View of General estimates for the linear positive operators which preserve linear functions

IT,IATHEMATICA

_

REVUE D'ANALYSB ^NUMÉRIQUE ET DE THEORIE DE L'APPIìOXIMA'TION

t,'aNALYsuIüuMÉRi*Jfi

,l$.ilTrii,t|T,o?lrtolttP*o'Yln{ar'o*

GENERAL ESTIMATES FOR THE LINEAR POSITIVE

OPERATO}IS WHICH PRESERVE LINtr¿\R trUNCTICNS

lìÀD u PÃr,r'Á¡{n;l

(Blaçov)

I-,et -I be, an

arbitrary interval

of the real

axis, J c I

^be a subinter-

val

^{and Ø}

(I)

rcspectively

ø (J) the

spaces

of

the real-valuecl functions defined

on ,I

and. respectively

on J. We

derrote

by

^Øo(-I)

^the

^subspace

of

øQ)

of those funcl,ions

that

ale bour-Lcled on each compact (subinterval

of l'.

^Tre1,

II c

^{Øo(1) be}

^a

^sultspace

^{and ,L:U} - ^Ø(J)l:e

^a

^litrcar

^posi-

tivc. operator th¿t

lllrcs(jlvcs

lineat' functìons. fn tho

prcsent

paper

we

give a new methorl

to

^obl,air-r cstimatcs

involving the

scconcl

olclcr

rnoclu-

lns of

continuil,y

for

l,he diffc¡rence

L(1, n) -

J'@)

l'hen f

^e

tI

^{a,ncl n e} J.

These estimates'are

in

sornc instances bel,ter anrl

tnole

general

than thc

cstimates

that

rve have obtained othern ^ise

in

f

4]. By

cornbining thcse 1,wo rnethocls ther-t¡ r'esLrlts a,n cstimir,te

tbat

intproves thc. earlier

oncs

_{(sce i3 l).}

Since

for

a fixed

point

ø e

I, I'(f

_,

r)

is a lint¡ar positivc ftrnctional,

in the first

section

we shail

express

the

esl,imates

in

terms of functionals, ancl

in

tht¡ seconcl rve shall

apply

these lesults

to tlte

operators

of S.

N.

Ilt¡rnstein.

1.

Estìmatcs

for linear

positive

functionals. In this

selcl,ion øo rviìl be a fixccl

point

of

the ar'bitrarJ'ilLtcrval

1.

For anyf

e

Ø(I) :ve

clenote

by àY anci àJ

l,he functions ot

Ø(I)

th.al,

are

clcfinei[

in lhe

foJlorving

mode:

(1) ( 8+/Xr)

: ^ft)-l@ò, teI,'t>ro

0 , telrt ^4uo (à-/xr): î(t)-lþo), tel,tlno

0 , tel,t2no

X'lom

(1)

there results the follorving

representation:

(2) _lCI) : (à*/)

^(¿)

+ ^(s7)

^(t)

^{*l@,), for}

^{I e}

^{I, Í eØ(I).}

\\¡e clenote by

^eoe

^Ø(I) the functions

^clefinecl

by

er(t)

: li. for

I e

I .

r\lso, f or an¡- r'óal nurnbcr ^n, e [p tve clenote

b¡'

]ø

[ the

gi'ea,test

il'te-

ger.

that ii

^{less llia;n u,, aucl}

by _[a] the

grca,test

intcger that

is less

than

a or equal

to

ø.

ùe

^give

ⁱⁿ

the following' definit,ion the generail c

nvilonment in

'which

we shall

v¿olk.

0 I-INEAR, POSITIVE OPER.AT.oRS

149

belorrg's

to

lz sirree

!r, gr$,, lr)) :

(r¡).).,

_îU) Fi¡rall) :

1,,r,

tl,^iç'r"(tri

i,J'ú

:

-.f(nù.

as,,irr

l)cfinition Z úht¡

:rsscr,tiolr of

folk¡rving

_{cqualit.y :}

(4',)

rìffi

_-J(,r'o)

: [ ^ç,

^f¿r,

^tr\

^dy,(r¡)X ¡r(ú2), L- >r J1*

'rvhele

p X

_i.,.is

thc plorltrct

tììeA.sut.rl

nf

_,u rviçr

itstìf o¡ I- _X I*.

fndeecl.

i;ji ^lH jÏlï'l,i::ì.11.,,n',,i.,"î'u,,'' r n;;'

'-'i"ä*i'':*rli''"ii ,,,';ir-

'r,^i"i,ì1j r i

( ,\[

¡)-r {

( _I ò*.r, i e")) ⁽ I

òf'

_/ ^(1, ₎₎ tly. (r") rlp Q,) ^-1.

\

I+

I

{ I,,r-.

^l

(¿,))riò*/i

_{(,,)) rlv-(,,)}

}

^{d,rírz) <}

" ^I ilf ^rrz f ^I l'(,r')t<

^oo.

";

ITrTìï,,Jit,iì,,;ï,ñ#"åi'l;,î].0t" ^ancl

^assertion

^{(4') can}

^ìre ot,r:air1¡{ Ì,¡y

*rrr,",li,lil"tl" l;,|ìi:*,:;, 'fttnctiottti

^{J1' tts it¿}

r)t:f'ínitiou, 2

tlta _Jouozoin,s

(re) )fp:e

(tt) I,'ff) :,1(a) for ^ut4

^,1.

