View of Some properties of the linear positive operators (III)

46 ^{IosrF KoLUMBAN}

erfüllen und die l(ostfunktion

10 RET/UE D'ANALYSE _NUMÉRTQUE

ET ÐE LA

TTIÉORIE

DE L'APPROXIMATION,

_Tome

g,

_N0

!, lÐ74, pp. 41_Gl

f(x, ^u) :

max

T

I

to

ù(x(r),

^wþ),'t)ctt zum

Minimum

nachen. Dabei

ist U c _R''

beliebig.

Mit Hilfe

einer Zeittransformation vo11

Dubovitskii

uncl

Milyutin

_[1]

lässt dieses Problem sich auf den Fa1l

zurückftïhren,

wenn lo

: 0, T :

I, m

: r, _e@, u) :

otÇ@,

ão¡, þ(*, ,,

^)

: uþ(x, ã0,

^) ^mit

^gegebenem

ã,0

tnð. U : {, = R lu )

^0}

^ist.

^Sind

^die

Funktionen

g,

^rl.,

und ,I

^stetig

und. besitzen sie clezùg7ich

x

stetige Ableitungen, so

folgt

das

in _[3]

erhal-

tene globale Maximumprinzip aus dem Satz ^4.

gOME PROPERTIES OF T}IE LINEAR POSITIVE

OPERATORS (rII)

by

AITEXANDRU I,UPAç (cluj)

LITERA'TUR

[1] Dubovitskii, A. Ya., ancl .4..4.. Milyutitl', Dxtrem.urn Pyoblems in the Presence

of Restriotàons, U.S.S.R. Cornputational Mathernatics ancl Mathematical Physics 5,

3, 1-80 (1965).

t2] ^{G i r s a Ìl o r', LV., Lectuyes or¿} Malltematical Tlteory of Extretnum Probletns, Lecture Notes

in Economiecs ancl I\fathernatical Systems 67, ]letlin - Heiclelberg - ^New York, 7972, Springer - ^Vcrlag.

[3] Gurin, ^{I,.G. and} E.l{. Stolj atova, Tlrc Mat¡itn.uno Princiþle in. a Miøi,max Pro- blent, U.S.S.R. Cornputational Mathematics and Mathematical Physics 13, 5, 1175- 11Bs (1e73).

[4] Hof ^f mann, I{.I1. und I. I(oltmbán, Verall,getnei.toerte Di,fferenzierbørheilsbegri,ffe

uncl ihre Anuend.ung i.ø d,er Oþli,tni,erungstheor'ie, Cornputing 12, 17-41 (1573).

[5] Köt1re, _tretg ^G., Toþologische li.ncare Räu+ne f. 2. Auìl,, Springer-Verlag, Berlin - ^Heidel- - ^New York, 1966.

[6] ^{L e nr p i o,} F., Ta+'rgentianzatanigfalli.gheitey¿ unrl infi.ni.te Oþti,m.i.erung, Habilitationsshrift, Univelsität Harnburg, 1972.

l7l I' ja sternik, L.,{,, uncl W.I. Sobolew., ^Elernente der Funhtionalanalysis, Betlin, '4.kadernie-\¡erlzLg, ^1968.

[8] ^{N a s h e} cl, I\'I.2., Dì,fJerenli.a.bili.ty at.d, Related Proþerties of Nonlinear Oþerators, Some As- pects of the Role of Differentials in Norrlineat Ilunktional Änalysis, in: L,B. Rall (ecl.) :

Nonlinear Funktional Aralysis and ,A,pplications, New York - Irondon, Academie Press,

103-309, 1971.

[9] Psherrichnyi, 8.N., Necessary Cond,itions for an Extrenntm, New York, 1971, Marcel Dekker. fnc..

[10] L W ar ga, Minimax _þrobletns 4m the calcul,us of uøriali.otts. Michigan M.af'n. J. ^{12, 3,} 289-311 (1e65).

tl1] ^{W e 1 n e} r, J., I.agrøngesche Vøriationsþrobleme, Vortrag anlässlich des Symposiums übet infinite Optimierung uncl optimale Steuerungen am fnstitut für Angewandte Mathe-

matik der Universität llamburg, Hamburg, 1972.

