f for functions / of class o, oo) and oo, oo), respectively. Analogously, letТГ

A N N A L E S SO C IE T A T IS M A TH EM A TIC A E P O LO N A E Series I : C O M M EN TA TIO N ES M A TH EM A TIC A E X I X (1977) R O C Z N IK I P O L SK IE G O TO W A RZY STW A M A TEM A TYCZNEGO

Séria I : P R A C E M A TEM A TY CZ N E X I X (1977)

P . T

(Poznan)

Differences, moduli and derivatives of fractional orders

1. Definitions and notation. Given any a

0, we have

( l + * r = J O ) z * il 1*1 < 1 ;

in the case \z\ < 1 , z Ф —1, the left-hand side is defined as exp (aL og(1 -j- +«)}. In particular,

2° = £ ( * ) ’ ° - i ; ( - i d ; ) -

k= 0

*=0

As well known, the penultimate Identity holds for a > —1, too.

The numbers

i / a \ . a ( a — 1 ) . . . ( a — Tc + 1 )

( - 1 )*(“) » ( ^ --- - are of the same sign for h > a > 0. Therefore,

k=Q

< OO.

It can easily be observed that

C(a) = 2 _ £ ( “ ) il 21 < a < 2 l + l (1 = 0 , 1 , 2 , . . . ) ,

C(a) = 3 ^ ’(ss.+l) U 2 1 + K a < 2 l + 2 (1 = 0 , 1 , 2 , . . . ) .

I Up

Write

OO

ll/ ( - ) lli« ,

= ess sup \f(co)\,

= { f \f(æ)\pda}V

—oo<x<co _ co

for functions / of class L°°( — oo, oo) and L p( — oo, oo), respectively.

Analogously, let

ТГ

/

l l / ( u2 n

= ess sup |/(a?)j,

__v J 11/(011 » = { f 1/И1 Pd®y I

for 2-rc-periodic / ’s of class —тс,

> and Ер( —п, тс>, respectively;

these Lebesgue classes will be signified by Lfn, Lfn.

Consider functions / of class either L p( — oo, oo) or Lfn (1 < p < oo).

Suppose that a>, h are real, and set

OO

+ for a > о .

7c= 0 ' '

Then, Theorem 11 of [2], p. 127, ensures that the last series converges absolutely for almost all æ. Therefore A^f(œ) is measurable and, by Levi’s theorem ([2], p. 126),

oo

U^hJ \ 7П

7c=0

if a > 0,

where || || denotes the norm in

or (1

The quantity Alf(co) is called the a-tJi difference of f at the point a?, with in­

crement h. Clearly, if / is uniformly continuous in (

its a-th difference is continuous therein. In the case of positive integer a,

A i m = J v i + +

k=0 ' ' *=0 ' '

Introduce now the a-th moduli of smoothness of the suitable func­

tions / :

ft>a(à,f)Lp = sup |HS/(-)I!

for a > 0,

|Л|<<5

ft>0 (ô ,f)LP = \\f(-)\\LP ( l < P < o o ) -

replacing || \\Lp by || || p , we get the a-th moduli coa(ô ,f) p . We shall

L2n L2n ,

denote by E n(f) p the constants of best trigonometric approximation' of order n, in Zf^-metrics. L2n

Suppose that

OO

« [ / ] = z

is the Fourier series o f/eZ fn. Given any a > 0, we define the a-th integral of / by the identity

OO

/ . ( * , / ) = ck(ih)-°elkx,

7 c = —oo

where

(ik)~a = \Tc\~ a exp { — (n/2) iamgnTc};

Differences, moduli and derivatives 391

the dash' indicates that the term with к = 0 is omitted in summation.

As well known ([5], p. 134),

exists eventually for almost every x, is Lebesgue-integrable and

If 0 < a < 1, the derivative f^ {x ) of / is defined by the formula

for positive integers r.

In the sequel the suitable positive constants depending only on a will be signified by Gp(a) (v = 1 , 2 , . . . ) .