^e

^Jr.

Ì)rttr¿\ .

On.

calr

illli.r.') +

¡¡,¡

ì¡\,r,lro'si'g

.!l

€ l,

rlt,l,irit,rl

h'I _í(l) : : l.(l

e

/-)arrrl g(t):

_{o 1l c i,r,o) v'}

_t

i' ).,J,lìr,n,ri/1)

:

t,.'fìi,,,r.rrî.t,r.t,r<rqìa)

+

* Ìl)i follou's ft'orn thc liacl; ilrat _{for, any n,2} j ¡ ⁰ :;lfn: [,r, _

-

^{ïnlo J}

^tli,

^>

^Qln)

^1t,(I n (]to1-gltù),ao)), _anc'l r:onñc.quenily

t.(1*) :0.

_{Ana.Jo_}

f,,il|tü,llr]li'i.:';.un,t* ,"(l-) : o'an.i' _{on ure} othel haiul r,(r) :1

anct

f,,,r"rifî;ïÍ'jl1;Så;ï,.;îi,i,,,,crf.obrain

^an analosous tcnul¿r,

fol _ilre

gcnclat

.,uo o,rllTTtNrrroN 3' (:u) 'Tror evcïy _J' e J"(r) arrrl eve'y

poi*ts

_of

I :

^ú1

{ ,:<t,

(õ) _{\(.f, t, tr,}

_,r)

: _!r-

-

t _{fU,) +} ,,' -h

_¡çtz)

_{_.f(t,).}

l.t

- l, '

_lz

-

^lrr

t'z'

_'/ \

4-c, ls'¡¡

.,

(,:lI ¡,)

-r

148 RADI' P,A,IT.4,NEA 2

Dnnrnrtrorq

l.

Let, T/ be a

linear

suìtspace

of

^Fuçy¡ such

that

(a)

^eu e

Ir,

í,

-:

^Ot

I¡

²

(b) tt !

^e

^{I/ then}

^{l.f I e V.}

(c) If f ^e I

ancl _{l e

Øb(I):rre

such

that _lgi < _]./l

^Ltrcn.

^geV.

Lt-1,

/I

_:J,'

*

[R l¡e

a linearpositive functional with the

pr.ojrerty

: ne) : '-n'å,i:0r1.

A

pzi,rticular

but sufficientl¡'

_¡cneral

for application,

cz-Lse

of l)efi- nifiolr 1, is

containecl

in thc clcfinition floln

ltelorv.

D¡tnrNrltoN

2. ttel: ¡r be a positive Bolcl

measurcì on

f

^{.u'ibh the}

(3')

1Í

\

^in,

^--

^{roeo I}

^d* : _\

_ler

-

^{;"rro i rl¡-r,}

JJ _rr+

propcrty

eu cl¡r

: rl ^for i :

0rl. a,ncl e, d.p

I æ.

\Ä,"tr clenote

by

I,-

: : _þLg(I)

tìrc spac-c of real-valuecl functions definecl on

f that

^¿ìre p-inte-

gral:le. Lct

^1,he

linear posltirre

funct'ional

-P; 7 * _R

l,-c

defined byt PU) _-- [./

^{cl¡.r., .7'c ]-.}

IJ

\{re shall givc

sorne L¡rnmas,

I-rB¡rlr¡. 1,.

For

a .fu,+tctional

X

^{cis in,} Ðefi,nitï,om

I

Me

hale;

(3)

'E'(

l8* hl) :

-F( i8-' ^ø, l).

Prool

. l-r'ottt

(2) thele folloll's a1

'- n6eo:

8+ €r

l- ^8-

^¿r,

^{and by the}

clefinition of X' there follows 0:1(r: -%eo):1(8+¿r) f ^1(S-er).

llhen ?(là*rrl) :7l(à+r") : -

Zl(à- et)

: X'(-

ò-¿,)

:?(là-¿rl)

Rentnrh

L Fol ^a

fru-Lctional :rs

in Definitiorr 2,

Letnma

1 can

be explcssecl

by

^1,he follo'r'ing cclrLality :

5'herc

I- : I [ì (- ^co, n)

nncl

I+ : I 0

^{(ø0, oo).}

lVc shall rvrite

,M¡

:f7(là*

ør

l). ^Also iÎ

^g(lu.

..

_,ln)

is a

funcl,ion of se'i'er'&l

variablcs

rve

shall

denote

by frukl(tt,...rt,))

l,ho va,lue

of

the firncLionäl tr' applietl to the

partial

function t¿ -+ g(tr, . . , ,

tò _with ,j:const"

(i + j).

L.,ul,rrru

2. Itr¡r

a, .funational

F

as

in,

Defini,tion,

1, iJ l'I¡ # 0

^tlten

(4) 7'(l) -|@) -'1",(Ih(q¡ftt,tr))Jor

^any

f ^eIr,

'¿uh,ere g¡ :

I x I

--'

[R is

defined, by :

?Átt,

^{tr) ==} (M),)-r{(18-Fetl(fr))

((8-./) (,,))-l-(jò-r,l

(r,))

((3n/')

(rr))}

fot"

(tr,

tr) e 12.

Proof

.