1, Introduation

þiugegarrgen am 12. XIL 1973,

There are many approximation

processes corstructed

by

_{means of}

linear positive _operators

ì"hict

enable

point of view. This

means

that

the elements

from the

domain. Likew remainder-term on the class

of

non-

"åiîi;ni# ^{" råil rãi,'1Ëi- ffiî}

ens

ln tlle

case

of many

variables.

we pfove

s a

linear

-¿l<0

iable

was some pfoperties

of the

sequence

It is

shown

that, in the

cãse of cfeasing on the class

of

non_concave

e

Bernstein operators defined

on

a

the

sense

of I.Schur). The

proofs

At the

end

of

thi,s, naner.we get

II

"rurnpte of

polynomial _operator y'hi9h interpolates

at thê'iertices ãf u

"oorr"xþorygoã-r'rrã

i, positiv; i;

its

interior.

Uni,uevsitalea,,Babeç-Bolyai" CIuj, Føoultateø de Matemøt'icà,-Mecøni,cd,,

Cated,ra de AnøIi,zd,

48 ^{ALEXANDRU LUPAS}

2. The

sign

of the

remainder-term We use

the

following notations and terminology:

K is a

^{comPact, convex}

^{set in} R*,

^nL

> l;

c:(cr, ^cr, ..., ^{c^),} x:(xr,

^fr2,

..., ^{fr^),} [:(t1, "',

er(t)

: l,

^e¡e(t)

: ¡0, _Q, x) :L.'o*ot

h:r

11

f

^:

K

^-+

R, then

th'e eþigrøþh of

f

^is

Epi(/) : {(*, y)lt( '. ^K, y ₌

^P.'

y

>

f(x)}.

K -' R is

called. notl-concqne

on R iff for

every

yi ₌ K,

þ,þ>2.

Í ^(D þþ

^o¡Y¡)

^"î D

^a¡Í(Y¡)

t':t i:l

whenever

ø¡e10,

11,

i:1,2, '..' þ' et*ar+ "'*%:l'

By B(K)

resp.

c(K) we

denote

the linear

normed space

of all

functions

k'-, n *ni"L ^aie' bounded, respectively

continuous

on K. They

^are normed

by

means

of uniform

norm.

An affine function on K is

d-efined- as

e(x):(c,x)lr, ^%eK,

where

r is a real number. Let E

^be

^the

collection

of all such

affine functions.

A linear áp"ratot L: B(K)

-+

B(I{)

_þreseraes ^the

ffine

funct'ions

iff

(U Le: e ^{for everY}

^e

^e

^8'

It ^is

^clear

^that

⁽¹⁾

^is

equivalent

wit

Leo: es' Lat'*:

^{e1'P' h}

: l' 2' ' "

^%'

rHEoR-ÞM 7.

IÍ L: C(K) + C(K) is a

I'inear þositiue oþerøtor uhich' þreserues

the affine

funct'ions, ^tken

/(ø)<(LÍ)(x), xeK

Jor ^eaery

Í = C(K) uhick is

non'-concøae on

I{'

PROPERTIES OF THE POSTTIVE OPERATORS 49

Proof.

The continuity

and non-collcavity of

/ ^{imply that Epi(/) is}

^a

convex bod.y

in

R'o+l. Iret

n: {@,y)lx =Rn', y e R, (c, x) i

^c*t_û

* c,rr:0}

be an arbitrary

closed hyperplane

in

R*+1

which

bounds

Epi(f),

say

(c, x) I ^c*+t! *

^cnt-z

) ⁰ for (x, y) ₌ Epi(/).

Becanse

for

each

t e K

the

point

(t,

Í(t))

^belongs

to Epi(f),

^we

^{may write} (c, t) I

c"+rÍ(t)

|

^{cn,-, Þ 0.}

By

means

of the

monotonicity

property of Z

(2) (c, x) i ""rr(LÍ)(x) *

^{cnrz Þ}

0, x ^e K.

In

conclusion,

if Epi(/)

^{lies on one sid.e}

of an arbitrary

closed. hyperplane,

then {(r, Øf)@))} lies on the

sarne side.

If we

^assume

{@, Qfl@))} )Êpi(f) :

Ø,

then,

according

to the

second. separation theorem

of

convex sets (see

[]

p. 58 or _[12] ^p.

65), there exists

a

closed. hyperplane

H, in R,+1 strictly

separating

{(x, Qfl@))} ^and Dpi(/).

^{Thus for}

Hr:

_{(%,

ùlx . ^R*, y = R, (ã, x) i

^õn+û

!

