| Approximation of functions in L- and O-metrics

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE X I (1967)

P. P ych (Poznań)

Approximation of functions in L- and O-metrics

1. Preliminaries. Let L[G] be the class of all 27r-periodic functions / Lebesgue-integrable (continuous) in every finite interval. Write

= sup max \f{x+t)—f{x)\,

(o2(6jf) = sup max \f{x+t)—2 f(x )+ f(x —t)\

О — rc<a;<Tt

for any feC, and

TT

= sup f \f( x+ t) —f{x)\dx, T U

<A l)(S,f) = sup f \f ( x+ t) —2f{x )+f (x —t)\dx 0 <(<<5

for any f e L . These quantities are called the moduU of continuity and the moduli of smoothness of / in G and L, respectively.

A 1, Z 1, Z 2 will denote the classes of all functions feG with moduli satisfying the inequalities

» , № / ) < « , юг(<5,/) < 2 й , ( « , / ) < <5*, respectively. The classes of functions /«X for which

«А'Ча.Л <-«, f> < 23,.

will be denoted by /1^, Z['K Z^K It is easy to see that Л 2 >= Z ,, Л*)* cz r/J'' and that these inclusions cannot be reversed.

By Wp and W$\ with an integer p > 1 , we denote the classes of 2 Tr-periodic functions f having the derivative f (p~l) absolutely continuous in any finite interval, such that

П

ess sup |/(р)(ж)| < 1 and J | f p){%) \dx < 1 ,

— л:<ж<гс _ —

respectively.

It is well known ([5], p. 216) that if f e Wp or f e W$, then vo cos{1c(t— х) + ^р-к} oo

/ W = T + T ... у

— tu k = 1

where a0 is the initial Fourier coefficient of / , xe( — oo, oo). Also, the function / conjugate to a function / of class Wv or (p ^ 2 ), defined by the formula

x 71

/ 0 ») = — ■ — lim f { f ( x + t ) - f ( x —t)}coątdt, A7Z e ->0 j

can be represented in the form 1

71

* t u u

?о») = - -X f f (p)(t) ^

—n fe= 1

sin{ft(/—ж )+ |р 7 т:}

for xe( — oo, oo). Moreover, J / (p)(/)d/ = 0.

In this paper we consider the linear operators

TC

f{x-, i) = — f f ( t ) 0 ( t —x, £)dt

TC

J

— TC

and

Я

f i x ; £) = —-X- j f ( t ) 0 ( t - x , S)dt,

— 7Z

with the kernels

O O O O

( 1 ) 0{t, £) = ł + ^ Qk(£)cosM, 0(t, i) = £ e&(l)sinfc«;

fc=i fc=i

we assume that the factors g*(£) are defined in a set E of real numbers having an accumulation point £0, and that series ( 1 ) are uniformly con­

vergent in t for any fixed £.

In Section 2 we find the connections between the quantities

TV

sup /I/O»; £)—f{oo)\dx, sup max \f(x-, £)—/(a?)|

f _n 1 — 7и<Ж <ТГ and

sup £)—/ 0 »)|<й», sup max I/O»; £)—/ 0 »)l,

/ „тх f --n^X^TC

where the supremum is taken over all functions of the corresponding

classes mentioned above. In Section 3 the exact and asymptotic formulae

for some concrete operators are given.

The following remarks will be needed below.

1° If / is of class L, geE, then T C

( 2 ) f ( x ; |) - /( ж ) = — f{f (x -\ -t )- 2 f { x )+ f{ x -t )} 0 ( t, £)dt n J ₀

for every x , and

7C

(3) /(ж ; 1 )-/(ж ) = — f { f { x + t ) - f ( x ~ t ) } { l c o t ^ t - 0 ( t , £)}dt 7C

0

for almost every x.

2° Under the restriction / e Wp or /e Wp\ £еЯ, we have

(4) /(ж )-/(ж ; I) and

(5) /(»; Ś)-f(oc)

T / / (P) (*) ^ 1 jp * ^ c°s (< -a?) ^

— n fc= 1

1

7ТГ

T t 0 0

J / (p) (0 ^ 1 ■ J p -— sin {A: (< -ж ) + |ртс}dt for an arbitrary x.