Since _l8+ ^er _|

(lr) :

(8+ et) (tz) ancl

là-

^{et I}

(it) : _{-_ (à*}

¿t) (tt),

it follcrvs that the furrction

tz

-.

g¡

Q, tr)

belongs

to I/ ^forany

^{t, e V.}

Èr

_{LTNEAR posTTïvn}

oȡnerons

151

,Ëf !^;,;,,(!,";

\,å:{,;í1r,,r,,î

(c) lf _./is unilolrrrly

conlintrorrs, _{u.c hrr,vc}

.c,t(.f2_p)* 0 (p_* 0 ^_l_)anrl

lliiJ'3it_,,1",tr"n,,'rlirf lrrovcr'L',iL'ioint tr) ifïí,ûri-

t"haL ot!¡J,,,p¡ _,

ariy

p

)

^{0. 1'\;e} have _obl,ained

a

cont;racÌiction.

(ct) fn

^or.d.er

to simplify the notation

_rve

put

:

(9) B(g, t, zt): L(f, _!/ _ _i¡y :_u,y):l!-fty _t)r

L+U

++r r+u t@-ru)-Í@l

fot:

y -- t,

_?J

f- ^ue I, t,n )

^0.

For'0 <h<l ^the folcwing _irlentity:

Tî;t@ -

^Ð

++; t@ +u¡ - ^t@) :

== l(qr-l_-lL J il h î, .

^ì

tr(t ltt) l;,,+,,JØ - ^{Ìù + -u-h I0} *"¡ -l(ùl+

[lt t,*Jt,, ,,ì

' _{tt(r I ,ð-1 ,}

^J@

- ^{t) 1-} :-=:"- _l@)

- ltv -

^h)

_|

^,

ployes thc

equalil,¡. :

(t0) B(!t,t, u¡ : ^J9.f-!!- _{h(t|tt) B(y,}

tt,

u) + - *yt- h(t+u)""

^B

^{(, -_}

^tt,t,

^_

^tt,tù)

for0(h<t.

lf we takc ir: (70) f :¡nh,

_and

_tt,:l¿,

_where

_h>O antl ziefJ{

tt-¿ Þ

2

^.we ci.btain r

(.LL)

Iì(y,rir,ti,tt):rfi _B(y,h,h) +;iB(y ^_ tt,(m_r)lt,,

tL).

1

(l-¡) For,{t1¡Lìïy .f

e9$\

^¿tnrl

^cy'clv

^{rernl rlutnL¡cl p}

} ⁰ llrf

u'Iitr-":

c,r1t'J',

o\: ^sup

¡J/(.r:

*

^l¿)

_-./'(p:i l, ^t,:'; *

^{å e}

l, 0 { ^/r 'i

^pj

(6¡ r,¡ri/,

^p)

:sup{i.f('tt ltl-LJ(t:) -I./(;"-tr) i ':t'- li'æi-heI'

u c ^fr <

^pl.

<,:f;(.J,9¡:

^Ñtlp

Ii ^A/'/' !'' t" ^:r:\l' lt' t"t' - ^t14pli

^'

Il,ent¡;trlr.l. \Are adrrrit,

Lht pos.sillilit¡' th"rt otr(/, p),

o¿(.fr ,.)) alld

coT(.f, p) atc'

itrfinite'

"""

ioororn

4. lf we

co.xsidi]' n,fí:raã fn,rtt{liott ,f etr'¡l,I) xüî l!ftre ^;

(al

^,l,tre "futtct,íott,

p -

c,:l¿(.,|'7 g)

is

n'on'decrwsíttg rtn (0,av)'

(l¡) ?'ol

attl¡ r'tnL nu'ntl¡er p

>

⁰

^ue

^hene:

rf(./,

^p)

(

¿¿ri,/, P) onrl

(tl2)

^ur(.f

,

^p/2)

^< ôf("/'p)

^-'.

o2(t,

^pl2!

R.ADI' P1ILT,1\NÚA {50

(1)

(et lÍ'l

^,i.y ctiti.forntltt cr¡nli'tì,u,tms.fttttt'l'ittn,

litt'tt

6i(f

, ?).:0 ^(p - ^{{) , )}

^ut'tl

i""ít".lrii,';i;¡,;r'tit,ltit

c',ístst,ttetí ^the ¡u^,ttion,J

ís

ccttttittctotts'

tll [f"t'r,"t',t"belon'gtof

^rtt¡tl

t. 1:r'<l'tltttt'fr'r'nttq

^p

] ^0'

^I

lÀ(/,

^{ú1, úr,,r')i}

( (t, - ^tt)-t

^{(r"

-

^n')(1

--i-lir:'-

lt)/pl)=-i

(8) *

^(,.'

