View of A new class of linear positive operators of Bernstein type

"r72 ^{IÙE,NES:T DAI\TI 1,4 Àf} S1'l IE _{]ì.I. DE} ì.f A'r' ICA. 'TIIÉORIE I)]1

-

^{l'.E\rUE D,} I,'ÄPPIìOXINIA'TION^{:\NÀ L YS E NUIIÉ Iì IQ UII}

,L';IN.{LYSB NUmúIìIQUII BT LA THÉORIE ÐE L'Apptrt0XIMATION

Tome 17, Nio

2, lfl88, pp. ll3-124

tor the arbitlation

^ga

of uavoff. onc obtains

-

o.g"6o,

i.+rz,

^2.07711

of

the c.g., considered

chapter VI t1l).

RI'FERENCES Ä NIìW ^CLÂSS OF I:INEAR, POSITIYE OITEF,ATOIiS

OF RT)RNSTEIN TYPE

L l) ^¡rrr l, Iì., ^ùlelode nun'rcrice ln ^Leorìa iocurilor (Nanterical I'Ielll.,tts in Game 1'ltcotg)'

litlittrla ^Dacia, Cluj-Napocn, ^1983'

2. }) a u i, Jì', Numerical l|4ctltotls in Ganrc .Iheor4 ('!clcLety4y:9:Gcncralized Cìlassical Coope-

' ,1,,i¡r, Go^r, St,t¿i" ^Univ. Brbeg-Bolyai, Oeconornicî' YIiXIJI' e' ^1938'

3.ìDjtrJrin,G.N.,S"r¿or,'v.G.,-Yuetlenieupriklactnuiulcoriiui¡¡r,Iz.dNauka'

J\[oslcva, 1981.

4. OdobIe ja, St., Psyclloloç1ie consonanlis[c, I, II, Libr:air'ie x.Ialoirre, Par'is, 1938, ^1939.

ó. owcrr, G., L-eoriaioeurilor (GameTlrcory), SetiaBazcleuratematicealeccrcetãriiopcla-

!iorrale, Iìditru'a 1'ohnicir, Ilucuregti, ^1974'

Iìeceived 10.III'1986 [JniuetsiLalett Clttj-Napoca Facullatect de lVlatemrtticd çt ^ltizicå

Str. IhgãIniceanu, No. 7

3400 Clui-NaPocø Românla

BIANCAMARIA DELLA VECCHIA (NaPoli)

Somrnario.

In ^questo lavoro si introduce e si stuðia una

nuova, olasse

di operatori

ìins¿1i

e po$itivi di tipo Bernstein. Si

ovidenziano xìumeÌose proprieta- e si dimostrano alcuni teoremi d.i convergenza.

Abstract.

fn ^{this paper}

^{a, new} class

of

lùrear

positive

operators

ot Bernstein t¡re ^is

introd.uced _and

studied.

several

propertios

and" some convergence

theorems are

given.

1. Iutroductiou. Let

_;l'

be a, oontiruous fuletion on f :

_10,

l-l (/e

Co(f

) ^and

^denote

by

(SiÍ)@) the corresponding Sta,ncu

polynomiálof

degree

¿. It is well known that

4 p?,,r(n) llf ,-ø)

lç

(,Si/Xø)

: _8Í(/; n) : _{En r} )' _neN

and.

øe

.fl-F

0 m

whero

pi,r@):

ⁿ

k ntn'-,)(L

-

n)lø-e,-ø!, # ^e

I

and

ütk,-a)

u(n

I ⁿ⁾

^{. .}

^.

⁽ⁿ

-l (k -

^1)ø).

This operator

was introducecl

in [16] anrl

^ßtudied

in _tg

_-11-rL7

_-201.

.- {oregver, from Bi

operator onc can

obtai i [17], aslimiting

^cases,

the following two

operators :

1)

n'avartl-Szasz-Mfuakyan

operator

Mn

M,(l; û):

^s-nx

Ar#, ^(*)

with ø ) ^{0 antl}

lf(æ)l

:

O(ns'),

whero p is a positivo arbitrary

fixecl

number

_14, 7 r 13r 22, 24).

714

uniformly on 1,

V ^c(,

:

ct/n

:

0(n

In

_[1-0]

Mastroianni

ancl operator :

BIANCAMARI1\ DELLA \¡ECC}IIA

-z¡

_ancl

Vf

^eCeQ).

Occolsio introducecl and

stucliecl

2 A' NIÌ\,l¡ CLASS O[' LINEl\rl POSITIVIJ OPERATOF'S r.15

2) Baskakov operator P,

p,ç; n):

å, (" * ,:,-r)_J.*r(+)

wil,h ø ) ^{0 ancl} l/(ø)ì :0(eß'), _vhere p is a positive arbitrary fixed number

_11,

6,23f.