^e

,¡2:

^0}

one has

(c, x) Idnrt! tõ*+z > ^{0 for (*,} y) ₌ Epi(/)

and,

(õ, x) -f

ê"+r(LÍ)(x)

|

^õ,*2

^1O.

Btrt this

contradicts

(2)

and

the proof is

complete.

As an

application

of the

above theorem we may prove

the

following two-dimensional

variant of a well

known

result by r. ^popovrcru

_t9].

Though

this

result was established.

by v. r. vorrKov U4]

^\rye

^present

^a

new, shorter

proof.

4 -- Revue d'analyse numérique et de la théorie de l'approxination, tome 3, oo l, 1974

2 3

tp, " ',

^tr,r)

A function /

j : l, 2, ...,

50 ^ALEXANDRU LUPA$ ⁴

'IHEoREM2.LetKbea'cornþaetinW2ønd'Ln:C(K)+C(K)' n: l, ^2,

^. '

', ^U,

^o"

-i,q*in"-

i¡. ^itiae

oþerators uloich' þreserae

in' i¡;l' irnition'' If ^{o e} C(K)

^a's

O(*,

_Y)

:

xz

I i'

ønd'

l1x ^lln - ^L,all :

^o'

th'en

lim ll/ - L,fll:0 ^{for euerY} ¡ ^e

^C(I<)'

Proof.

¡"1

^{Ç{'z)(K) be the subspac.e} of.C(K) formed

with all

^functions

which have "orrti,liJrr!'ö;i;""ä"ri""ti""I ãt the

^{second order}

on /('

For

/ ^e

^Ç{z)(

⁽⁾

^1et

^us

^d'enote

¡r*: i ^vn'+

^Íl't"

⁺

st : _; lÍx,+

^AL

-! ^{(il,- Íì)'*}

^4ÍyÀ

as

well

^as

(3)

ßor ^(x, Y)

^atbitrarY

in K

Put

<pl^; (x, !)l:

^À2

- I ffh@' v) +

Ít'""(x'

v)l +

i Í\,(*, ù

^Í1,@,

^v) - ^{lÍ *@, !)f''}

It

may

be

seen

that

glmrt

_{@' Y)7}

> o' çlMIt;

^@'

^{Y)) >}

^o

(4) ^qlÍl'(x,

^Y)

^{; (x' Y)l <}

^o'

An

element

g from

CPI(K)

is

non-concave

if its

Hessian matrix

6iyoll by,øll

is oositive

semi-definite

for

^every

^{(x, y)} = x

^(see 1111,

p'

²⁷⁾

Let¡eÇ(z)(K)

^and

s: _+M¡ ^'e - f' s:Í - I*''n'

5 PROPERTIES OF TIJE POSITIVE OPERATORS

According to (3)-(4) one

observes

that g, h are

non-concave For,instance,

this may be

motivated

by the

equalities

8t),: ^M¡ -

^{"ft)^}

^>

^o

st),' sl,,- st)i: çlM¡; 'l >

^o'

Therefore, theorem 1 implies

84L,9

lt,

4 L,,h,

^{ort K,}

which are

of

course equivalent

with

51

o11

K

ul"-llù'+4Í

₍₅₎

f,m¡lL-o-ol ^< L,l-l ^< ₊ Mil,L,Q-ol, n:t,2,

If the

hypothesis

is

verified,

then

(5) furnishes

lim ll/ - L*fl :0 ^{for everY} Í ^e

^CP'(K)

Fina11y, the fact

tn^t

çtz\(K) is densein C(K)

andllL*ll : l, ⁿ :

L,

2, .'.,

proves

our

theorem.

Another

consequence

of

theorem 1

is the

followi11g fepfesentation of

the

remainder-term

in the approximation by

^means

^{of the}

^operators

L*:Cla, bl->Clø, bf, n:1,-2, ...'

Some

similar

^ideas

^were

^exposed

by tte

^present

^author in

_[4].

A-n operatot L: Clø, bl

-+

Cla, ål is called

strictl'y þos'itiue relative

to K, ^c. _{lø, bf,} it|

Í=Cla,bf, Í>0,Í+0 on _la,bl implies

LÍ > O on la,

^bf,

^{LÍ > o on}

^Kr.