-

^{l¡ ) (1 -ì-}

^l(r, -

^{e')l pl)'lorf (,/, 2 p)}

Prrtr,f

' ^{(a) It is}

^obyittus

^ftr¡tn J)efinitioti

_"l'

llr)l,r'l /' ( ,r 1lz, ^It,

^{lo e}

I llt' ^stlt'l¡ tllitl lP.- [' ( p' 'l'ltc

^itrcqua-

",t ,,iíÍ,"el"k

^ì,17,

-pii'oii"*"

ItoLtt

tlrtr

int'qrLnlitv:

i

\(1, t,tz),)l < /'

,,t

ⁱ

.i,,,)

^{.f ('")}

^, , ,,:ir1r,J'(trJ -./('i')i'

If :r--lt,:r: !heI ar(Ì stt{'lt thnt' 0<i¿<p/2 t-l't't'-\1¡c 1,a"",-l'1,r,

F ^/r¡

''- _zilr¡

-i--¡1"

- tù:Z\l'j' ^r - ^{h' r )- h' t:)'}

lhert"fore

(r12)

!;2('1,

p/2) <

^o,f

(./'

_P)'

\ / 'f,,n,t,'il,

<fr'rtr, ^/re {lrestlcltlhri tr:lt (.pl"trtl lt'l te(Lt'{"z)

1,.,

,, ,iiii,ilrtr' Ioirrl-.' i,r't-rrs

t'oltsitl'"r'

llrt' liolvrit¡rrrial2l(r¡ --

'lU")

+

ykt:ì

-.f(nl :- ^2(lr')'

^{\1'e hnvc a(lt)}

: nial

o1'<ltgt'l'e

onc lhelrl follorl's:

:rrirl

crr(./, ? 2)

: ttr(g,

_ç12)' ^flincc

Ig('t)l,l

^è'

ltt,

^fr

ll. ^\\'c

^har-c

^onl)'to I¡or,{)eaÅl

12:t't,L¡ill'arilrÚltr¡st,rr1]rt.r*i.isu,c={1,,12)Su(.lltlìat

Itt4,t¡l'> li ., \ìä'r1rij¡-c,.i,,riLitr o*l;' ¡re

^{r:Íi*. ri.h,err' ø(Tr¡}

>0

^nt1

LINEAR POSITIVE OPÉ]IIATORS

A("i, ^¿,

, ta,

^ro)

if

¹¹

< to {

^l¿

iItL>üoortz<fro

153

The sy'rirrcti'¡r sf ^(t4) pro'es the following ineq'nlity:

(15) _iß(y, t,

?/)

I ^<

I, n;n el ^ttlp

L

.i _{Oulptt,t] .;-}

if 0<I ( pand0<r¿withU--t,yIu,el

Fi'ally let us considrr. ilro c^sc

rvhe'e

t ), p

^{ancl u,}

) ^p.

^Lel,

nt,

-ltlpl

^and.

lt':tf(nr'

-l-

1). Ry tsi.g (l0),

(18) ancl

(Lj) ilre'e

lciltorvs I

B(y, t, u)i { 4+++l

_h(t

_{-l u,)} ^{B(tt, h, ,,)l1-} _t(i ,,"."! _{_l u)} ^{I B(l-tt,t "\./ tt,tt)!{}

. H++l't ,hQtutpl*(lølpD') l.r,r, ^zp)-r

* 'lT^ ^'f('f'2nl :{tffi:+

.t --.L-" 't*w (zlulpl# 1ul _-'el)')f

^ì

^o:[(f.2P)'

I'lorn tho last.

irLeclua.lit¡',

by tarring int,o

accouirt

(14)

ancl (-l-5)

we infer, _t,hat bhe follorving inc,qualitv

(16)

_lß(!,t,t, ?¿)

i ^< Q i- ^u¡-t

{¿¿(J

l- ltlpl)'

^{-,1 J(1 "j-}

ltrlpl), t:!(!,DpJ holds.in

l,he

genelal

rìâ,se- rrhere _lJ

-.t, ^{y f-}

^u,e

I,

u,,i

) û;

p

), 0.

llhen (8)follon's frour (1ri).by tahing _{U_}

J *, t :

c¡

--

ú, ancl ^,tL

: _t, *

r;.

Our nrain

rrsult

is

thc fóìlov'ing

t,heorent.

llur¡or¿nu 'l. _{Lat þ'}

_{be tt}

Ju,nclionar ns

in

Defi¡ti.t,íon

i

^,'rLa'.¿ tltut

ltr /- 0.

'l'h,en,:

(7't) _{!tt(Í) -. /(a'o) I} < ?(00)

,of

(f,

Zp),

J:or

tion: 0o:

^a,n|¡

1u v (1*l

^and _I ^a'nE et

-

real ^uoeo

nunlbel ^I

^lpl)z.p

>

0, where

by

0o zue c.enoteíite ,þntc- Proof

. Jìy Definition 1 it results

1,hat 0p e

z.

Iù,om r,crlrnra

2

rvtr

b.a,ve:

lI!(Í) -f@ò l:

i]1,,

(I,"(qÁt,, rr))) I ^< n,,(Ftu(!pltt, _iz))).

Sircc ^:

(1, 2 p),

çv

fir,

^{tr) =-'}

it

results

frorn

(8)

that

^:

I

q¡Qt,tr)l < {glflI.) ⁽ⁱ

^8*¿,

l0ù F,i- I

I à-¿¡

i$r)lpL)r *

* ^{(LlMù (lô-¿r} l(r¡))

⁽¹

* I ^I

^ÐtB,

l0ùlpl"),

^co!

(f,2p).