3° Let Tp be the class of all trigonometric polynomials g snch that j \g^(x)\dx < 1. Obviously T x с Л^, T 2 a Z (2 \ Tv c= Wp* (see [11],

— Tt

p. 180, ex. 9). Applying Theorem 3 given in [ 6 ], we easily obtain the fun­

damental inequality

(

6

TT T t

sup ГI f g(p)(t)K (t—x)dt\ dx o^Tp

^ j \K(x)—K ( x Jr ii)\dx о

for any 27r-periodic function K(t) Lebesgue-integrable over < —тс, 7 i>.

2. General theorems. Let us start with L em m a . Given an integer p ^ 1, we have

(i! sup

feTp _J f ^{I№ ; —} _7Г

oo fc =0

1 ) 1 £ 2 fc+l(£) (2&+l)p+1 ’

(ii)

T T

|/(ж; £)-/(ж )| 4

^ж > -—

тс

00

2 V 1)

fc = 0

kp ¹ @ 2 fc+l(£)

( 2 & + l f +1

for every fixed £eE.

Proof. Denote the left-hand side of (i) and (ii) by Sp(ś) and Sp{łj), respectively. Representation (4) and inequality ( 6 ) imply

7C TC

&p(|) = sup-*- J| J сов {к (t—x)+$piz}dt

— TT — TC k = 1

utv 2

dx

— TT — T t

T t o o

I ^{1 @ 2 fc+l(£)}

tvj

0 fc =0

v '

cos{(2ft+l)a? + |p 7 i} dx.

Therefore, in the case of p oddJ

TC o o

0 k = 0 v '

1

k = 0

1 Q 2 k + l ( £ )

(2Tc+l)p+1 ’ and in the case of p even,

TC o o

c . 2 Г I V-! l — ^Q2k+i{£) /07 ,

$«(£) > — > --- ,, cos( 2 fe-fl)a?

pV ' 7Г J \ ^ J ( 2 fe+l)p

0

&=o ' ' n/2 oo

> — f cos( 2 fe + l)xdx dx

к f 2 (21c + l f

= ± V ( - D

0 fc=0

4 y ^ ^ y f c 1 —&*+i(£)

TV

(2fc+l) ^p +P'*

Analogously, by (5) and ( 6 ),

71 TC OO

,V,,(s) = s u p -1 f f > W У L .

>’ tp 71 A A H 1

dx

/ 1 2

1 (? 2 fc+l(£) (2Й + 1У

0 k = 0 ' 7

sin {(2 A: +1) x + \p tv } dx.

If p is even, Spd) > ²

TV ' II

1 £? 2 fc+l(£)

( 2 k + l ) p sin( 2 &+l)#da? 4 TV

OO

1 @ 2 fc+l (£) (2Jc+l)v+1 ’

if p is odd,

Sp(S)

(2fc+l)p cos ( 2 ^ + 1 ) a? I dx > 4 TV

1 @ 2 f c + l ( £ )

T 2 fe+T f+r *

Thus, estimates (i) and (ii) are proved.

T heorem 1. I f Ф(/, £) > 0 for t e ( 0, тг>, £eS, then

7Г oo

sup Г|/(ж; £ ) - / И 1 ^ = — V 1 f 2fc+-1^ = Slip max |/(a?; £ ) - / И 1 *

/eZ(i) _■£ 7Г ^ ( 2 A;+ 1)2 /eZl

P roof. In view of (2),

тс я

sup Г |/(ж; I ) —/(ж)|йя? < — Гt<J>{t, £)dt = Л ( 0 ; I)—Л ( 0 )

И Р Л " o'

= sup max |/(я?; f ) —/(a?)|, /eZj — тт<а:<7г

where / х(ж) = \x\ for a?e< — тс, тс>. Since тс 4 cos(

2 &+l)a>

Л И - Y “ 7 Ż ( 2 /c + l )2 ’

Л И £) = @2k+l(£) cos( 2 & +l)#

2 ( 2 * + 1)a

for же<-тс, тс>, л\ге have

oo

Л(0; ^)-Л(О) = — У -1-~ g2fc+l(^-.

J J У ’ iz (2 & + 1 )2

On the other hand,- by Lemma,

n

TC oo

sup Г ^I f ( x ; £ ) —f ( x ) \ d x > sup f| f ( x - , l ) —f ( x ) \ d x У ¹

/.zW J , Ti Z j (2 * + l)>

and the required formula follows now immediately.