From the proof of the theorem 1

we

see

that

for such an,operator t,:Clø,b)

n Cla, ó]

which

is

moreover linear and preserves

the

linear functions ^one has

Lh-IL>0 on

^KL

//Lr

: min 8i(ø, l'), ^MÍ :

max.

8/(ø'

_Y)

' (r,y)=I< ' @'Y)- K

øll 6nyoll

7

+

Let us

denote

PROPERTIES OF THE POSITIVE OPERATORS 53

52 ALEXANDRU LUPA$

whenever h, is a convex function on _{lø, bl,}

i.e.,

%1, %2,

ø, being arbitrary distinct points on _lø,

^b

F,oi Clø, bf +R be

defined ^as

F""U):Øil@o)-f(xo)

6

Then

F,"(h)

>

⁰

for any

convex

function

k

e _{Clø, b).} An earlier result of t.

^popovrcru

f10l

^asserts

that

there exist three

distinct points (, from _la, bl

_iso

that F^(f) : F*,(er)1\r, 4r, ("i ll,

^er(t)

:

¡2.

In this way

we have proved

TrrEoREM

3. If L:Clø, bl

^-->

Clø, b) is

a. I'ineør, strictly _þositiae oþø-

vator rel,atiue

to R, c. la, b),

^and,

Leu

: eu,

^k

: 0, l,

^e¡(t)

:

¡i,

tken Lo eøch function

f ^{e Clø, bl}

corresþond.s

a

system

4r,

Er, l,ò

of

d,is- tinct _þoints

from, fa, bl

suck thøt

(IJ)(*,) -

^f@o)

:

l(Le,)(x¡)

- ",(xr)l . [{,, ^{1,, er;} .f], ^xo = Kr

By [€r,

Er, E"; "f] we have denoted. the divid,ed. d.ifference of the second

order at the knots 4r, 4r,

Er.

It this way

we see

that the

remaind.er-

term, in the

approximtaion

by

means

of linear strictly-positive

operators rvhiclr preserve

the

linear functions, has

a

simþle from,

3.

The

behaviour

of

Bernstein's operators

on the

elass

of

non-eoneave funetions

of two

^variables

Let ^us

^denote

Kr: _{(x,

_-y)

₌

^R, I

x e

_10,

ll, y - [0,

^1]]

(6) Kr:{(x, }) =R¿lx ^Þ0, y >0, ø*y <

^1}

b*,i(t)

:li)t' _{t, -} t)"-¡

(7)

_þ*,*,¡(x,

_r) : (1)øl0)

^*o

,'

^(L

- ^* - ^y)il-k-i

Th" ^{Bernstein operators}

^B*,,,,:

^B(Kr)

^->

^B(K1), n, m: l, 2, ..., B": B(Kr) + B(K2), n: l, 2, ..., are

defined. respectively by

(8) (8,,*Í)(*, y):åå b,,¡(x)b,,,¡(y)f(+, k),

^(x,

^{y) = K,,}

n n-þ

(e) (a,Í)(x,r) :

Ð

D^Þ,,0,,(*,

Ð f l+, ;),

^(x,

^{y) = K,.}

TrrEoREM ^4_.

If _{f e} B(Kr)

'is

a

non-conca.ae

function

^on

K1,

tken

for

eaery

(x, y) _{= K,}

(8",*Í)(x, !)

^>-

(Bn+r,^*-tl)@,y), n, m:1,2, ...

Proof. We

have

* ,1,

(8",*Í)(x,

_"y)

: _Ð

_{h:o i:0}

_Db*,*(x)

b*,n(y)

t(t -

^ø)(1

- ^{y) + y(r} - ^x) *

: Ð Ðu,*,,a(x) b*n,,,ulLål#¡i)¡l*, _;l *

n tu+l

+ D Ð

^{u,*,, n(x)}

^{b^*,,,U)} ffi_u+)¡ ^(*, #) ^*

tt+1 n

+ Ð Ð

^bn¡1,e(x) b,,+t,¿(!)

#iffir(T' ^{;) *}

n+1 n+l

Ð Ð

b,+,,0(x)b*+,,n(y)

,.h. rf (+'+)

-.(r) (n-h*l)(tn-i*t) (nlt)(mt1)

^'

-.(3) h(øc-itll

⁽⁴⁾

qik:

@+r)(*+t)'

^dìP

,l!*'

: _(+,

;),,1?t : _(+, +)

^{aih t3) -}

(T';)

(4t lh_l, r_tì, rro:( h , ,

I

ziþ':l-

m I _\2¿+r

rn-frJ

D*(f) :

oll'ÍþÍlut)

+

_"Í1) Íkllo))

+

^otî\

fþnl\ +

_"lît

ÍþÍî\ -

^.f(zoo)

i, k: l, ^2, ...,

^n.

lxt'

I.