762 RADU PALT.A,NE.ê,

B(a -

^(/,;

^{+ 1)h, h,lù) +}

0

Thc following equality

^:

(12)

.B(y, nth,,

lrl :3:

_'nx

_"l

.L

j:o h

^(n¿

- j) ^{t¡(y * jtt,}

^{Ít, tr,)}

^l- + y--+ _{r)(y * (h}

_-r,

_1)lr,

_(m

-

^tn

- ^r)h,

^tt),

?tù a-

r

for 0 {

^lt;

4

^nt,

-- ^2,

^h,

}

^0,

^nt 2

²

can

be proved by induction

^ov(')t'

/r.

fndeecl,

for

/c

: _{0 tho} equality in

(12)

is

cqtivzr,Ient ^1'o

(11).

AfLerrvalcls

if

⁰

< k<

^m

- ^{2 the inrluction}

step

h

^--,lc

-l1

^results

from the tollowing

ecluality :

'Jt -

^{1" ßQJ}

'- (k :, L)h,

^(rir

- Å ^{- l) h, lt)}

^-=

m"l-

^l.

^'rn-h--l

m,

l-7 tn*. I¡-

^|

¡ ' -""-- _nt,|.L ':-t _R(u -

(lt-12)h,(nr,

- ^h -

211t., lt),

that is

âr cons€eüonce

of

(10).

Ifrom

assertion (12)

for h :

tn,

- ²

^{rve obtain :}

l2n-2\

:B(tt,ttt l,h) I ^< I-"-: X

^(r¿

-i) ⁺ -=-; l

^t,lf

U,zh):'malj¡2lt).

\lr. f lfi

^nt

.l Il

Consec¡uen1,ly 1,tre follorving inequalit,y

is tlue

^:

(13)

I

B(y,nth,,lt)l <

^{m<rf (,f}

,2p),if 0<

^/¿

(

pancl

rn2 I.

¡1;11',

lct f >0 ^{antl 0} <u, ( p be

such

thatlJ -tt ^{U)-tt, EI.} If

ü

( pthen ißQl,t,

^rr,)l

(,'t[çf,zp). Ilt ] plel,us'rvlitc tn,:ftlpf

and

IL

: t

.

I{cnce tn >

^{1- ancL}

lt ( p. By applying (10) :lncl

(13) rve tut,

-l-7

have :

lR(y, t,

?l)

I < _'!y.

_h(x

_{.l u)} ! ^o).

^I

^B(y,

^{tt, u,)l}

^-l

^-,,

_h\t ."!-,

^lßkt

^*h,t _- ^h'h)l

^<

1- u)

-

^t(tr,

l-

^{It) -l-}

tttttt -.*/{

^o

^, ^{_ -l} f_:!,(7 r- ,rr)'

_..*

\ -

ú),¿\Ji à p/

-- ---l-

___--_- _t,ti $r

^2p).

h(iftt) t*u

'I'hcrcfore

in both

l,hc ca'ses

I ( p and I Þ

p

rre

^ha,'te:

(14)

_I

It(y, t, u) I *

{t * ; _* ^{(21 tlpt -t Qtlp lr,} .f $,2p),}

if 0 <iand0 <{/ <

^{p rn'ith'!J}

-ltE -u'e

^L

from Jttit:r:t:t"l'e

^b¡'

F tlte functional ,f ^{-' L(Í,}

^ao)

tltat a,!;;;ctionu'l

as

in'

DeÍitt'í'tiott' 2 ctn'd' we srL[),ose

(2t) _{IF(I) -f(æ,)l ç min {Blz} + ^{p-,r(ü), r}

^-l_

^Bp-,r(V)}

azff,?) Jor

aty

_{J e}

C(I) î V

^rutù. rn1¡ r.eul nu,tnlter p

)

^0.

;'f,'åi_ ,fil;l'î,ål

hese csl,iuratcn btt

It'or any r, e lfrf rve clenol,e b.v ßn ; Ø

t _L}]I

_-, ^Ø n ^tTte oper,atol, clefinecl LINEAE POSITI'r'E OPERÄTORS

bY B"(Í,

_ø)

= I ^{lØln) ?nr(n),}

^a;

e lo,1l -

fr)"-x and,f e .4-0

Øt _L}Jf.

o

t55

rvhele pr¡ (n) :

^1L

It,

ür

(r _-- ff

^we

^fix

^nç

:

øothe

f'nctional Í- ^{p,ff, ro) is a}

pr'r.ticularcase _of

Definitioir 2

and

ttren

^.n'e can

apply "C"rJiäíii: ^-" - '*'

'Trroonnivr

B. Iot,

a,jLA

nelN,

r¿

) ¹

cnt,tl,cr,ny

JeØt, _lAJl

we høae

(2r)

wh,ere

ilB"ff) _/ _{ll < r,4B}

_{lo,r(f ,}

_n-T)

't

il ^.

ll is

tlte sup-nornt.

Proof

. From

Corollilr,y ^.w¿ _o'btain :

i B"(Í, r) -"f(ø) I

^<

i

^_].

.E^

Q\V"i r - ^{t;ln;l +(l}

^{Vu i}

^* - ItlnlL),)p,r(r)

À-0

r

I

-klnl>n

²

(22)

-;-

I

(l,n')<

ô2

<i1 -r-T' _QVn i ⁿ -

^Ttfni

ï

^n(:.c

-

^hln)')

^{p,o (r)\ *r(f , n-*¡}

where ['clenotes

^1,he

^suin i:i,ken'r,er _ilrosc

_{inclices /¿}

for whichlr

^__

In

_{16l aucl}

_l7l

the

follorving

irrequality _is pror,:ecl ;

4'

^I

^* - ^{kln !. p,*(u,) <}

⁴

^{-- 1,}

^\r,here

4306

+

^{s3?.v 6}

_{< 1,09 is the}

Sikkema's constant.

