1. Preliminaries. Denote by В* the space of all 27r-periodic functions / integrable in the Denjoy-Perron sense on the interval <0, 27

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE X IX (1976)

P. P

(Poznan)

The D en joy integral

in some approximation problems, IV

1. Preliminaries. Denote by В* the space of all 27r-periodic functions / integrable in the Denjoy-Perron sense on the interval <0, 27

). Consider the points æ for which

h

(1) (B*) J (f{œ + u ) —f{œ))dv, = o(h) {h->0 + ),

- h

and write

w(f,oc-, h)D = sup {—

0< t < h l 41

for any f e B*. Evidently, the function w(f,cc', h)D is non-decreasing in h and, by (1), w { f , x ; h

( h 0 + ) almost everywhere.

Let 0 < t < 7u and let

t

(2) f x(t) =/(<» + « )+ /(я» -« )~ 2 /(ж ), Fx(t) = (B*) j f x(u)du.

0

(-£>*) J (f{<D + u)-f{a}))du

We have then

t

(3) \Fx(t)\ = |(D*) f (f{a> + u ) - f ( x ) ) d u \ < 2tw(f, X) t)D.

-t

In this paper we consider some linear operators defined by I)*-integral.

We give estimates of the rate of pointwise and mean convergence of these operators. In order to show th a t our results cannot be essentially improved we introduce the class B(Q) of all functions f e B * such th a t

w ( f , æ : h ) D

sup --- < 1 at a fixed w, o</i=ot В (Ji)

where Û is a non-negative and increasing real-valued function defined

on <0, 7i> with i2(0) = 0.

For 27t:-perio(lic and Lebesgue-integrable functions, in symbols fe L, the corresponding estimates will be given in terms of the expression

t

\ 1 Г

J i ) l = SUP Ьтг \f(x + u ) - f ( x ) \ d u 0 <t^h I J

used in [1]. Clearly, w{f, x\ h

(h—> 0 + ) at every point x for which

h

f \f(x-\-u)—f( x ) \ d u = o ( h ) ( h 0 + ) . - h

Moreover, w(f, x\ h)D ^ w ( f , x; h)L.

In the sequel M x, M 2, ... will denote the suitable positive constants.

2. Pointwise polynomial approximation. Given any

D*, we consider trigonometric polynomials

UJx-, f )

71

= - ( - # * ) f /(«){* +

7Г J

— 7Г

'S

A^ao&k(t — x)}dt,

*=i

where the factors Яр* (k = 1, 2, ..., щ n = 1 , 2 , ...) are some real numbers.

W rite

ЯР = 1 (* — 0 , 1 , 2 , ...),

z u r } = Я Р -Я [% (к = 0 , 1, 2, ..., л - 1 ) , л я р = яр,

^ 2я р = М и) - M i l (* = о , 1 , . . . , и - 1 ) , ^1ЧЯ) = 4 4 Я 1 *яр = А Ч ^ - А Ч ^ и (к = 0 , 1 , . . . , ^ — 1 ), 21 «яр = 4 И).

We start with the following

T

1. Let f e D*. Suppose that for all n П

(4)

and fC— “ 0

(6) (» + 1 ) 2 '( * + 1)M*4*,I

fc=l

Then

(6) |17„ ( « ; / ) • iH3

/

—/(#)| < --- > w ( /, a?; —

^ Л w + 1 Z

V

’ к k= 0 x P ro o f. Retain the notation (2) and set

(n = 0, 1 ,2 , .. .) .

ФкУ) ^ {(sin\ ( k + 1 ) «/sin\ t Y ).

Applying twice the Abel transformation, we get

ТС П

#»(®i Я - / 0 » ) = ^;(-z>*) / ^ л {к )со$ЩМ

&=i l

2

tc

Tf n

0 *=o ' л '

By partial integration ([3], p. 244) 1 n

P„(®! / ) - / ( * ) = — A4 % \m n \(k + l)TzY-

k=0

1 Tt n

- j

^ f ^{F x(t) £} А Ч Р ъ (t)dt.

* О * = 1

I t is easy to see th a t

f e W K S T c f i + l ^ r 1 if «e(0,7ü/(» + l)> ,

and

n 3 » n

y 1 и ! 4я)1 ⁺ ~ j ^{^} ( f c + l)/J 4 » > sin(fc + l ) f

f t = l fc= 1 A := l

if ïe<7r/(w + l ) ,

). These inequalities and (3) lead to '

79, IT 99.