Xs, 2Çt, i

For

øo

e hl>0

K, let

* ^x(r - ^{y) + ,rylf} l*, ;):

(2\

dìn

(ntt)(mtt)

h¿

(ni|)(rnjr)

54 ALEXANDRU LUPA$

I

^o PROPERTIES OF THE POSTTIVË OPERATORS 55

Since

A

similar result may be established-

fot

the sequence of operators whose

images

are

defined

in

^(9).

THEOREM

5' For øn

ørbitrary

Í = B(Kr) uhick is

non-concøue

on K, (B*Í)(x, y) Þ (B,+tf)(*, y), ^{(x, y)} ₌ ^K2, lL: l, ^2, ....

Proof.

One introduces

the

numbers

^*¿(fl: ".-t¡(*'+)* "*¡( ^{" '} ;)*#r(+' _"t-]l-

"ÍL'\

+

"Í?)

+

^olil

+ olî:1. ^i, h:0,

^{1, .}

..,n +

^r,

we write

D _oo(Í)

: o,

D"e(r)

: # ^{¡ (*,}

^o)

^*,

^*_]_rr

^{(+, o) _ r ("*,}

^o)

h:1,2,,,.,n-fl

and similarly

f _nll

^lì

_n*l

D*(fl:4f

_{?n+ | (0,}

_; )*;?¡(,'+ ^-r

^0,

^rnll

^where

^for

instance we have

tacitly

assumed.

that

'i: l, 2, ...,

^rn

₊ |

Dm+t,,+r(/) :0

lor

^'h

:0,

_nll^h

f l+';) ^:0, i:0, l, ".,

^%,

and

D¿,*+t(n:+f[ _/n+ |

I

1,

,, h)*;¡r(r,+)- ^r(r, ;u)

^Aoo(,f)

^:

Ao,,+r(.f)

- ^{0, ..}

^'

We have

(B*Í)(*' _v) :

^{n n-þ}

þnn,¿(x,y)t(l - ^x - ^y) *

^tc

* nt(|' ₊₎ :

h

l, 2, .,.,

^m,

Ð

_þ:0

n

D

h:l

b

D

i:o

D

Dnt+l,'(,f)

:

#¡ ^{t) +} j-r(r= , ^t) - r(4-, ^t

+

"f^' _{" -}

1 -r+' r(+' ;)þn+t,h,o(x, v) t

h: l, ^2, ...,

n.

In the

same

time, fot i:0, l, ..,, %+1,

^h

-0, ^1, ..., n+l

"ll,\

,ÍI\ *

_"Í?t ^zl'ì

⁺

_":1\

^,Íit + _"lî)

^zÍ?

:

^zoto.

I1

f

^:

I{1-t R is

non-concave on

its

domain, then

/: _{0, 1,} .,.,

^rn

*

^1\

(10) D,r(f)>o l'^-_'l \--0, l, .,., nll)

^I

But

from

the

above eclualities and. taking

into

account ⁽¹⁰⁾

(B*,*Í)(x, !) - ^{(8,+r,*+r f)(x,}

^Y)

:

fl+1 m+l

=

D D

b,,¡t,n(x) b,,+r,n(y) DnoU)

2

^o.

h:o 'i:0

#¡(0, ,")þ,*,,o,{x, y¡ ¡

*ä-

_,Ð^'

"./(#' |)Þ-*,,n,a(x, v¡ ¡

i

þ^+t,*+r,o(x,

y)/(1,

o)

+

*Ð -. ^r¡(0, T)þ*+t,0,¿(x, ^v) t

+ É "f' _{--:--re, '-} ^tl

þn+1,þ,a(x, 3t)

:

h:l i:0 n+r \1x n ) n n*l-h

: Ð D

^Lo,n(f) !)nt,t,h,,(x,

y) I

^(B*+t

f)@, y) *

h:r i:o

*1-l

+

DAon(,f) þn+t,o,n(x, ^Y) i:o

In

other words

fr

(8,Í)(*, !) - (B*+tÍ)(x, Y): Ð

But

/ = ^B(Kr),./

non-concave on

K2,

assüfes

the valid.ity of

the inequali- ties

Lou(Í)

> o, (uo::oo,t', ..',i +, -,)

Taking into

account

that

þn+t,h,r(x,

y) Þ 0, (x, y) ₌ Kr, the proof

is complete.