5832

{n

n-

ïrence lB"(l,n)-Í(æ) _/ < í+Z(r¡_-1) løe_.n)jazff,rr-T)<

1

<1,43

^rar(J,

rt-T¡ lor any

n

e

_10,11.

Remurlt 6. _'rhe constant ifì Theorern B irnploves thc

constant qqual

¡o 1-J2- :

L¡62ß... givcrr

in

_[5].

115

L54 NADU P.A.L?.Ä,¡IÌEA I

Consequently :

u,, (8,,(l _w ftr,rr)l)) < F((1 +

^:Jj

s- ¿rllpi),

-,j

+(r ^-j- ll

^{8* e,}

I lpl),) of

^(.f

_,

D?)

'rhe

^thcorem

is then plovcd if

rve tar<e

into account lhe equality

^: (1

-i- I

I 8*¿,

I lpD, *

⁽¹

l- I

^I

^8*

^¿,

i lpL),:

-

⁽¹

^{-l- I}

^{I e,}

-

^noeo

^I

^lp])r.

Conolr,¡nn 1-.

^Ir! ^n,

^{be a} functio+t,ul as ^,i,n, Defil,tilio,¡t

1

^{sucJt, tii,t¡t,}

M, *

^{0 ot' us i,tt,} Dcfì,nítion

Z.

Tlrcrt we hruuc

:

^"

i) _lPff) -l(ro)

ⁱ

^ç

-E(0n)or7(p)

ii) lx,(Ð -"/(øo) i

^<

(18) _{< {1 + zx(lt}

e1

-

^noeol

^/pl) *

p-.zrt(ü)}

.¡(e)

iii) P(l) -.f(nò i ( _{1 - ^s

p-r-E (,1)}

'r(i'),

for

_-cuty

Í ^eIr

^{and, any.}

^reril

n,wrnl¡e^r p

) ^0,

tohere oy(p)

is

tínuone of.the n,¿oduli

of

eontinuity

a[(f ,2p),

^u,r(Î,

dfl or

'z(f

¡ ?)

,ìr:a' ,¡,i,s tí,e

¡utiAti,t þ:(et"-fioer)2,

,P.r9of..Coroll¿ry results

flom the inequaiity Q

I et

--

^uoeo

I lp[)i

^<

<

p-' üforj :1,2,flomI-,emma ₄

- ^(7),

^t,örnma

^à-ani't

f.höoïerir'i.

Remaùc

5. For the

modulus

of continuity of the

second

ordeï

*,.r

iii) with an

csbimate 1,hat rve ha.vt:

osed itrtc;:r'a,l

of tho leal axis

anc-I.

,

ä3l¡ilnf,

JJ

îi:?iì

"Í"å'"'ìiå

^å

"J;

that'

Lprcser,ves

rinear

runcrions

th"P?f:'t;ii'""-"Jå tÏ"åi'jíî'"incl

^such

(1e)

_I

L(1, n) - ^l@) |

^E

^rnax

{714, Bl2

+ I ^I'

((e, --

ûeì2,

n) lh,zj

a, (!,

^tr,)

holds

for'.\ny I

^e

^C(I),

^IL

> ⁰

^und

r

^e

I.

_.rrolever, 'we mention that the ploof of (lg)

uses

neither.

bhe:

condition

t]nat CQ)

is

includecl

in the

clorüain

ot z, uirt oniy the

concli-

tion that

tJre functions et

j :

0r 1,

2

belong

to this

clomzrin, nor

the

corL-

dition that the

^yalues

of thl¡ opr:latol L

^are, contirruous llunctiorLs, Ali:<r

tlre

conclition

/e

^{C(-I) can}

ìteleplaccd by the

1vea.k

conrtition /'

^eØ6(I).

tl

LINEAN, POSITIVE OPER.A.TORS

151

rlencc _Þ:o irr:ilr+z _-d"or J-a í I _n

o:-o

\

n-2 k-7 ^{ù O --}

^n)o-k

^:

(" ;'7or'(7 -p1o-'(, ;=)" + _Ã(;, _i). ^{(L --}

^slo-,,.

e

---

i-l'îiïå _t?l i-t?":'L tj"ïrt5; _{in ea) n} : r ^-- _!¡ r -

^,tL

-

^st

É r- ^y * ^ptn),(_':

,),(1 - ^lt)n-nye:

.-=

(:, _l) ^,r _- !/)n-,.,tr" (,F_ì ^_ ,) *

(1 --

g ^i-2

n--2

ru-i-2

l-

TL i:s- I

x _{(I -}

^,¡1¡n-t-z _Ot ^.

!"

^{t ho}

last cquality putting

17

:

n , ^{{.!ooÍ of} the tneoirin.'itã""í fis ^ol+ i :j rvc

obrain (25)"

obtain for r _{e [0,1]}

^'anh

fr¡?:lt

)-ii

^we

I

B"(r,4 ^_ _l@)t<

<{1 + 2'PY*

|

a - telnltn(n._tr,ltt¡z¡?^t(n)} o>zç, n*}1.

rvhe'e

{' ^{cle'¡tc's ,re sunì taken ovr:r. those i¡diccs} È for

which

læ

- hln,l>n-T

.