1 2

< w

|J»(")I ] ? \ A 2W \ + ~ ^ j |.F«(<)l|J^24“V*№

Л;=0 О &=1

W 7Г n

( / , * ; / *«>(/,®j « ь Ц > + 2Я(Л>*(г)

fc=o о fc=l !

n n n/(w+l)

<w(/,®î 7t)2,^,M !AP| + 2^l(ü + l)2M 24”)| J »(/,»; tb<ff +

A=0 &= 1 0

n 7Г

+ 7Г2^ И2я(^| J t~2w{ f , X ‘, t)Ddt +

к — 1 тт/(?г4-1)

л n

+ ^ J ^ас, Од I ^{^} (fc + 1) 42Aj^sin(fc + l)t

Tt/(M+1)

= I 4- Y + Z + 1 7 . _{n I n 1} n ^ " n

In view of (4),

г < М г тсЬ < т г т т ^ +

^{\ 1} \ ^/ />®; - r r r ’ ^ + 1 /п

^\

- У ( й + 1)2И*Л?>|«;(/,®;

ю + 1 ^{\ n + l JD}

п + 1 27Г

ft=l

I тс \ 2тсМг

П

тс

<2тг JfjW / , я; — —- ^ У.ю />®; т г г ,

\ w + 1 /Х> ^ + 1 \ D

п n+l I \ тс М

Zn = 71 ^ _*=i И24и)! J wl/,a?; _{1 '} ^y ) _{'D г '} тИ _'

Т с М г V I / тг \

< —--- > w l f . æ : ---.

п + 1 Z

Г А; + 1 I d

Applying the Abel transformation to the sum in Wn we obtain

n _ n

(lc + l ) A

Psm(lc + l)t = (w + l ) J 2A<?> J T s i n ( i + l ) i +

*=i

+ ^ { ( к +

) ЛЧ ^ - ЛЧ ^ } ^ sin (г+ 1) if

*=1 г=1

37Г

T

71 — 1

k = \ k = 1

for #€<7т/(»г + 1), 7т). Therefore, by (4) and (5), Зт с Ц М .+ М ,)

w m<

2(w + l ) / t w ( f , a?; t)Ddt

n/(n+l)

3tc( M 1 - \ - M 2

2(w + l ) 7 K(f’e>rl Зтс(М1-\-312)

dt ^ 27 > w / , a?; -— - 2(^ + 1) ^ V fc + l / p Collecting the results we get the desired estimate (6) with Жа

= Н (2 + 9тс) Ж^ + З

Ж,}.

Consider now the class D(Q). I t follows immediately from (6) that,, under the assumptions of Theorem 1,

sup I Un(ar, / ) —/(я?)I <

felJ(Q)

П n

k = о

The last inequality, for some operators Un(æ;f), cannot be improved.

This is a consequence of our next result.

T

2. Suppose that zJ2A ^ > 0 {k = 0 , 1 and that П

1 У (Й + 1)М 24 ”>> Mt > 0 ( « = 0 , 1 , 2 , . . . ) . n -f 1'

k= 0

Then for all 2 we have sup I UJæ-, f ) - f ( œ )I >

feD(O) П + - 1 k= 2

Ё“Ш

( Ms< м г).

P ro o f. Let g(t) = Q(\t — x j) if \t — æ\ < iz and g(t +

iz) = g(t). Then g

)*. Moreover, for

< t < h < тс we have

« t

(D*) J [д{со + и) — g (æ)Jdu = J Q(\u\)du ^

t û ( h ) .

Consequently, w{g,ec\ h)D^ Q(h) and geD(Q). Next, sup \Un(œ; f ) - f ( œ ) \ > \Un(æ’, g ) - g ( æ )| = | ü n(ar, g)\

VD(Ü)

where

= i f a W ÿ A n t f ^ + ^ î m

tz J \ sm*t

0 k = 0 x 1 '

ü(t)t 2(sin|(fc+ l)/)2d/

k = 0 n/(n + 1)

g+hn

n n+1

f йт*~г(»™№+Щ 2 ^<я)

k = 0 г=1 in

n+l

n n

4 т г ^ ( и + l ) 2 J T + 4 “ ’ | У f i P ” ( * + i r 2 ^ q ,

k=0 i = l ' ' '

(г+1)я

n + l

A U = f (sin|(/i: + l ) t f

(г + 1)(/с+1)тг

2(n+l)

dt k-\-l ^l(fc+l)TI / ^sin

^dt

~2(n + l)

(*+Ря

4(n + l)

k +

sin 2tdt > tz ( A: -(-1 )2

12 (n + 1)3

Since for all n > 2

И + 1

_ / г7С \ „ 7C /* 1 л ,

£ --- (г + 1 Г 2> ---- --- — Q(t)dt

U + 1 / 6(n + l) , J 1W

t2

(see [1], p. 28), we find th at

d-i)rc

n + 1

snp \Un(w, f) - f ( c c ) \ > 2

/£

»(

) 15тг(п + 1)2

157

:2(^ + 1)

^ ( f c + l ) 2/ !2^ J

± û ( t ) d t

fc=0 n/(n+l)

w M+l . .

-^(fc + l)2zl24“> J fl(y)<B

-Мд У

7

157т2(тг + 1) ^ \г-| -1/

.-.and this completes the proof.

R e m a rk . If a function / is of class L, then ЗЖ, \~1 / П 7C \

Я - / И К —T T Л ад( ^ а?;1ГГТ' (я- = 0 ,1 , ...),

™ + l \ 7 c +

I l

provided th a t assumption (4) holds. In this case Theorem 2 remains valid for the class L{Q) consisting of all functions f e L such th a t

w (/, ®ï h)L . _ sup ---—--- < 1.