Remørh.

Any

non-concave

function from C(Kr) or from Ç(Kr)

^m7v

rc "iitãi-

appíoximatecl

by a

non-increasing sequence

of

polynomials.

4.

The Bernstein operators

anrl

(S)-convexity

I,et S:

llsnoll

i, h: l, ^2, ',., ^{ffi, be}

a ttoubly-stochastic

matrix, i'e'

%%

s;o

)0,Dtro :fs,u: l, i, h:1, ^2, "',

^rn'

i:L h:l

I1 x:

^{(x1, Nz, .}

'.' ^{x*) e R. then}

t]ne sckur-trønsfonn

of ø is the

poin

!:Sx:(yr,!2,...,!*)

where

fl

lt: Ðrs;nxn, ⁱ : l' 2,

_'

"'

^nt'

A

subset

D

from Ro

is

called an ød,missibl,e d,ornain ilL

it

^verifies:

i) x:(xr, *r, .,.,

^{x^)}

^{eD implies %n:(xng¡,}

^finp¡,

.",

^tcn@r)eD,

rc being

an arbittaty

permutation

of _{1, 2, '. ',

^m}.

S

^and

^{any point} x ^{e D, the}

Schur-transform ^Sø amples

of-such

admissible domains

in R2

^a;re

bv

^(6),

RowsKr [7] a ^function f :D +R, D

being an m

>

2, is õaited S-conuex

(in

tlte sense of

I.

Sckw)

if ^{for every} matrix S antl ^any point x ^e

D

f(sx)

^<

f(r).

ll

PROPERTIES OF THE POSTTTVE OPERATORS 57

He notes that a

S-convex

function must be

symmetric

on its

domain.

A1so,

if _{f :D} -rR ^{has on} D

continuous

partial

d.erivatives

of the first

ord.er,

then

a sufficient condition

for

S-convexity is

(1

l) (*, -

^*¡)

^ôf _ðf

ôx¡

ôni

Þ0onD If D is

open

then

(11)

is

also a necessary condition.

L

^{e m m}

a The

Bernstein oþerator

Bn: B(Kr) -* B(K2)

_þreseraes ^the symrnetry, that is

Í = B(K,), _Í(*, ^y) : _l(y, ,)

ALEXANDRU LUPAS 10

56

tt+l-þ

D

^Le¡(Í) þ*+t,n,,(x, Y)

i:0

'irnþl,ies

(B"f)@, y) : (B,f)(y, x), (x, y) _{= K,,}

Proof. From

å ti A;n:äïo', li)(";'):(i)1";

we get

(B*r)u,,) : _{þ_-80( ;o)}

ro *,

t, - ^x -

^vY,-r-,

^{r (+}

:h$on,h,¿(x,

^þ=

ùr(+, +): ^(B*fl(*,

^y)

,)

where _þa,p,¿

was

tlefined.

as in

^(7).

Further we

show

that the

Schur-convexity remains

invariant

under

Bn. In the

case

of

one

variable

such preserving

linear operators were exposed

in [2]- ^{[6], l8l. It}

^is

the

convexity-preserving

property for the

^usual

(see

[8])

^{was used}

in

statistics

by w.

wDcMürrLEn _[15].

THEoREM

6.

Let

Í = B(Kr)

^be

ø function uhick is

S-conaex

on Kr, Then B,f,

n

: l, 2,

. .

., a./

^S-conaex

functions on Kr.

Prqof.

We find, ô22!