It is well

knorvn

that SÍ is a linear positive

operator

verifying

the

following relation

_[11] :

ttf# ^{siff; n)} :

_Jtøt(n),

n> p >

^o

B;tt"Í : _f'nl,

from

(1.1),

u'e have

also

B;tl'l:r,nf, vÀ>0.

rrinally, a positive

again opelator.

frorn the definition, ii;

follo'rvs

thab, if

^), e (0,

11,

¡sÍi,r is

\rorv we want

_{_to-}

rglre¡en]; SÍ.i

(V

),e

-Z?r)

in a matrix

form.

lile

clc.note by,

sl ul{ ri tlìe stirling-'numbers of'first

ancl second t

tili;

respectively, _clefinetl

by

nttt,,,l

- 4

1

,

BtlrtL-ttlf â,trd #?,

-

^. s,jn(tt-iJ4lri, V lr,e

IJ

ancl, n,

¿

^L the

Being

(2.2)

0,4

,11,

: ^til-l-

nh-t

17- _,l-

E

o

j ii

1

ì I ,l

Si,r$; n) : _{lr -} (r -,SÍ)l(/

^;

ⁿ⁾ : _É,

t

-t)'-' ⁽ 'i)

rUrr'rt, O

where

/c

is an integer number greater than or equal to 1

ancl

(,ï)t

is

the i-th iterate of

Bji

operator

definetl

by

(BÍ)t : Bf

ancl

v i > ¹

^(S;:)n

:

BíÍ(SíÍ)'-1, ((Si)o

: _/).

BÍ.r is not

posil,ive

in

general,

but,

n'hen

/ is sufficiently

smooth, S,i,,,

approximates

the function / better than ñ,,/

_[101.

'Îhe ^airn of this paper is to sturly l,he follor,ving rnore

general nperator

(1.1),s'i,À(/;n): tr-(r-E)t^(/ ir): Ë,, ^t)'-'(

^{1)tsÐ'Cf ;}

^rt

with /e

Oo(1) and. u

e

R-t.

We

point out

many properties

of SÍ.i

(Sections

2

a"nd 3).

Finally, in

Section

4

rve stucly the convergence for nb ^-->

@ anil

À -+ oc, separatelyo

P.

Genelal properfies

of SÍ,r opertrtor. BÍ.^ operator

clefinecl

by

(1.1-),

for a : 0,

coincicLes

with 3,,,i operatol,

inl,roclucecl

ancl

studietl

by Mastroianni and Occorsio in

_i121.

In

pa,rticular we have

ñÍ,r : ñi,

^BÍ,r

: ,SÍ,

,Bíi,r(/; 0)

:

"f(0),

^,Si,r(/;

¹⁾ : _/(1)

and.

(2.1)

^Sfl,¡(e¿)

: e¡, i : Ot1-¡

þ,,(u)

-

^{at', lt, e}

^N).

So

^¡SÍ,a

has

exactness

degree l- ancl interpolates the function / ⁱⁿ

⁰

and

1.

We

d.enote

now by L*f

Lt.e l,agrange

polynomial interpolating the function / ^{on the knots} !, _i : }rL, ...,

^tr,.

n

One can

verify

easil¡'

that,

^y /c

2

^1,

it

is

T .D

,."., ,wh IL

r,r,he.re

[0,*, ...,-'rrtr) is flre

ctivictect

differe'ceof _{order Z,} of

flre funcl,ion ø¡,,

with

respect

to the

nodes 0,

!,

_{. .}

., t . We recall

norv

that B,/

can

be

expressecl,

in the

follorving

tä"- tt6i

(2.3) ^*i(l

;

n):

Ë, ^{r,ñú,-o'tlr, *,

^{. .}

^., *

^,

,l

with

l(k,U ^nl

"(i,,t

:

11r-"¡r v

/c e _-rY,

(yi,o:r).

Ilrom this, _{b5' Q.Z),} lt

follo.ws

(2.+J ,Sfi(ø^,;

rl :

T

1yl,¡nÍ'-*tsf;_y1¡r-t

fhen letting

lùn,-o

fn,-. :

(rt'rt*

;,)

(ü, ¡¡\2'-ø . . , ulk'-"!¡,

ur :

(0t

0,

. . . ,

I)r

e -ED,

BIANCAIIARIA DELLA VECCIIIA

ltk sl*t

7¿'-1

sÍ1 Te,n:

116 4

antl

we

^obtafut (2.e)

Ð ,4 NEW CLASS OI¡ LINEAII POSI'IIVE OPERATOTìS

tt'|

By

bJris

relation,

(1.1) becomcs

(2.10) ,s,,r(/; nl :

Ë,

⁽

- ^1),-, ( l) _i Ð,yi,*[o,*, .. ,+;/lx

X hn,-oW*,,,,,t'r,lWl,î,,**rl

: _Äoú,rt

u,-.^Yn,,,"{

F,

^O

^{-- q,} (¡ì _{.} "*,,}} .