0<Л<тт iQ(^)

3. Approximation by Abel means. As well known, the Abel means -of Fourier series of fe D* can be w ritten in the integral form

7T

-Р Д * ;/) = — (-»*) f f M j z

1

—Г

2

rcos(t — æ)+r - dt

(0 < r < 1).

We first give an estimate for the difference P r (o>; / ) —/(&) at the Lebesgue-Denjoy points æ defined by (1).

T

3. I f fc D* and n = [1/(1 — r)] — 1, then

n l \

((7) \Pr{œ; f ) - f { x ) \ ^ M

{ l - r ) ^ w ^ f , æ i (0 < r < 1).

The inequality cannot be improved.

P ro o f. Using the symbols (2) and integrating by parts we obtain

1 - r 2 r . 1

РЛ *; /)-/< » > = ( * ) f f M j z 2r cos i +r , о

Tzl(n+1) n

r ( l —r2) j Г Г \

+ -( ^I + f ) * ’ m W = dt

1

— r 27

(1 + r)

1

—r

sin t

nl(n+l)

((1 — r f + 4rsin2|tf)2 dt 27

(1 + r)

In view of (3)

1

— r

F X(K) + A(r) + B(r).

2

n{ l +r )

( TZ \

(1 - r ) w ( f , X ‘, 7C)jr><(l- r ) nnd

W l+ r) r „

u ( r ) K 7T (1 —Г)3 J ^,

^{j i p » ( * ) i td*}

2 r(l + r)

7Г ( 1 —

r)S

n/(n+l)

4

2

<

3(1 —r) (w + l) 64тг2

wi f , a?; 47Г2

n + l J D 3(1 —г) (тг + 1)

J* t

w( f , a?;

Ê ” ( f t æ ;

i t ï )

*=o n

a - o J T ' - j / , . ; ^

whenever 0 < r < 1. Moreover, B(r) = 0 if 0 < r < % and

2 r(l — r2) r . . tz

^ :zr(i — r*) r

\ B ( r ) \ < —-—- J «>(/, 7

<

7T/(W + 1) (1 —Г2)тС3 ,

((1 — r)2 + 4rsin2^ ) 2

8r / t w( f , a?; i^dtf rc/(n+l)

(1 — Г2) TC2

8

r

П

71

J d £ < - ^ 2( l - r ) ^ w | / , œ,

1 ^ k

provided th a t % < r < 1. Collecting the results we get the estimate (7) with M

= (6 + 131712 )/ 6 .

In order to show th a t inequality (7) cannot be essentially improved we consider the class D( ü) . By (7),

sup IPr (®; f ) - f { x )I + M 6(l — r)

f*D { Q)

*=0 '

1

On the other hand, using the function g defined in Section 2, we get s u p \Pr(x) f ) - f { x ) \ > IP r(x-, g)-g(®)l

feD(O)

1 — r 2 /■ 1

= --- O(t )---

7Z

J 1 —2rcccos t + r

dt

— r — гй r

— m

TV J

0 < n/(n+l)

Since (1 — r) < l/(n-\-1) < t, we have

(1 — r)2-f rt2

n + 1

dt,

sup

UD(Q)

\Рт(щ } ) - № \ > f r *Q( t ) M = A ( l - r ) j q ( j ) dt 7T/(n+l)

> --- (1 —r) У q [—^ — and this completes the proof.

B ern a rk . In the case of f e L we can get the following estimate

I-Pr ( ® ; / } - / ( » ) I < 1 2 ( l - r )

when 0 < r < 1 and n = [1/(1 —r)] —1.

Following DzvarseiSvili [2], let us introduce in the space D* the norm

2 T Ï

\\ f\\D = sup J (D*) Г («)/(»)<&» I,

<PeV 1 o'" 1

where V denotes the class of all 27r-periodic functions of bounded variation in <0,

n) and such th a t

sup \<p{x) \к 1 and var 9 ? ( # ) < 1 . 0<а:<2тг

Denote by Ev ( f ) D the constants of the best approximation of fe D * defined by the formula

Д .(/)л = in fj|/(-)-^ (* )lln (v =

, 1 , 2 , ...),

where the infimum is extended over all trigonometric polynomials tv(x) of degree not exceeding v.

Arguing as in [4], we easily obtain

T

4. Let feJD * and let n = [1 /( 1 — r)] — 1. Then П

\\p, ( •, / ) - / ( • )llz> « 94 ( 1 - »■) £ E, (J)D (0-< r < 1).

v = 0

References

[1] S. A lja n cic, R. B o ja n ic and M. T om ic, On the degree of convergence of Fejér- Lebesgue sums, L’Enseignement Mathématique 15 (1969), p. 21-28.

[2] А. Г. Д жварш ейш вили , Интеграл Данж уа и некоторые вопросы из анализа, Труды Тбилисского Матем. Инст. 25 (1958), р. 273-372.

[3] S. Saks, Theory of the integral, New York 1937.

[4] R. Taber ski, A theorem of the Steckin and Leindler type connected with Abel sum-

mability of Fourier series, Demonstratio Math. 8 (1975), p. 215-225.