2. Estimate for а-th derivatives of trigonometric polynomials. We shall now give a generalization of the inequality of Steckin’s type ([4], p. 228, 230).

Th e o r e m 1.

Consider an arbitrary trigonometric polynomial

CL П

(1) Tn(x) = +

{akco$kx + bk$mkx),

“ *=1

with real coefficients and n > 1. Suppose that a > 0, 1 < p < oo. Then, /«(») = *«(®»/)

£[/«] = / e(0) a.e.

f->(x) = 4 - J),

provided the righ-hand side exists. We set

if 0 < h <

n P ro of. As well known,

П

where 2c„ = av — ibv, 2c_v = av-\-ib„ (v = 1 ,2 , ..., n). Purther,

OO

n

У 'c,e‘" (

n

Therefore, putting

,,( « ) = (

i n | i ) , »(*) = l3(ùl(M /a) г j (1*1 < щ g(o )= л °), we have

(

® -|-A = E ' V {v)c' e,’X’ ^ )(æ) = E î vW sW o.J’

^ / v= —n v=—n

for

&e(0, 2п/п).

The function </(C is continuous and even in < —гг,гг>, g '(t )^ 0 , g " ( t ) ^ 0 for te (0 ,n >. Hence

OO

^r(i) = ^ dkeikntln uniformly in < —n , n} , k= — oo

and the coefficients of the series satisfy the condition ( —1 )kd±k > 0 (see [4], p. 230). Consequently,

n oo

Tio)(®) = Е ф ) E dke<i‘"’in-e^ ’x

v = —n k = —oo

° ° n , 0 0 ^/ t. \

= E E f (v)e’ eHx+h’ ln) = ^ + - ^

).

k= —oo v=—n Thus,

№ 4

8 HiV/

_

* 4 - 1 * ) k— — oo

= д{п)-\\А1Тп{-)\ I

rJL L2

and the proof is completed.

Co r o l l a r y.

Taking h = n/п and writing |] || instead of (2) \\T(E ( • )|| ^ (nj2)aC{a) \\Tn( • )|| /or a > 0 .

гое 7moe

/гг particular, if 0 < a < 1,

l|2î»(-)IK21- “» “ ||ï’„(-)ll- Moreover,

I Æ +1)( O I K C-2a^ r - t f + ’ l l T j • )|| = f ± i » « + 4 | T „

< 21~ana+11|T% etc. (cf. [4], p. 230, 266).

An argument similar to that of Theorem 1 leads to the more general

Differences, moduli and derivatives 393

T h e o rem

2. Let /u(t) [resp. y)(t)] be a complex-valued function of bounded variation [continuous] over an interval <( —a, o}, a > 0. Write

a a

/ ( * } = / , ( ( ) Л М * ) .

—a —a

where g(t) is as before. Then if either fe L p( —

) or feLf^

<

<

we have

11^Н1К</И11/(-)11,

with the norm || || used in the suitable IP-space (cf. [4], p. 229-232; [1], p. 49-50).

3. Properties of the moduli with positive indices. We suppose that a is a positive number, 1 < p <

and that the functions f , f x

f 2 are of the class L p( — oo, oo) or LfK. We write coa(ô ,f) instead of coa(ô ,f)Lp or coa(ô ,f) p • The norm in one of the spaces L p( —

is denoted

L2n

T h e o re m

3. The following basic properties hold:

(i) a>o(0,/) = 0,

(ii) <na(ô ,f) is a non-decreasing function of <5>0, (iii) ft>a(<5,/i + f

)< o > a {à ,f1) + (üa(ô ,f2) for <5>0.

The proofs of (i) and (ii) are trivial. Inequality (iii) follows at once from the identity

Л*(Л(»)+Л(®)) = Wi(a>) + Aif,(*).

T h eo r em

4. I / a > 1, then

\coa(ô1,f ) - c o a{ô2,f)\ < Ol (a)œ1(\d3- d %\J ) for each non-negative dxi ô2.

