On the rate of strong summability of Fourier-Chebyshev series

(1)

Problemy Matematyczne 14 (1995), 27 - 36

On the rate of strong summability

of Fourier-Chebyshev series

W ło d zim ierz Łenski M aria Topolew ska

1 P r e lim in a r ie s .

Let us consider th e s y stem of Chebyshev polynomials

Tq(x) = 1, Tn( x ) = cos(n arccos x ) (n = 1 . 2 , . . . ) ,

which are ortogonal w ith th e wełght g(x) = (1 — x 2)~* on th e interval [—1,1]. Let / be a real-valued function continuous on [—1,1], an d let

OO

(1) S [ f ] { x ) - ^ ckTk{x)

k=0

b e its Fourier series w ith respect to the system t h a t is

(2) co = - [ f ( ł ) g ( t ) dt, ck = - f f ( t ) T k[ t ) g ( t ) d t ( k = 1 , 2 , . . . ) .

7r J- 1 7r J- 1

D en o te by S n (x; / ) th e n - th partial sum and by <r°(x) = c r " ( x ; / ) th e n -th C esaro (C, a ) - m e a n of o rd er a of the series (1). Intro d u ce th e m o d u lu s of c o n tin u ity of / defined by

u { 6 ) = u ( 6 , f ) = sup ( sup \ f ( y + h) - / ( t / ) | j ,

(2)

28 O N THE RATE OF S T R ON G S U MM AB IL I TY OF

an d th e best ap pro x im ation of / by polynomials Pn of th e degree a t m ost n given by

E n{ f ) = inf { sup | / ( x ) - P „ ( x ) | ) .

Pn

l -!<*<!

J

T ake into account a regular su m m ab ility m e th o d d e te rm in e d by a trian - gu lar m a tr ix | | a nfc/An || w ith a„jt > 0 and A n = I X = o Qnfc- W rite, for each n o n -n eg ativ e integer n,

1 i/p

(3) i2n ( x ; a ) p = 1 ( x ; / ) - / ( x ) | P| , p > 0.

T h e aim of this note is to estim a te th e ą u a n tity (3).

In considerations, th e suitable positive co n stan ts in d e p e n d e n t of n. f a n d

r are d en o ted by K j ( j = 1 , 2 , . . . ) , and

7 (j/) = m in (2‘/ — 2 , n ) , 71(1/) = 2t/~ \ 72( 1/) = m in (2,/+1 - 2 , n ) .

2 A u x ilia r y r e s u lts

We s t a r t w ith an analogue of th e B rudnyi and G opengauz th e o re m (T h . 4, [1], p. 892).

P r o p o sitio n .

Let

OO

(4) F ( z ) = ^ c kz k, 2 = r e “ , 0 < r < 1,

Jt=0

where ck are defined by (2). Then

(5) ^

f

“ 4 ~ dt unif ormly in x € [—tt; 7t ] .

J l - r t l

(3)

On THE R ATE OF STRONO SUMMA BI LI TY OF 29 1 rr = — j f ( c o s y ) { P { r , y + x ) + P ( r . y - .r)} dy + 1 f r + / f ( c o s v) { Q ( r iV + x ) - Q ( r , y - x )) dy = — 9 i u (r i —x ) + u (r ’ x ) — i v (r, — x) + i v (r. x)} ,

w here u ( r . x ) . v ( r . x ) are the Poisson and th e Poisson co nju g ate integrals of

f { c o s y ) and P ( r , s ) . Q ( r , s ) are th eir kernels. respectivelv. T h u s

— F ( z ) ■ ( — d z 1 ’ \ d x O F ( z } d x 1 2 r d ( u (r, —a:) + u ( r . x ) ) , d ( v (r, — x) + v (r, x) ) d x — i d x

Therefore, it is sufficient to show th a t th e b o th term s of th e r ig h t-h a n d side of th e above relation are of th e order

Since we have 0 t 2uo{t)dt^j . l - , F s P { r ' s ) i s = 0 '

du

(r,

±x)

1 [*

dP{r,y±x)

di

=

i L f(c° sy)

di

dv =

=

±-

f

{/(cosy) - /(cosx)}

7r J - i r o y = ± -

f

{ /(c o s (f =F ■?)) ~ f { c o s x ) } - - -- - - - dt. 7 7 J - 7T o t

B}' th e m ean-value th eo rem , for arb itra ry real x , t and some 6 € (0 ,1 ), |/( c o s ( f + x ) ) — / ( c o s rr)| = \ f (cos x — t s in (x + 9t)) — f (cos x) \ <

< sup ( sup If ( y - h) - / ( y ) | | =

| / i | < l h L —1 < y » y — a < i J

(4)