Also, a related result holds. Namely:

T heorem 2. I f W ( t , £) = ± c o t% t — 0 { t , £) > 0 f o r t e ( 0, тг>, f e S , th e n

7Г OO

sup Г ^|/(ж; £) — /(a>)|<&& = — V (— 1)*-—g2fc+1^

M ?)- k тс A J _{k= 0} (21 c +1)2

= sup max \f(x-,£)-f(oo)\.

fеЛj — тг<х<тс

Proof. Identity (3) can be rewritten in the form

tt/2

/И ^i)-f(oo) =— f {/(a?+t)— _{TC J} /0»— £)d$ —

о

-- Г {/(®+*+"-<)-Л*+я-я+«)} _{TC J} тт ⁹ ,((, {)й«

тт /2

5 — Prace Matematyczne XI (1967)

for almost every x. Therefore,

тс тт/2 тс

sup Г |/(ж; £) — /(ж)|йя?< — I Г |)й < + Г(тг— £)й/1

К ) Л 71 У- тс% j

= /

(

; l ) - / 2(0) = sup max |/(ж; £)-/(ж)1, /еЛ^ —

where / 2(ж) is 27u-periodic, odd, such that

/ 2 ( ж ) =

ж if 0 < x < 7 г /2, ти—ж if 7 u /2 ^ a? ^ 7u.

Since

4 v i / 2 (ж) = — > ( - 1 )

7U fc =0

& sin(2&-f l)a?

( 2 &+ 1)2 ’

~ 4

f 2(x) = ---> ( - 1 ) 7Г fc =0

vA. cos(2&+l)&

(2fc +l)2~ ’ and

^ fc

cos(2&+l)a?

(2 & + l)2_

for each x e ( — ж, тс}, £ e 3 , we have

/ г ( 0 ; ! ) —/ 2 ( 0 ) = — Y ( - l ) 4

7U k=o

к

( 2 &+ 1 )2 Applying Lemma, we obtain

sup Г|/(ж; £ ) - f ( x ) \ d x ^sup Г|/(ж; £)-/(ж)| V ( - 1 ) к—О

&4 @ 2 fc+l(£) ( 2 &+ 1 )2~’

and the conclusion is evident.

T heorem 3. I f <P(t, £) > 0 /or U<0,7u>, £еЯ, i f

fc=i

&+i 4fc(&+l) —(2&+1 _{_ _ _ _ _} )2 {>i(£) + £ 2 fc+i(£)

0 { 3 —4ei(^ )+ e,(f)}

as £ -> £0, Йе?г

7Г

sup / |/(*; = l - ffl(f) + 0 { 3 - 4 ffl(f) + ffs(f)}.

*44 - i

Proof. In view of ( 2 ),

TC * TC

sup f I/(ж; I ) —f(oc)\dx < — Г^ 2 Ф( 1 , £)dt

uz\f) Л 71 ^

7Г T U

= — Г ^{( 2 в т ^ ) 2} Ф(«, £)d<+— f ^{{«2 — (2} sini«)a} Ф(«, £)dt

TC

о ⁰

TC TC

= — J*(l—cost)<P(t, £)dt-\-0 j ^{J ( s i n i)dt} j

1 — gi(£) + ^{3 — 4^ i (^) + ^2(^)}- By Lemma,

snp f \f(x-, £)- f(x )\d x / 4 1}-

sup f I/(ж; ^)—/(а?)|йя? V ( - l ) ;

*T 2 _i те «

1 g 2 fc+l(£) ( 2 fc + l )3 4

7T

00

I ’ f - u fc = 0

l - g l ( g )

( 2 &+ 1)3 4

7Г oo

fc= 0

g l(£ ) — g2fc+l(£)

( 2 &+ 1)3

4

7Г oo

к— о

l - g i ( D

2&+1

~\---- 4 7C _fc

= 0

■ 1 ) fc-f-l 4ft(fe+l)-(2fc+l)2gi(g) + g2ft+i(g)

( 2 &+ 1)3

= l - e i( f ) + 0 { 3 - 4 e i(f) + gs(f)}.

Thus, the proof is completed.

B e mark. Under the assumption <f>(t, |) > 0 , the asymptotic rela­

tion

sup max |/(ж ; |)- /(ж ) | = l - M ! ) + 0 { 3 —4e1(£) + M £)}

ftZ^ — tt <£C< t c

can easily be deduced, too (cf. [ 1 ]).