,-J_,ntr':f n,n-fn¡anas:ln,r+V" J_J]. Ilence ln_klnl >

íì'j;rîrï-.åquivalent ^rvith fr ( _r or tt > s. rn

_16l the

rolowing

_equalfty

(26)

l, ^{i * _}

ttftt, iztn¡(n)

: n-L

?"

*'+t (1 --

^fr)n-r

+ ⁿ -7

s-l

^{Ø" (1. - Ø)n-e+L}

l'rom

l_¡emma

5 we obtain:

\i ⁿ⁽ⁿ -

^kln\2

^?**(n) :

n

to, ')

^",l.1

^(r -

^n),,-:

^{( n-} _n-7 ^r _I ^I

^-]-

{27)

In _'s--1 ⁿ

^.._1.

)

^{øu (1}

^-

lû)"-"+r

{

s'.--1

n,-l

^{-- 1) ]-}

* ^r(7 - ,

{'Ð' !^ ^z1@)-r

rF*, p"=r,@)l

156 RÀDU PALTÄNEA f0

Tunonplr

4"

For

an,y ;f e

Øuf)rl]

tlt,u,l 'is

not

lin,zar u:e ^{'lta,le t}

(23)

lim

sup 8,,(.1)

-

^{,f I} _< 1,30

'æ

_<tr(f

, n-T)

\Ye shall use the

follo-.ving

L;mma. fn what

folìo.¡'s ^-,,¡e use ^l,hr,' rroiaiion

(i")" ^au

extenclecl scnsc f

or ît)

^,til, e

Z ^ïhart. ₍ ; ) :0 if ø<0 arm, 1Aotn1m,

Lp¡n¡rÀ 6. 1'or n,

Þ 2,0 < r 4n ^and,0 ( s < n

^{we l¿a,ae:}

eÐ t ^{(r --}

^k,ln)z

^o,r

^@)

: ₍ ^{n -7}

r

^,,r+t

^{(I *}

^î;)'-r(

ü--- r

n'--I ^-f

n(I -

^{a) r-l}

I

j:o \ ^?n-ti

^@),

fl

't1,

n, ^-.-1.

I--1)

s*1

(25) 5 þ --

^kln)2

^p,* (t) : n-I

hÈt

n"(7 - n)il-a*t

*fr _-r

^I

fr-z

\-

J:s-1.4 P"-z¡

(r)

1

u¡here r:

e

_[0,1].

Proof, Irct, nundn fixed.

\\¡+: rl'r'ibe c¡.-..

(r -

l;¡n,)2JJnr(t:) and, d¡.

:

ll,-2

h--2

^{. fro (7}

^.-

^n)n-'t;+z

fol' 0 ( h {

^n.

\\'e

have

'r : _(T)

r.*+z

(r -

^rú),.-k

- ^z ₍ ⁿ _k--r ^*7.

"..t'+I(1

_,

_xt)*-r

-f

*+(i,_")

^u,

^(L - ^n)n-*

^----

("

ff,

(.0

h;

2 ,rt+z

(I

^__

,r)"-r +

, rl;, _{_-1).r,,} ⁽¹ - ^r)r-r

^1- ₂^{o ¡;k+2} (I -- t:)ø*rt ^--.

- rft:

T) "'-' ^(L - ^a\Ir-t -

²

_{_:)}

^{or+t (L}

^_

^{fr)n-k - ..}

- (; _"- 'r) . ^(r _-

^ç¡n-*

. _+(-

-?)

^nþ

^{(r - r)"-*} -

tl

ìi ll il

:iI*+ø-2il*+t ldr++( ^n*2 _lr-L ^{'ù (L * û)n-',}

^bccause

^d,s:0 - ^ilt

LÌÑI'¡1R ^POSITI1¡E OPtrEATORS . ¹⁵⁹ S

"

Jlldrillis

t¡c ,srrrti takrr¡ oyet' tþosc

i.ntlicres

j fot

rvhich

l)

i,

l:t- ^jlþt, -2)l> @- ²⁾ 2. _Ry using Stirling's folrnttla it can

l:e

¡)ä'oYetl

that

^:

lìulx ìì-tl}K ['t¡-2¡(;t):0(1)t (n

^'.>

^æ)'

1e[0, 11 0<i<il-2

firerr ìl¡' applfilrg

(22)

IoÌ

n'

- ²

^{inste¿rcl of 1¿}

^$'e

^{cibtai;t¡ :}

l3

whtrre

ä(1

-

^rr)

(3r)

{,},

^,po_z¡,,)

^),-

^j:s-l

^f'

^{?^_r,(,*)}

l ^{- + (¡ - r)+}

^{o(1) (rc}

^*

^oo)'

trrt'n¡tr (2?)' (:ìf))' (31)

it

^{fgllou's :}

,(32) \'

^tt(:c

-

^{h'ln)'pu *(.r)}

*

i)

; ⁽¹ - ¹⁾

^-ì-

^0(t),

^(ø'

-

^oo\'

Finallv, if

wo

talic into

accou rL

(22)

we oìltaiu ^:

11ti,,(.f)

-/ ^{ll < (.} * # ^(r -

¹⁾

^{+ 0(1))}

^o,,(.f

,n-*¡, ^(rt

^{--+ æ),}

but

u,e

rr*.c 1

^,.