30 O N T H E RATE OF STRONG S UMM AB IL I TY OF A nalogously, If (cos(t - a-)) - / ( c o s x )| < (tt + \ ) u f — ; / j . H ence d u (r, ± x ) d x sin 11 ( 1 —2r cos t + r 2)2 sin t dt = 7r J ((1 — r ) 2 + 4 r s i n 2( t / 2 ))2 t \ t

=

2(1

+ ; ) (1- rJ» r

~

^ " i '

^ ./i

\nj

((1—r}2-f-4r(£/5r)2)2

= 2,(» + l )(l-r

dt = A r g u m in g fu r th e r as in [5], pp. 150-151 we o btain du (r, ± .r) 8 x an d th is im m ed iately implies 1 d v ( r , ± x ) d x < A 3 f J l - T t 2ui{t)dt.

Now t h e desired assertion is evident.

R e m a r k 1. W h en x = arccos ?/, estim ate (5) holds uniform ly in y (E [—1,1],

L e m m a 1 . I f a > k > 0 and fc(l — a ) < 1, f/ien , ^ i/* (

6

) 1 2n . n + 1 Ś S u n i f o r m l y in x and n = 0 , 1 , 2 , . . . . < I<3

E

n + 1 *=0 a;

1

(5)

On T H E RATE OF STRONO S U MM AB IL I TY OF 31

P r o o f . Let us consider th e power series (4) and den o te by ~ "(y ) its m - th ( C . a ) - m e a n s for z = e ' arccosy. Easy calculation gives

i v a r c c o s y , u- 0 O -4“ « - ' ( ! / ) - < ( » ) ) = E - C - l ^ e " whence. for z = r ' v" (0 < r < 1), a £ / ) ~ T ° ( y ) ) z m = Z g , a r c c o s y F ' ( Z g i a r c c o s j / ) ( j _ m=0 F u rth e r, ta k in g a n u m b e r p € (1,2) such th a t p a > 1, by Hausdorff-Young in e ąu ality w ith q = we get

i 1.

' C O n } <> ( r * \ F f ( r f W p t a rc c o s y \ \ P ) P

- ki - r e - * r

M

■

.. 777 —0

In view of P roposition, th e right-hand side of th e above in e ąu ality does not exceed

7 i/p

K‘ r L i L rUt)dt)

\ p ((1 — r ) 2 + <d<p i22)°'p/2

S e ttin g 1 — r = 1/(2?? -f 1) and taking th e real p a r t we o b ta in our thesis (see [5], pp. 152-153).

L e m m a 2. Suppose that f o r each q > 1, the condition

( i 7(*d ) 1/,? "yW

(7) O E ( a , , ) * < A 6 ( 2 - ) £ a „ t

I, “ J r-=7i ((/)

^ 2"n - 1 . Then, for

i/p

P r o o f . P roceeding analogously as in [4] and using T h eo re m 3 from [6] we get th e desired result. holds f o r v — 1 , 2 , whenever 2"" 1 < + P € (0, oo), ' / M*/) \ - 1 7(p) ) i

E H

_{5 3} _{I M * ; / ) - / ( x ) | p}, V = 7 l ( " ) / k=~ti (v) J

(6)

32 ON T H E R AT E OF S TR ONG S U MM AB IL I TY OF

3 M a in r e s u lts

Now we present two a p p ro x im atio n theorem s which are analogues of some L eindler's results ([2], [3], [4]).

T h e o r e m 1. Under the assumpti ons o f L c m m a 2, we have

j , yn / ~A,A \ 'i X/ P

R n { x ] l ) p < ^ £ | Y . Qnfc) (£'-yi(^)(/))P | f o r n = 0 ,1 ,2 ,

An u = l \fc=7)M

P r o o f . T h e above es tim a te is a consequence of L em m a 2.

In p a r tic u ła r if p — 1, a n* — A„Zl, & > 0 th e condition (7) holds. T h u s we o b ta in

C o r o l l a r y . Under the a ss umpti on 8 > 0,

-Je f l A n - l I5* (x ; / ) - f ( x ) I < K 9 — ~ T Y E ^ f ) f o r n = 0 . 1 , 2 , . . . .

An k=o n -r i k=0

P assin g to th e m ore generał m eans (3), let us p u t 1 J L / 1

y

K") =

-v k=o ^ + 1

T h e o r e m 2. Suppose that a > 1/2, p > 0, p ( l — a ) < 1 and that

f o r each q > 1, whenener 2m < n + 1 < 2m+1 — 1. Then

1 A t/p

(7)

On T H E R AT E OF STRONC, S I M M A BI LITY OF P r o o f . Clearly, { R n { x : a ) ) p < S 2' ( 7 - Ż k r ' ( i ) - < ( • ’■)[ + E - /< * ) ! ’ I !/=0 !/=0

= ^{E, + E

j-C h o o se two n u m b e rs p' ,q' > 1 such th a t pp'{ 1 — a ) < 1 and p + ^ = 1. view of e s tim a te (6) and condition (8).

