A N N A LES SO C IETA T IS M ATHEM ATTCAE PO LO N AE Series I : COM M ENTATIONES M A TH EM ATICAE X V I I I (1974) RO C ZN IK I PO LS K IE G O T O W A R Z Y S T W A M ATEM ATYCZNEGO
Séria I : P R A C E M A TEM ATYCZN E X V I I I (1974)
L. B
e m p u l s k a(Poznan)
Approximation to continuous functions
1. Let Л = ||Aj£ II (Тс = 0, 1, ii; п = 1, 2, ...) be a tringular matrix of real numbers, in which A” = 1 for any positive integer n.
Consider the Fourier series
Since 2 AX(k ] = 1 , the quantity B n(œ-, /, A) — \Ln(x-,f, A )—f(œ)\,
00
( 1 )
of 2jr-periodic function /e _L <( — tz , tz ), and the linear operators П
B y the Abel transformation, П
L n(x) = У M">S*(æ) ,
п
к —О
considered below, can be rewritten in the form П
Й „ (*;/ ,Л ) = | У М ’,) { « * ( * ) -Л ® )}| (» = 1 , 2 , ...) .
Let us introduce, for any positive p, the related expression П
Н Ц Х )
= H * ( x - ,f,A ) = { у |М “)П « * И - / М Г } 1'3’ (» = 1, 2 , . . . ) .
Evidently, B n(x-, /, Л) < /, A) (n = 1 , 2 , . . . ) .
Given an arbitrary positive
(3 <1, we shall signify by
A pthe set of all these matrices A for which
(2) n f £ ( M n,)! = 0 (1 )-
k—0
Putting (px(t) = f(æ-\-t) — 2f(æ) + f ( x — t), we shall denote by Z a (0 < a < 2) the Zygmund class of all functions f e C 2n such that
max \<px(t)\ < 2 Г for te <0, тс).
Write
An(a, /8) = sup sup т а х В п(ж; /, Л), ЛеЛ^ fe^a x
B%(a, /?, r) = sup sup тахЛ ^ (ж ; /, Л ),
ЛеЛ^ /M<=Za ж
where/(г)(я) is the r-th derivative of f(x ).
In the present paper the suitable estimates for A n(a, (3) and ^ ( a , ,8, r) will be given.
2 . Beasoning as in [1], p. 21-23, or applying our Theorem 2 (with p = 1, r = 0),' we obtain
T heorem 1. Under the basic restrictions 0 < a < 2, 0 < /3 < 1,
(3)
0 {n ~ a+{1- m ) i f 0 < a < 1/2 0{n ~ pl2(\ogn)112) i f a = 1 / 2 , 0 {n ~ m ) i f a > 1/2.
We shall prove that estimates (3) cannot be improved.
1° In case a > 1/2, consider the matrix ylj = ||A(fcw)|| in which
>p-T cp
4 ”1 = i , 4 " (& = 1, 2, . . . yn ; n = 1, 2, ...), A simple calculation shows that condition (2) holds. Further, if f*(t) = 5-1 cost, then
f * ( x ) — L n( x ;f * j Af) == (1 — Я(1и))/*(ж) = (5^ /2)-1 cosæ
for all real x (cf. [1], p. 23).
Since /*<r Za (0 < a < 2), the desired result.follows.
2° If 0 < a < 1/2 we introduce the Fejér means of series (1):
= — У Ski®',!) n
k= о
«о
2 У i Jfc
n ( cos Tex + bk sin t o ) .
Approximation to continuous functions
73Denote by Л2 the matrix with
= 1 --- n(l~P)l* ft for ft = 0, 1, m , n
№ n — m
where m = [n/2] + l* Then, / & —- m \ -
( 1 ---1 f o r 1c = m + 1 , m + 2, . . . , w,
\ n — m J
1 v -1 A(w) ^
АЛ#;/, Л2) = П(1+Р)Ц 7 , ^ ) / ) + ^ ^ fc
= 0= ~ (T+ff)/T g m ( ^ ; / ) + K ( ^ ; / ) - W <rw ( a ? ; / ) }
n — m whenever fe L < ( — 7z, 7t>, see ( — oo, oo).