How, writting

Ap(£) = snp max |/(ж; £)—/(a?)|

and

= S lip Г|/(ж; £)—f(x)\dx, feW§)-n

we shall demonstrate the following T heorem 4. Suppose that

00

Qi(t} i) = —

vanishes only at the points t = hn (Tc = 0, ± 1 , ± 2 , ...) for any £eE.

Then, given an integer p ^ 1 ,

4 4 (f) = ± y (__ i )'''(>'+

fe=o

1 {?2fc+l ( I)

(2*4-1 f +1 P roof. By identity (4),

with

4 4 (f) = sup uwm *

7Z П

J | j f v){x+t)Qp{t,

— TC — TC

dx

0 0

Qp{t, i) = ^ COS (Jct + $pn).

k= l

The function Qp (t, £) is continuous in t it p ^ 2 , and Q'p{t, £) = —Qp^ x(t, £);

hence it can easily be shown that Qp(t, £) possesses at most two roots in every half-open interval of length 2n.

In the case of p odd (p = 2 r + l, r — 0,1, 2, ...) the function

Qp(t, £) = ( - l )r+1 ^ sin Jet k= l

vanishes only at the points t = Jen (Jc = 0, ± 1 , ± 2 ,...) . Then

TC

4 l,(f) < — _TV f I Qp(t, 01 dt J

2

TV 0 fc= 1

sin Mdt

oo

d ~ g 2 fc+i(l)

( 2 f c + l f +1 ■

If p is even (p — 2r, r = 1, 2, ...), then

OO

Q„(t, £ ) = ( - ! ) ' £ с о Ш,

and the difference Qp{t, £)—Qp{n/2, f) vanishes only at the points t — ^ic-f-hic (J c = 0 , ± 1 , ± 2 , ...). Consequently,

71 TC

4 4 (f) = sup — f f f v){x+t){Qp{^ £)—QMiz, £)}dt uw$ n J J

dx

n/2 n

~ j ^{I Qp(t,} f ) - e » ( K f)|d« = -=-| (J ^- fj{Qp(t, D - Q p O tz , l)}dt

n/2 oo

0 /Г A =0

1 £? 2 fc+l(£)

( 2 k + l f " ( Я + 1 ) , Й

0 n/2

4 TC TC _fc

= 0

fc 1 g 2 fc+l(f) (2Jc+l)P+1 '

On the other hand, by Lemma, 4 4 ( f ) > sup

utv

TC

*/ TU

— тс

oo

Jfc=

1 @ 2 fc+l(£) (2A;+l )J,+1 ‘ Thus the first part of Theorem is established.

Analogously,

1 "

4 (f) = SUP 1 / ( 0 ) —/ ( 0 ; f)| = sup — f f p){t)Qp(t, £)dt

uwp uwp 1 тс

sint)Qp(t, £)dtI for p odd,

— 7T

71

sign (cos t){Qp(t, £)—Q p {\ tc , $)}dt\ for p even, i.e.,

4(f)

k-= 0

1 ^ 2 fe+l(f) T 2 fc + l ) pTr ? and the second part is proved.

Applying identity (5) instead of (4) and arguing as previously, we get T ^{h eorem} 5. I f the function

O O

- V l l — Qk(£)

Qiih i) = 2 , — с о Ш

fc= 1

has at most one root in the interval ( 0 , тс), then

тс 00

4 V"! . ' . .ftp 1 g 2 fc+l(£) sup f |/(ж; i ) - f ( x ) \ d x = — V ( - 1 )A

(2&+l): ^{г >+1} sup max |/(®; £)—/(®)|, /еИ7», — r:<a:<7T

where p is a positive integer, p > 2 , £eS.

S. Examples. ¹ . It is well known that for Abel-Poisson integrals

7C OO

f(x-, r) = — J /(<) fa + J ^ r fccos&(ź-a?)jdż (0

^ -n ' fc= 1 *

< Г < 1 )

the assumptions of Theorems 1 and 4 are satisfied. Hence

sup I \j(ar,r)-f{x)\dx = — V

71 ^

1 — r2k+1 2 1

--- 2 = — (1— r )ln --- [-0(1—r) тс ( 2 &+ 1 ) Ti 1 —r

as r -> 1 —. This is an analogue of a formula given in [9]. Further,

r 4 ^ l _ r2fc+1

sup I \ f ( x - , r ) - f ( x ) \ d x = — у ( - 1 ) (i>+1) = 0 ( 1 —r)

/.irWjl 71 t i <M+1)

if p > 1 . The asymptotic formula for the last series is known (see [9]).