72(0

H

_Y1

q"" <

-*1" ;=o W=2' - i < 72(') PP < 72(0 < _{i - E B »j 5?}1 m / 1 _{E K “ - ‘w - < ( x )} -*1" /=0 i/=2( — 1 pp <

A" ^ ° '

" (2/— E

_-

₂

)jp <

< 1=0 (2A'3)p lri /=o £ 2 . T T T T T w - J < ( 2 A 3)p A i 0 r T T X > ( i O ) , 18 w w h e re #„,/ = ( a £ (^ + 1)* 1 U/=2'-l

(8)

34 On T H E R ATE OF STRONO S U MM A BI L I T Y OF r- m | 72 (0 1 2 e | 2 ((s- ,) Z < i / =2' - l A 1=0 ^ h 12

j

y ^ a n v ( lT ? ( , / ) ) PS 1 / r r r - 1 y y / >vp -

x S i £ ^ T T ) ^ )

S A . O A . - ^ E M w r

S u m m in g up these e s tim ates we o btain (9).

R e m a r k 2 . C ondition (8) is satisfied in th e special case when a nk = A SnZ\

an d fi > 0.

Indeed, observing th a t

*?(&)< <p{k') when k > k' an d

tp{21 - 1) < 2v?(Jfc) when 2' - 1 < A- < 2,+1 - 2

an d also th e fact th a t a nf: = A„Z\ (/? > 0) satisfy (7), we o bta in

^ ( A g : " ) 9 ( ^ ( ^ ) ) qp | 1/9 ^ 1—0 l i / = 2 ' - l

<

2 £ | M 2' - l ) f . 2' ^

Z ( A t l ) j

" ' U

m ( 72 (0 \ < 2 . A '„ t , M * - 1))” E < - ) < ^=0 \ u=2l —1 / m / 72(0 \ < 4 ■a' „ e e = i'=0 \i/=2! — 1 ) = 4 • A '13 £ A f c i (*(<'))” = i / = 0 /'[?] » \ = 4■ a', 3 E + E U S : l M O ) p < 1"=° «/=[Sl + l /

(9)

On T H E RATE OE S TRONG SIWIMABILITY OF 1Ź A i-i i / = 0 (n + 1)3- 1 D ^ + U - U ^ I + i J J ' Z < - l \ <

Hł

]+1

v

_—

n

_0-1

< /v15(t7 + 1)'3 f - ^ - r E M ^ r + i ^ D T

\n

+

1 tZo

)

R e fe r e n c e s

[1] BpyjjHbiH K ) .A ., r o n e H r a y 3 H .E . . 06o6w,eHue odn ou meopeMu.

X a p d u u JIu mmA beyd a. MaTeM. CSopHHK 52 (1960), 891-894.

[2] L eindler L., On stronę approiimation o f Fourier seińes, P eriodica M ath. H ung., Vol. 1 (2), (1971), 157-162.

[3] L eindler L., On the stronę approiimation o f orthoęonal series, A c ta Sci. M a th ., T. 37, Fasc. 1-2 (1975), 87-94.

[4] Leindler L., Uber dit Ap p ro ii ma t io n in starken S i n n e , A c ta M ath . Acad. Sci. H ung., T. 16. Fasc. 1-2 (1965). 255-262.

[5] Lenski W . , Generalizations o f iwo Leindler theorems, F unctiones e t Ap-

p r o x im a tio III. (1976), 149-155.

[6] R em p u lsk a L., Twierdzenia aproksymacyjne dla szereęów Fouriera-Czeby-

s ze wa , Fasciuli M a th e m a tic i Nr 5 (74), (1970), 63-69.

Streszczenie

O rzędzie m ocnej sumowalności szeregów Fouriera-Czebyszewa

U zyskano oszacowania uogólnionych mocnych dewiacji sum częściowych i (C . a ) - ś r e d n ic h szeregów Fouriera-Czebyszewa funkcji ciągłych od tych fu n kcji. B łęd y oszacowań wyrażone zostały w term in ach najlepszych przybliżeń i m o d u łó w ciągłości. Udowodnione twierdzenia stanow ią uogólnione o d p o w iedniki wyników L. Leindlera.

P E D A G O G I C A L U N I V E R S I T Y P E D A G O G I C A L U N I V E R S I T Y I N S T I T U T E O F M A T H E M A T IC S I N S T I T U T E O F M A T H E M A T C S

Zielona Góra 65-069 Bydgoszcz 85-064

Plac Słowiański 9 Chodkiewicza 30