: "E ^{-Ð' þn-th,i} tr(+, ^{+) - r(+, +)l}

ôr

F5-o

i--¡ t

ry : " rÐ-"är

^þn-,,h,¿ V

(+, +) ^{- r (+, +)1.}

+):

t2

13

we

get

súccesively

PROPERTIES OF THE POSITIVE OPERATORS 59

5B ^ALEXANDRU

LUPA$

On account'of

the

above lemma we ^shal1 use (11)' Put

D(n,,f, (*,

Y))

: _{*@ -,) (Y} -'4)

and

LÍ(o,

x, v) : _Í(*, y) - fþx +

⁽¹

-

^{o')y' (1}

-

^e)%

I

^ø.v)'

It ^is

^easy

^to

^{see that'}

f ^: K, -+R is

^S-convex

^on K' iÎ

^anð'

^{only if}

Lf(a, x, y) >

^O

for

every

(*, y) ₌ K''

^6¿

e _l0' 1]' Fot n fixed- 1et us

^denote

q,,*(x,

!) :

l" ;')(" ;:

^o

o)

*o'o(t -

ⁿ

-')))"-þ-L (* -

^{v)(xo -z;}

-

^vh-zi)'

(*, Y) t K,

D(Bnf,(x, y)) :(x- _{r) p:.} ^"-.fn

O,-',o,n(*,

y)Vl+' _+)- f l+'+)]:

tçl D

_þ:t

io,,*(*,y)V(ry +)-r

_n

^{-¿ i,+r}

_n

)l *

w]

_À:0

D f,

Q,,"0*,{*, rùV

(ry, +l - ^r (ry ^¿+1

+

?L

t?l

^þ-

_{2h-2i 2h-¿+l} i

D

^Dqo,'u(x,

^y)LÍ

h:l i:o

2h-2i+l

^t)

nn +

These functions have

the

^properties

l+l

^h toh oi+l

n¡za¡{x,l Y)Lf

G;fr,

^2k

^{- i +2}

^tn1,

I

I

DD

_h:o i:0 ⁿ

qn,o(x,

y) 2 0,

^(x,

Y) = K,

(* - y)lþ,-t,rn-t¡(x, ^y) -

þ1'-1,i,2h-i(n'

y)) =

h'za(x' y¡

(; : o, l, ^...,

^h

- l, ^{h: l, 2, " "} t"-])

(rc

- y)lþ*-t,zh+t-i(*, y) - þn-r,i,2h-i*'(*' y)l:

^qt'zna(x'

!)

(o

: r, l,

^.

.., h, ^{k :0,} 1, 'l+ll

Therefore

(12) fr-L

(13) D(B*f, ^.) : _D

lr:1

t"l

D

¿:0 q¿,n(')

' Lf ^{h-2i h-¿+l}

^i.

h-zi-ll n

n

Now

the

S-convexity of

/

enables us

to

write

By

means

of the

summation-trick

^Í

h-2¿ h-i +l

^i,

h-2¿

+l n

ⁿ

h: l, 2, ...,n - ^| ) " o, ^{i : o, r, ...,1?]}

n-l n-l-h

v+)

^þ-7

P.

^{Ð--, ouo ¿itLt}

^{-r- \- D}

^{i:o (An,ro-n}

^*

^Arn-¿,¡)

^*

Combining these inequalities

with (12)-(13) we

conclude

with D(B*f, .) > 0 ^on K,

and

(11) finishes

the

proof.

+ l+1 _-Ð

^h

Ð

(An,"u+r-o

¡ ^A'e¡r-t'i)

60 ^{ÀLEXANDRU LUPAS}

t4

_t5 PROPERTIES OF THE POSITTVE ^OPERATORS 61

5. A

^method

oi

^positive interpolation

I.et Pr, Pr, ..., ^{Pnbe the}

successive vertices

of a

convex polygon

C'Cp¿, *itt i

^sid.es.

^If /: ^C, +R

then we

may

^formulate

^the

^following

iitJpolttion ^problem, ,,to

^fittd.

^{a linear operator} L,-zi

B(C")

-'

^B(C")

with the

properties

r) (L"-,Í)(Po) : _Í(Po), k: t, 2, ...,

^n,

2) (L^-rfl(x, y) is a polynomial of

degree

n - ^{2 in x}

^anð'

^y'

3) if Í ^>

^O

^{in C*}

^then

^{Ln-rf ) 0 on the}

^same

set"' ^A

^method

^for

constiucting such an interpolation

^operator

is

^as

^follows:

^{Iet d'¿(x,}

y):

:aitc*bni+cð, i:1,2, ..',

^{n', such}

^{that dn@'y):0 is} the

^equa-

tion of the hyperplane

^(PoPr+r),

i: l, ^2, "', ^{n, (P,P,*r):}

(P"Pt)"

Putting

REFERENCES

lnn(x,

Y) : n

_d¡(x ^{d¿(x' v)}

n' ^yn)

, Pn:

@0, yo),

h:1,2, ...,

^n.