P roof. Since

\k /a à l +J(< e )- A tf(x ) = У ( - 1 ) u

k= 0

we have

oo

+ (a — h)(h + r])) —f \ x Jr ( a — h)h^f

k =0

oo

/c=o

^ CO

сох( \ а - Ц \rj\,f)

“)| (| a -i| + l)« )1(|4 | ,/)

.«h i,/){(«+D i ,|(;)|+ « ^ |(;:i)|}-

Hence

Мл+ч/ ( ‘ )11 < +

•with

Сг(а) = (а-\-1)С(а)-\-аС(а—1).

Consequently,

sup 1Ил+ч/ ( *)II < IHÂ/(, )ll + Cfi(«)o>i(<5>/)>

l*i<*

and whence

e>a($i + <*,/) < oia{à itf ) + C 1 (a)oj1 {ôff ) .

Putting ô% = Æi + Æ (<5^s 0), we obtain

a>a(Ô 2,f)-coa{ô1, f ) ^ C 1(a)œ1{ô2- ô 1,f ) if <5а><Зг > 0 . Thus the proof is completed.

Th e o r e m 5.

I f

a

o>p{àff ) < C { p — a)œa(ô jf) f o r ô > 0 . P ro o f. Writing

Ф(Х) = a if( x ) = ^ ( —D fc (°)/(® + (a —fc)A),

*=o ' 7

we have

OO

Л 1 -Ф ( х ) ⁼ 2 ’ ( - 1 ) ' ( / , 7 ° ) ф ( Ж + ( ^ - а - г ) Л )

1=0 OO 00

= у ( - i ) ' ? 7 l {_ £ ( - i ) * (“ ) /(® + (|8- а - ï ) Л + (а - Й)h)\

oo oo .

- 2 2 < - i>! 7 “ < - 1»*( ; / ( « + < / » - * - *>*)

Z=0 & = 0 V V '

oo v / \ ' / \ ч

- 1 { 2 v=0 >=0 < < - 1»" t 4 f{æ + ( ll- r) A) ,

00 v

= ^ ( - l ) 7 ( « + ( ^ - v)^) for almost every ® But

a |г|

fc=0 W Z=0 X ' »l=0 ‘ 7 V = 0 V = o ' ' ’ Consequently,

< 1

й = х е к ; : ;

Differences, moduli and derivatives 395

and whence

А1~°Ф(«) = J ; < - 1 >'(f)/(® + (iS-»)A) = Ai№ a.e.

r=0 ' ' Thus,

1147(011 =

г=о

/9-а 1|Ф(‘ )11:

and the desired result follows.

T h eo r em 6 .

I f r is a positive integer, then

cOa(2r ô,f)^(C (a)Y <oa(Ô J) for 0 > 0 .

P ro of. Obviously, it is enough to show this for r — 1, only.

Let and

Then

ij/(æ ) = Alf(os-dh)

v i m = J 1 (“) i»/(® - Щ ( « > « ) •

г = о ' 1

00 00 ^{/ \ / \}

w w = у У ( - i ) * ffl m л® - » - » ) i=o *=o \ / \ f

v=0 /«=0 4 v

---« /4=0

for almost every

—

Since

/1=0

0 if v is odd,

( - 1)W2(v/2) й v is even, we obtain

i.e.,

Consequently,

v i m = J ’ t - i ) 4 ( * ) / ( » -

» ) , Jfc=0 ' '

Vlf(<n) = ij»/(® ) a.e.

-)ll = Il4\f(0ll = М Я 0 И

°» ,

< Z \ l ) 1147(011 =<?(«) 1147(011.

b=0 ' '

15 — P ra c e M atem atyczn e 19 z. 2

Therefore

coa(2 ô ,f)< C (a )c o a(ô ,f) for <5 > 0.

Co r o l l a r y.

Given any

let us choose the positive integer r such that 2r_1 < X < 2r. Then,

ш„(Л<5,/) < o>a(2rS ,f) < {C(a))'<oa(0, f ) . In particular, if 0 < a < 1,

<oa(X Ô ,f)^ 2 ra>JÔ ,f)<2X ioa(ô ,f).