Similar results for biharmonic operators can be obtained as in [3].

2. Theorems 2 and 5 apply to the conjugate Abel-Poisson integrals

7Г OO

f(oc-,r) = --- j f ( t ) ^ r ksinic( t - x ) d t (0 < r < 1 ).

-ТГ f t = l

Therefore,

Л OO n J ,

i X

r ~ ~ 4 v i Д — r + 4 rarctgo;

sup I \f (x -,r )- f(x )\d x = — У ( _ 1 ) /ox. 7 i \ a = ~ ■ — “— dx

, ,(n J 7 Г ^ (2 a ;+ 1 )2 7 z J x

/ел9)Л (cf. [4]), and

it oo

sup Г \ f{x-, r)—f(x)\dx = — 'S1 ( —l ) kp ¹ — r ² *+l

(23fc+l> ^{, 21 + 1 = 0(1 —r).}

In this case the asymptotic formula can be deduced, too.

3. Now, we shall verify the second condition of Theorem 3 for the Jackson integrals (see [ 6 ])

f { X )

n) = ---3--- [• ( Sini,,;, 2 тг%( 2 % 2 + 1 ) J j \ sm |(2 — x) I

— TC

T U oo

ł i 4 ł + 2 Qk (n ) cos Jc {t — a?) j dt, where

Qk{n)

i ( 2 n — i ( 2 % —

—fc+l)! (n-Tc+1) 2 %( 2 % 2 + l ) [ {In—1c— 2 )!

{In—&+ 1 ) !

- 4 i!

2 %( 2 % 2 + l ) { 2n -T i-2 )\

0

' { n - J c - 2 )

when 0 < 1c < %— 2 , when n —2 < 1c < 2 n —2, when A: > 2 %— 2 .

Clearly, 3 —4g 1 (w) + £ 2 (w) = 0 { l / n 3), and

2 i ^

fc+i 4fe(fe+l) — (2fc4-l) 2 gi(^)4-g2fc+i(^)

( 2 ^{& +} 1 )3

\n-2j

2 %( 2 % 2 + l ) V ( - i ) * +il i --- -— 1 + f - i l (2 J t+ l)« |T

f t = l П— 1

+ 2n{2n

3

1 fc+i | 4 %(%2 — 1) 2(6%2+% — 1) 12% j

»a+ l ) (_ 1 ) \ ( 2 ft+ 1)5 ( 2 ft+ij* h 2 f t + l _ 1 ) +

* - F i V

+ 3 V ( - p ^ - J - - V ( - p * + - - j— = о Ш

2 » 2+ l Z j 2 f t + l Z j (2 ft+ l)» \ я 3/

Hence

n

sup I |/(#; w) —/(a?) I da?

j */

3 / 1

---

(-0

2%2 \%3 = sup max |/(a?; %)—/(a?)|

/е ^2 — п^Ж^я

4. Consider, next, the Cesar о means

п П

Г (я i п) = - i j ^ A yn_kcoslc(t-x)^dt (у > 0 ),

’k = l

where

A l _{( T )} (y+ l)(y+ 2 ) ...

V

I

Write е*(тг) = An_kjAn for Те < тг, р£(тг) = 0 for Tc > n, and

00

0 y ( t ,

w) = £ + ^ e f c ( w) co s f c < . fc=l

If у ^ 1 , the kernel Фу {t , w) is non-negative ([11], pp. 78, 88 , 94).

Then, our Theorem 1 and Theorem 1 of [2] yield

r 2 1 п п I 1 \

sup \\f(x-j n ) - f ( x ) \ d x = — у ---\-0\ — )

fczwJn ■ 7z n \ n j as n -> oo.

Now, we shall show that the last formula remains true when 0 < у < 1.