[1] ^Fa t Ky, ^{Conaex sets and, lheir} aþþlicatioøs. Argonne National Laboratory, 1959.

l2l \up aç, A., Sowe þroþertàes of the lineør þositiue ^oþerators (I). Mathematica (Ctuj) e-(3¿),

l,

77-83 ^(rs67).

l3l Lupa9, '6., ^Sorne þroþerties of tke lineør þositiue ^oþerøtors (II). Mathernatica (Cluj) s-(32i, 2, 2es-298 (1967).

[4]LupâÐ,A.,DieFolgederBet'aoþcratorcn'Dissertation,Stuttgattl9T2'

[5] Meyet-König, w., zeller,7(., Bernsteinsche Poten¿teihen. sl.udía Math., 19, B0- e4 (1e60).

16l Mäller, M. W., Aþþroximation du,rch lineare þositiueOþeratoren bei gemdsckter Nortn.

Habilitationsschrift, Stuttgart 1970'

[7] Ostrowski, ]\., Sur quelques aþþIication^s-cles foncti'ons conuetes et colxcaaes axl sens

ile I. Schur. J. ^{Math. Putes Appl.,} 3l' ^{253-291 (1952)'}

[g] ^{p o p o} vi ciu, T., ^Sur t'aþþroxinnation ^des fonct'ions ^conue*es d"ordre suþérieur. M.atlte- matica (Cluj) 10, 49-54 ^(1935)'

[9] popoviciu, ^T., Asuþ.rø demonstraliei ^teovemei lui Wei'evstrass cu ajut-orul'

!o_I!1'o11ry'212r

dà interpotøre. I,ucrärìle Sesiunii denerale $tiinfifice ^Acait. RPR., 1664-1667 (1950).

[10] popoviciu, T., Sur le reste d,øns aertaines Íor*y]?t li,néaires d,'øþþroúmation de

l'airalyse. Mathematica (Cluj) I ^(24)' l' 95-143 ^(1959)'

tlll Rockaf ellar, ^R. T., Conuex analysis. Princeton lJniv' Press, 1970'

ifZl S"haef er, lI. H., Toþologicø|, aeotor sþaces. Springer-Verlag, ^1971'

[13] T ^{e m p} I e, W.8., Stieltjes integral, reþresentation of aonaex functions, Duke Math. _J., 2r, 527-531 ^(1e54).

[14]

Volkov,

^{V. L,} ConuergenaeoJ.sequences of .linear ^þosi,tiue oþeval'ov-s in ^{tke sþaoe} oI^"9!.- tinuous functio,ns of luo aøriøbles. (Russian) Dókl. Akad. Nauk 115, 17-19 ^(1957).

il5]

^{W e g m ü 11 e}

r,

W., Ausgl,ei.ahnng durah Bcrnstein-Polynont'e.

Mitt.

^{Verein. Schweiz.}

\¡ersich.-Math., 3G, 15-59 (1938).

i+

i:r

h,þ-l we have

l,u(P)

Þ

0

^for

P:(x,!)=C^

1 for i:h 0 for j+k,

I^o(P¡)

:

Receivetl ^10. xII. 1973.

and

if

we define

Lu-zi

B(C")

-' ^B(C")

^as Institutul de aq'lautr d'in ^CI,uj

al Acailemiei Reþubl'iaii Soai,øIista Româ'nia

(14) ^L,-rÍ : Ln-zlc*; Í, ^.l : _{Ð/(P o)t,ol),} n :3, 4,

_{' '}

"

the ^{pfoblem is}

^solved..

^{'we want to}

^use

^this

^operator

^{in the}

^following

approiimation

^{problem, which}

is yet

unsolved:

let K : _{(x' y) ₌

Rzlxz1'

+ r, * fÌ

^and.

^Bd,.K : _{(r, !)

^eF{zlx2

* ^y' : l}.

To find.,

if it is

^possi- U1e,

a ,,âense" system oi distinct ^points P¡,, Prn, "',

^Pn,

oî Bd'K'

such

that

lim L*-rlPy,

P2n,

...,

^Pnn;

Í,

^(rç,

!)7:Í@, y), ^{(*, y) = Bd'K'}

ør cp

whenever

f ^{e C6)}