I n

the case 1

a

2,

<»„(M , / ) < (2аГ (Ô, f ) < 2Xa'ma(6, / )

s; 2аХа',щ^ < iX a'^-’ o jJ S J ) , etc.

Th e o r e m 7.

Let Tn{x) be an arbitrary trigonometric polynomial

of the order n ^ l . Suppose that 0 < a < 1,

— 0,1, 2, ... Then,

®e+. ( « , T n ) » < 2 λ) i f 0 < Ô « тс/».

* J2 n ^ 2 П

P ro o f. By the identity for the a-th difference obtained in the proof of Theorem

Æh+aTn^ x - - ^ ~ ^ - h j = J ? ^ 2 isin v jj cveivx = F(x) and

AQ hT ^ ( x ~ ^ h \ = ^ sin r-^ -j (iv)acveivx = f{x).

Therefore, putting

I \h \e /2 h \a

(p(t) = I2isin*j <1 (it)a, g(t) = (y s11* — 4 ( -w < t < л),

we have

and

/(®) =

F(x) = £'<p(v)g(v)cveivx.

v — — n

The function #($) is positive, even and g '{t)^ 0, </"(<)<(> for

if

h

Consequently,

oo

g(t) = JT 1 dkeikntln uniformly in <

k=—

Differences, moduli and derivatives 397

and

d0> 0, Further,

( - l ) * + 4 > 0 , d_k = d k (k = 1 , 2 , . . . ) .

Aknvln

Ща>) = <p{v)c,e»* £ ^<**«“

v=^—n k~ —oo

oo гг со

=

£

X

k— — oo v = —n k——oo

whence

РЧ-)11 - < У 1<У 11/(, )11га> •

&=—

£2

But

к= — оо fc=1

i.e.,

si» ) = J 1 <*»«“ * = j r ( - i ) 4 = «i0+ 2 ^ ( - i ) 4 ,

&=*=—OO

oo

à0-g {n ) = 2 J£ |d y . Hence fc-1

OO 2 71

J T 141 = 2d0 — g(n) = — J g(t)dt — g(n)

k = — oo

< 2<jr(0) — <jf(w) < 2</(0) = 2&“ (0 < ü < Ttjn).

Thus

11*4 O IL , < 2 Л “ ||/(-)11г, ,

•^271 ^Tt

which implies

]И Г ° 2 ’. ( - ) Н г1, < 2 A “ | I ^ 2 Î ’ ( -) I L 3. a 0 < A < T t / » .

Ь2ТГ Ь2тг

Observing that the index h can be replaced here by — h, we get the desired assertion.

T

8. Suppose that the function

r](æ ) = l x_ a { œ , f ) ,

with a certain a e ( 0 ,1), is absolutely continuous in <( —тс, тс) and that rj'(æ)

— f a^(x) is of class Lfn. Then if

= 0,1, 2, , we have we+a(ô ,f) p

•^271 ь2п for Ôe(0,

where A {

, a) denotes a suitable positive constant depending only on

, a.

P roof. Let Tn{œ) be the trigonometric polynomial of best approxi­

mation of /, in Z/fjj-metric. Set

( 3 )

0>a(Ô,f) = 0)a(Ô,f)

B J f ) = E J f ) .

By Theorems 3 and 7,

< 0 ' +A â , f ) < а>„+ а (д , T n) + a t+, ( ô , f — T n)

^ 2 ô a<oe(ô,T £') + C(e + a)E n(f) (0 < < » « * / » ) . In view of Theorems 3, 4 of [3] (see also [4], p. 274),

K (f) ( ^ - f j ,

we(S,

l£>) <

< С ( е)С4(а)Д,(/<«>) + а>,(.

« C(e)C4(a)C3(e)«,t ( ^ r i / “lj + «>,(*,/'■ >).