Clearly, 1 0 y{t, n)\ < n-\- 1 for у > 0, źe < 0, л:>. We also have 0 y(t, n) — + sm{Nt — ъу-к) 2 y { l —y )0 { t,n ,y )

n -\-1 ( 2 sin ^)2 А уп{ 2 &тЩ\V ⁺¹ [n + 1 ) (n + 2 ) ( 2 sin -|£)£

for 0 < у < 1, te(Tz/n, 7u>, where iV = w+-J + |y , \0(t, n, y)\ < 1 (see [11], pp. 94-95); hence, by (2),

f l f ( ^ ; n ) —f(x)\dx

— 7V

TT TC

= ~ / I / { /( ^ + ^ ) - 2 /(ж )+/(ж -«)}Ф у(«, n)dt f ( x + t ) —2 f( x )+ f( x —t)

- n 2Tcjn

Y

dx-\-0 ( i )

- П — Г 7C I I W+l J W+l

— tc 2 n / n

( 2 sin ^ )2

+

7T

■Jy j { f { x + t ) - 2 f ( x ) + f { x - t ) }

n 2njn

dt +

sin(jW—%уп)

( 2 sin \t) UW+1 dt dx-\-0 ( i )

TC

1 f ^{n -\-1} n , f ) + — 12(x, n , f )

-An. dx-\-0

Ш-

uniformly in f e Z ^ . Obviously, we can write 412( ж , n , f ) = J 2 { f{ x +t )—2 f{ x )+ f( x- t)}

2 ir/łl

s i n ( B -

( 2 sin -^)y+1 dt —

■n—n/n

- f \ f [ x + t + — \ — 2 f { x ) + f ( x — t — —

J [ \ n I \ n

njn

Я + Я / П

— J |/(ж+г- — ) - 2 f { x ) + f [ x - t + —

sin JV< + ^ (l+ y )- ^{---± y}

n

Znjn П

h*K)]

sin

НН)Г ^dt

STZjn

я

= 2 i f + / ) t f ( » + « ) - 2 / W + /(» -» )} dt-

2я/п я— njn'

Зя/ft.

- - " Г / \ / u sin ^ + l ( l + y ) ---|77T

/ {/ e + * + T - 2/(e)+/(e _ ,_ ^ f— 2 s i n | l * + — n r — * -

TC + n/n

я— я/m /I/

я - я In

+ I _{3 jt // i}

sin(jV<— | ( l + y ) — - | y + ж + ^ - ^ - ) - 2 /( ж ) + / |ж - / + ^ -)[--- —--- j--- -d« +

M-C

2 № + / ) — 2 /(a>) +/(o?—<)] sin(JVi —^y 7 r)

/ ( Ж+ Н — I —2f(x) -\-f\x—t ---

n j \ n

( 2 sin ±*)y+1

8 ш | ж £+-| 2 ( ¹ + ⁷ )— — ^{1 у ъ} n

f ² sin ⁴ +^)r

Bi n m - * (i + 7 ) - - i y7c

- |/ |® + * - - ) - 2 / ( » ) + / |® - « + - ] j -

П TC \ y ⁺¹

t — WJ

dt

= 2( + iK *b/) + Y a(a?, n , f ) ) - Y 3{x, w , / ) - Y 4 (a?, w,/) + Y 5 (£t?, n ,f) .

It is easily seen that

7X

Л f I Yi{x, n, / ) Idx = o l —

ny J \ n

for i = 1 , 2 , 3, 4, uniformly in feZ^K Further, TZ—TCjn

n, f) = J ² /(^ + ^ )—/ |^ + ^ + f | Ж-Н — I \ l sin {Nt — \y, /J ( 2 sin |/ ) y+1

+ 1 2 / ( ^ 0 / ( * - * - " ) ] 7 a S r ,łt) +

/ | а ? + < + - ^ - | — 2 / ( ж ) + / | ж — < — X

7Г)

• / лт-ł 1 ч s i n m + i ( l + y ) — - | y :

sm(NY—£y7r) \ n

( 2 s in i( r [ 2 gi n l ( « + ^ ) ]

+ [/ (ж+(_^ )_2/<а:)+/(ж_(+^ )]x

+

X s in ( ^ —■JyTC) ( 2 sin ^)y+1

s in J ft—$ (l+ y) — ---iv ТГ 71 i V

n f

[2sini(*-^)] ⁺

>dt.