Taking the 27u-periodic function
/а(<) =
\*\аfOr
t e< — 7U, 7Г)
and using the asymptotic formula of Nikolsky ([3], p. 205), we obtain 4 , ( 0 ; /«, Л ) = в*1- » ' 2 {<rm( 0 ; / J - o , ( 0 ; / J + O | - b - j j
( 2(2a —l ) JT(a) arr 1 ±n
= { --- ~---7~^ sin — + 0(1)1 n~aHl m .
{ 7C(1 —a) 2 J
The matrix Л2 satisfies (2) and/ae Z a. Hence
A n i« ,P )> №п(0;/«,Л2) - / в(0)| = №„(0;/a, Л2)|.
3° In case a = 1 /2 we choose the matrix Л3 with the modified Baska
kov terms ([1], p. 23, (B))
1 for 1c = 0,
1
Vк
11
;(«■) _ I 1 / g - / . ^or ^ == 2, . . . , m,
Ak — Vn^logn V r
1c —m i . . 1 --- ) &
n — m for 1 c = m + 1 , m + 2 , . . . , w ,
where m = [w/2]-f 1. Then condition (2) is fulfilled.
From estimates of [1], p. 24-25 follows that i M(0 ;/1/2, Л3) is of the
order 0{n~ pl2 (lo g n )112), precisely.
T heorem 2. Assuming that p > 0 emd r = 0 , 1, 2, ... (0 < a + 2, 0 < /? + 1), we have
0 {n ~ r- a+(l- m p ) i f r + a < 1 2 p (4) в р (а, P ,r ) = ■0 {n~r~a+(1~p)l2p\ogll2p n) i f r + a 1 2 p
0 (n ~ m p) if r + a < 1
2 P P ro o f. By Holder’s inequality and [2],
n П
fc=0 л=о
1 1 v~4 1
= O inV-W *) {— — > .
Now, the assertion follows at once from Theorem 6 of [2], p. 150, slightly modified.
3. Besides of (2), we shall further suppose that > 0 (к = 0, 1, ...
. . . , n j n — 1 , 2 , . . . ) .
In this case the first estimate of (4), with r = 0, can be improved correspondingly. Namely,
T heorem 3. I f p > 1, 0 < a < 1-I2p, then
B p (a, /?, 0) = 0 (n ~ a/}) as n oo.
P ro o f. In view of the Minkowski inequality,
< Р х У ) sin(A; + l/2)#
2 sin#/2 dt
p\i i p
I +
sin (Je + 1/2)#
2sin#/2 V + W.
The assumption f e Za implies
V = 0 ( 1 ) { ^ ^ 4 W)( J dt)P] llP = 0 {n ~ ats).
k=0 0
Approximation to continuous finctions
7Applying the Holder and the Minkowski inequalities, we obtain
»c Я 7T
W < { 2 < M - > ) f P £ /*,<«) sin(fc + 1/2)2 2sin£/2 dt
k= 0 ' k= 0 1 n
iiSife—rr LSI J>—«П
2pi i/2p
n ТГ
2 p l 1 jip
к— 0 n
Jp 2 tan </2
k — 0 n= 0 ( ^ - /î/2î,)(TF1AWr2).
By the Hausdorff-Young theorem
~~ 2p ^ 2prc
J l 9)x (0 t2p~1^ } 1 = 0 (1 ) I J I 1 — l/2p
=
0
(1
) ,?*(*) 2 tan t /2
2p A. ) ll^l/2p
2/>-i dn 0 ( г а^+ № ).
The thesis is now evident.
I t is easy to see that if 0 < p x < p 2 (zU(fcw) > 0), then
Indeed, by Holder’s inequality, П
= { у M ’W (*)-/ < * ) I*1}1" 1
A = 0
& = o
= H *H x-J, Л).
This fact and Theorems 2, 3 yield
T heorem 4. Under the restriction 0 < p < 1, we have О (n~aP) i f 0 < a < \, Б £(а, 0, 0) = 0(w-/3/2(logw)1/2) i f a = |,
0(n~P/2) i f a > \ . Finally, let us consider the Euler means
П
E n( v ;f) = ^ r ] ? ( l ) s k(x-,f) (я = 1 , 2 , . . . )
k
=0
of series (1). They may be treated as the linear operators L n{x\f, Л) with factors
П
4 n| = y « ^ ( “) (* = 0 , 1 , . . . , » ; » = 1 , 2 , . . . )
v = k