+ ^(r - ¹⁾ :

1,9938. . ..

+

t,ìIiF'El'.ENCES

1.. E

'nrìrrvi, Yu. A ,:On a meltrcttof ap¡.troluirrl,líott ttf'ltottntle.dfur.tcliorts clefined irt cn

-- itlìrual ^(iri ttttssial), stuclics irr Oontomporât-'¡' rotr]"tttt coustruc[ive'flÌeor'y or Functions' P|oc. ^Secoltcl All-t-illioÌì Conlercllcc' nai<u 11-OOZ¡, l. I llrragimov ecì' Izclat' Akarl' Nattk Azerbaiclzan S.S.tì. Baku (1965), 40-45.

2. lf e y ^{o r e} , R. Ã..: ^'l-lte' appíóa:ín'raliotr of cortlinuous futtcLiotts by posilirte lítrcar opu'ators

- ^Splingcr, Bcllitr, Ilcitlelbelg, Nerv York, 1972.

B. G o ìr s Èa , tf . il ., O,, ap"prorimation b¡¡ li.near opetalors : Itnprooed eslirnales, Aual' nurnór. ct théor. approxitn. 14 (1985)' nr. 1, 7 ^32'

4. Iråltirrrca,R.:Im¡ùoucdeslintitlcsuitltllrcsccondortl¿rmodulsofcotúintilginappro-

¡lintatío¡t by li'near p6siiiue opualot's, r\rral' rL¡r¡ér'. et théor" appl'oxim' 17 ^(1988)' nr' 2 17T-179.

5. Påltärrea, ^Ll .: Itrrytroùc(Ì conslr.rrt itt npprorintaLiott by l).crrtsleitt ¡tolynomictls' Prepr"

Eabes-Bolyai ilnir'. Fac. Àfatìr. Res. Scmin. nr'6 (1988), 261-268' _.

6. Sikkema, P. C.: ^{Ülter clenGrad cIeL} Ap¡trotintalionnit Bcrnsleüt-Polgnorne,n,Nunter.

i\datlì. 1 (1959)' 221-23[J.

Z, Siklrerna,'1>. ^Cì .: Dcr ÍIrert eüúger l(ons[anlen intlctT']rcorierlerApprot:intttlio¡tntit Betnsleitt-Polgttorirett, Nutucr'. 1\'Iath. 3 (1961) 107- 116.

F.eccived 15. I\'. 1989 (Jnìuct'sitatca tlin Rrttçou

C ^a tcù e ^clc Allalcnta ticit' Str. I{arl lllcrtÌ il. ⁵⁰

Brttçou, Ro¡ttîutitt

I ..

1õ8 ñADU PÁ,I,TÄÑEA 72

11

The cclualiTi

r :1nr -- /ø

¡rncatts

n-T ¡rfn,1n 4 n^T _¡--î

_{1-]_)ln.}

\Yc haverl(n,--I) {rfn

1_tr,

2 a,lclrlþr,-L) j-n*t

-l-

n-L> n-T -f t- ^(r

^{-l- 1)/ø}

^Tlicn

the follorving staterncnt is

tmc ;

^,

(28) o<æ[email protected])<tt,'l-rr-r.

1

In a similar

mode

the

er¿uality

s: _lnn j-1t"

_-l-

tl

^rlrea,ns:

1tl

(s

- ^I)ln, - ^n, ' -{

^e,.

lsfn - n

2.1,\¡eha-ve

(s - ^{L)lþt,- I)>sln} - ^rr-i'

ancl

(s -I)l@-I)-n

^?

-r,-7 4s-tfn,-n ^2.Ilence:

(2s)

,{(

(30)

1

0 <

^(s

-1)

^/(,tL

*I) - ^{n <} n ^z ¡,¡t-t

I¡rom

(26), (28), (29) ancl (22)

we olitain

^:

çt,

-7

î I

^{I r:,+L (L}

^*

^n)n-r

t ^{{ 11,-I}

JT

tt,-I

s--1 ^{r" (1} -

^n)n-sa1

s-1

tt,

-

^7-

)l-

71- n*T (n * l) :

ⁱ¹

-

1-J- 0(1Xø ^,-n cç).

,-1"

_ ¹

nit

Iì'romtheinequalii,y(r -- B)l(n,--2) | (n-z) t ^<

^rln,

! ^n,

^z

<

1

results that for 0

<

j (r-3r ln- ^jlþt,-- 2)l>@- 2)-ã,anclflom

11

the incqtraliíy sl@ -2) -

^(rt,

^--Z¡-TÞ

^sfn,

-rr-T { ⁿ it

¡esnlts

lø

^.--

jlþ, -- 2)l >

þt,

- 2

1

^2)fors <j ^{< n} *2.

Ilren: n(I - ^r, _{F* ?"-zín) I _X

J=s-1

{

^ø(1

^-- r)

_{

I " ^In-ziln) * ?n-2,-z(n) l'Itn-2,-r(n) -l ?,-2,-r(r)} (

J

-< ø(1

- ^n)tltrL - ^z 2," I* - ^jl@ -

^2)1.

^?,-zÁn)

^1-

* 3' max ^max p"-a@)j

,elo,7) o<i<n-z

P.-r,@)l

^<

lr l'

l