Next, an argument similar to that of [10], p. 422, leads to 7t

"T f \Y ^ X’ ny J — rc О l—), \ n j uniformly in feZ^K

Consequently,

П T U

sup Г [ f (ж; п ) - / ( ж )| а Ь = - . - - sup \ { I ^ x , n , f ) \ d x + 0

2 / 71

/

* (w + 1 >J»<2sinW , dtf + O Ш

2 Inn

= — у---\-0

71 П

1

П J

Now let деТг. Then,

7t

gv(x’,n ) - g ( x ) = — fg' (t ) K ( t- x )d t,

TC J

with

к м = — + 2 j - j ~ ~ i 0 { t >n)dt-

k=l

It can easily be observed ([11], p. I l l ) that

тг, x У 1 , i . 2 0 !(a>, n, y) B ( x , n , y ) К (x) --- cot \ x +

n -\-1 2 n Avn(2sm %x)y+l n2x2

and

K {x - \ -tz) =

У 1 ,! 2 в г{у,п,у) B { y ,n , y )

п -\-1 2 соЦу n A yn{2$m%y) 14ЛУ +1 п 2у 2

for ж е < 7 г /п, тс—n /пу, where у = л —ж, |0 х(ж, п, у )| < 1, |В(х, п, у )| < С, С being an absolute constant. Thus, by ( 6 ), we get

sup I 1/У(ж5 n )—f{x)\dx ^ sup j Igv(x\ n ) —g{x)\

An geTi

dx

— TC

Ti—nln

- f \K{x)-K{x+ic) TU J

тс jn

dx

1 у ТС—П/П

tu т г - | - 1

J c o t ^ ^ + o | —

n/n ' ^

2 hm / 1

— y — ---- bO — i, 7U П n

and the desired formula is estabhshed.

5. Finally, let / у(ж; n) be the conjugate Cesaro means, that is,

TC 0 0

f v(x\n) = ----— ff(t) ^ g l ( n ) s i n k ( t —x)dt.

-n k=l

If у > 3, the difference

П n) = |c o t \ t —

is non-negative for <e(0, 7 r>, n = 1, 2 , .. .([ 8 ], [11], p. 78). Then, Theo­

rem 3.1 of [7] leads to

(cf. [ 8 ]). In view of Theorem 2, the left-hand side of the last formula can be replaced by

[1] Л. И. Б а у с о в , О приближении функций класса Za положительными методами суммирования рядов Фурье, Успехи Мат. Наук 16, 3 (99), (1961), рр. 143-149.

[2] Guo Z h u -ru i, Approximation of a continuous function by Cesaro means of its Fourier series, Scientia Sinica 11 (1962), pp. 1625-1634.

[3] G. К ан и ев , Об уклонении бигармонических в круге функций от их гра­

ничных значений, Доклады Акад. Наук СССР 153 (1963), рр. 995-998.

[4] В. Sz. N a g y , 8ur Vordre de Vapproximation Tune fonction par son integrate de Poisson, Acta Math. Acad. Sci. Hung. 1 (1950), pp. 183-188.

[5] С. M. Н и к о л ь ск и й , Приближение функций тригонометрическими поли­

номами в среднем, Известия Акад. Наук СССР, сер. матем., 10 (1946), рр. 207-256.

[6] Г. П. С аф р он ов а, О методе суммирования расходящихся рядов, связанном с сингулярным интегралом Джексона, Доклады Акад. Наук СССР 73 (1950), рр. 277-278.

[7] R. T a b e r sk i, Approximation to conjugate function, Bull. Acad. Polon.

Sci., Ser. sci. math. astr. et phys., 10 (1962), pp. 255-260.

[8] — Asymptotic formulae for Cesaro singular integrals, ibid., 10 (1962), pp. 637-640.

[9] А. Ф. Тиман, Точная оценка остатка при приближении периодических дифференцируемых функций интегралами Пуассона, Доклады Акад. Наук СССР 74 (1950), рр. 17-20.

[10] П. Л. У л ь я н о в , О приближении функций, Сибирский Матем. Журнал 5 (1964), рр. 418-437.

[11] A. Z y g m u n d , Trigonometric series, I, Cambridge 1959.

TU

sup f |/ y(^; n )— f(x)\dx.

M ( / ) - 7 Г

R eferences

KATEDRA MATEMATYKI II, UNIWERSYTET A. MICKIEWICZA