" we have the following bounds
\--2r H~(l,a,[~,x)=
Proof.
0 ~
,
if
[t>O,--l~r
o 2T~'
~f
~>o,-i<~
o -~T z(~ ),
~f
-i<~
If ot = /3 = 0, t h e b o u n d
tt~ (/, O, O, x ) = / ( x ) is Lebesgue
~].
W h e n oe = 0, fl > 0 t h e b o u n d
H~(/, 0, ~ > 0 , x) = ~ ~ _ 0 was obtained independently by Leindler Further, the bounds
[4] ( T h e o r e m
~_, I I ( x ) -
8~(/, x)l = 0
3) and S u n o u c h i [1] ( T h e o r e m
iff(x) E HT, from the bounds for general linear summation
! (z) - ~ (/, x) =
840
& (l, x) = O [~n k T~/~
/
\n"-]
1).
methods, we can easily obtain
(see, e.g., [5], Eq. (7.4)). U s i n g t h e s e b o u n d s , we obtain
H n(], (z~O, O, x)-~/(x)--v](f,x).~
(')
~,,(l,~
-~
0 ( ~i ) ,
,
= / ( x ) - - ~ (I, z) = 0
,~
.
Further,
t Z ~,=0 A~51~1/(x) -- ~ (], x) J H n (/, a ~ 0, ~ ~ 0, x) = A--~ --- I-~.,,Z = o
=o
A~-,I:(x)-,(:,~))+Z=[~]+,~'-' ~ ,
"-' ....
1:
~i c:, } ,~)
[,,] ( -7 ' / " Z,=o ('-'+~)'~-'('+~)-'+~",= [~]+, ("-" +')'-'(" +')-'}) = 0 ( t~ 'Tb'-I "-Y+I-~ "-'t''}) = 0 ( ~ ) ,
Now i f - l <
oz < 0 , / 3 > O , t h e n
Hn(f,g,
/z["]
x ) = ~A,}, t
, = [ _ ~ ] + , A~(--~li (x) -- z: (S' x)l
-~ A~-I ,=0 --, I l (x) -
:,a (1, x) I
=otT{E,= 0 ( - - , +
(,+
,=[-~]+, S i m i l a r l y , if ~ > 0 , -
1 < fl < 0, then
H.(t, ~>o, ~
0
[~] ("-~+'?-'('~+~)-" ( 7' ( "Z =o
,-,.,>)=o(') Finally, i f - 1 < oz < 0 , -
1 < fl < 0, we have
H,,(t, ~ O ,
~O,x)=O
+ "=[-~]+' The theorem
is completely
7t
(n--~-~-, l),~_~(~ + t)_~_~
~,,,~,
+ n~....
proved.
We note that the bounds obtained in the theorem for the class HY of functions in general cannot be improved. That they cannot be improved when ~ = fl = 0 is Lebesgue's result, while in the remaining cases, it is easily verified for the function ~(x), defined in (3), at the point x = 0. LITERATURE
1. 2. 3.
CITED
G. Sunouch[, "Strong a p p r o x i m a t i o n by F o u r i e r s e r i e s and o r t h o g o n a l s e r i e s , " Indian J. Math., 9 , No. 1, 237-246 (1967). A. Zigmund, T r i g o n o m e t r i c S e r i e s [in R u s s i a n ] , Vol. 1, M o s c o w (1965). H. L e b e s g u e , "Sur la r e p r e s e n t a t i o n t r i g o n o m e t r i q u e a p p r o a c h ~ e des fonctions s a t i s f a i s a n t a une c o n dition de L i p s c h i t z , " Bull. Soc. Math. F r a n c e , 3_88, 184-210 (1910).
841
4.
5.
842
L. Leindler, "t~ber die Approximation i m s t a r k e n Sinne," Acta Math. Acad. Sci. Hung., 16, Nos. 1-2, 255-262 (1965}. A. V. Efimov, " L i n e a r methods of a p p r o x i m a t i n g continuous periodic functions," Matem. Sb., 5__~4,No. 1, 51-90 {1961).