Assuming that

те/(«-Ы) < <5 < n/n (n = 1 , 2 , . . . ) , we obtain

» ,+ .( « ,/ ) < 2 a « {C (e)C4(a)03(ff)+ l}«> e(3 ,/<“>) +

• + 7 1- “ 0 ( е + а ) С 2( а ) С 3( е ) й « Ю г( й , / < « » ) ,

and the proof is completed.

4. Direct and indirect approximation theorems. Suppose that

(1 < p < oo), a > 0. Betain the symbols (3).

Th e o r e m 9.

The following estimate holds

E n(f) < for n = 0,1, 2, ...

P ro of. In the case of positive integer a the result is known ([4], p. 274). If

Jfc—1 < a < h, where h is a positive integer, we have

(Л = 0 , 1 , 2 , . . . ) . By Theorem 5,

CO,

( ^ T ï ’^ )<C(fc_a)a>a(^T T ’ ^) = 2 “ °(» Т Г ’ ^)’

and the assertion follows.

Differences, moduli and derivatives 399

R em ark . Considering functions fe L p( —

(1 < p <

and a > 0, we easily get the inequality corresponding to (16) in [4], p. 274.

ISTow an estimate of Timan’s type will be given ([4], p. 344-346).

Th e o r e m

10. I f n = 0,1,

, then co„

\ w + l , /

Сб(«) ( n + iy

n

v = 0

P roof. Let Tn(x) be as in the proof of Theorem 8, and let m be an arbitrary non-negative integer. Start with the obvious inequalities

“ • («УГ (^T T ’ f ~ + ( « ^ 1 ’ T* “)

< C ( a ) E 2m ( f ) + œa ^— — , T 2mj .

By Theorem 7 (see also [4], p. 116),

||T$(-)|| if n + 1 ^ 2 ™ ,

\ л + 1 / \ n + l j

where || || denotes the norm in the space Lfn. Since m-l

T $ (a ) = T i “» ( œ ) + y {Tÿ+A <*)-T ÿ (.**)},

V = Q

we have

(

\ / \ а 7/1 1

( ^ г ) K ’

+

; p ^ . ( - ) - 2 « ? ( - ) i } . In view of (2),

ll24; )+i( ')-2 l? ( -)IK C ( a )-2 ~ | | 2 ’2>+I( - ) - 2 ’!,(-)| K O (a )-2 ~ + 1E a.(/ ),

||2V»(-)II = ||21а)( - ) - 2 ’Г ( - ) 1 К О ( а ) - 2 1- « ® 0(/).

Hence

It is easily seen,

29

< C7(a) У ( f - 'E ^ f ) for » = 1, 2, ..., H=2v~1 + l

with C7(a) = 2a+1 if 0 < a < 1 and C7(a) = 22a if a > 1. Consequently,

4(7 (a ) / ^

2a W + i

4(7(a ) , ( 71 2“ 'U + i ,

4 0 (a ) ( 71

+

m zw

+ C7( a ) ^ £ Z'”" 1-®»*/)}

”=1 л=2’'-1 + 1

2Ш

Щ - й г ) > о < я + а д 1 > “- 1® , < л }

/4—X

2m—1

2 С , ( а ) ^ ( г + 1 Г ‘ ®„(/).

2“ \ю +1

Choosing

such that 2m

2W+1, we obtain

\ w + l / (W+1) ^

^ ( / ) < ^2»-l (/) < 7 - 4 ^ - У ( v + i r ^ A f ) , (тг+ l ) and the proof is completed.

References

[1] H. M u sielak , Inequalities for the norms of some functions defined by Stieltjes integrals, Comm. Math. 16 (1972), p. 45-51.

[2] И. П. Н а т а н с о н , Теория функций вещественной переменной, Москва-Ле- нинград 1950.

[3] R. T a b e rsk i, Approximation of functions possessing derivatives of positive orders, Ann. Polon. Math. 34 (1977), p. 13-23.

[4] А. Ф. Т и м ан , Теория приближения функций действительного переменного, Москва 1960.

[5] A. Z ygm un d, Trigonometric series, II, Cambridge 1959.