, nn 1t_ ^, "',

^,

1, h:0

("î" ":o),k )

^0,

(2.4)

oan

be written

^{as follows}

Bî(er;

u) :

hn,-'oln,oDa-r¡¡*ttrr

Now, letting

Ih'-o : filh'-a) anct

fotlorving [10]' ^we

^obtain

12.5^)

(,S,)o(no,-' ; fr)

:

hn,-o(ln,oDn-r¡¡nDll¡-o)oü"

: : hn,-oM1,,t¡t,-Jh

^(V

ⁱ

^{€ -¡f)}

llhe matrix

(2.6)

^Nu,rt*,-o

-

In,oDn-t,r¡nDllr,-a has as eigenvalues

the

numbers

-{ß,¡:

_-

#4 ^< t, ^{(i : r,2,}

^{. .}

^.,k).

Tlt'-ø)

Then, we denote

by

Wn,*,othe upper

triangular,T'lt1T, ^havÍng

^{as columns}

the

eigenvectors

"t Ñi,iìr,-", fdrmalizecl so that the elements of ^the

main

cliagonal are

1.

So

it

results

(2.7)

^Nn¡¡n,-*

:

W¡,r,oln,,W r,h'*

Therefore

(2.5)

can be

written

as follows

(2.S) (S,)i(t*,-* i u) :

hu-oWq*,'Tl^Wl,l',our

From

^(2.8) ancl being

(^gtI(f _;

ù: hoYÍ,r(SÍ)'-1(l o,-.; *¡lo

oL

,-,J

lo^

m såot

1. rl .ç:' ' - ltz

It

V'h,eR'

(0 e n'*)

x

^{Wr,L:,ou, 0}

D*,n: 4

On the other

hand, being

(2.r.1) Ë,(-t),ûì _{,)u:¿-,[1 -(1} -,)À]:

^q.,(,,

^{À), vúe(0,}

^1.1

and,

yÍ,0

( 1,

/c

:0r 1, ...,n,

(2.10)

can bc wribien

as follorvs

(2.12) lf,^Uir): _Ë,

^ri,nzä,*(xt À)

Lnnl l-0, t-, ...,]'- ,lf

where

zi,,u( a,

)') -

^{7 ¿ o,} -,,1y' b,r,oT t,o,xl[ n,l,,n'ì-{ t

and

Ir,o,x: Ii(-

æ ^1)j 0

{

L¡

¡.

I1 (1,,,à¡

is a

c.liagonal

matrix lvhose elernents âr'e _*(fi,,r, ]'),

^i,

:7,2,

. .

.,

^lc.

We notice

LTtat ø7,,t

is a

polynomi¿r,l

in r of

d.egree

not gleater than

^/¿, Thereforc, t'"om (2.72),

it lollorvs i;hat

ñ|,x

Í is a polynomial of

degree nob greater 1"Ìaan î?,. -r\Ioreover,

if / ^{is a}

polynomial

of

rlegree

not

greater

thnn p(p ç

tz), 1,hcn also ^8fi,1

/

^is.

R,ecalling

(!.L)j Q.Iz) ^can

^also

be ¡yrit,ten

as

follols

S:i,,x(f ; ^ro)

: _),

_{opi,,n(n) E} [1(",-ø)]-1

0 n

lc

),

t__n

,(2.13)

where

(2.741

y(n) : _Ë, (-

',' ^(, ì ^{,) ,ur,' (Í;}

ⁿ⁾

(Bi)u(/ j æ)

: i-ovß*lo,*,

^{. .}

.,+; /] ^x

X _{h n, -,W ^n, ",I.'¡-,LW ;,!,,"u

¡l

Now

we

d.enotc

by IÍ,i the remainder of Bf,¡ operator, definetl by

Ri.x: r - ^sl,i

118

So

l.c

have

(2.75) Ríi,¡.

:

(1

-

^,SÍi)t

: I, (-

^{1)' (,Síí)'}

'7 ¡\ Nlrl\\¡ CLASS OF LINIT;\R POSITIVE OPÌjRATORS _11g

.PIìOPOSIIfON 8.2.

SI.),

olerator

(À €

(0,1))

1)el.i.fies

ttte

_fottoeoing

raltttiort,Vøe(0,1),

sii*t,t.([;

r) _

ñÍÍ,r(.f ;

r): ^{r(r _ ,rxr} _+,) ^{[]r++ia]^

(8.1) _{-l*#.]^l ,-', Í", Ít).ff,}

_v

_f

_e

_c"e)

talt'ere

rr, Í,

^an'd,

r, tre

su,itabre poi'u,ts

oÍ (0,J)

generatty d,epe*d,ing _ou,

f.

*,u n#uool.

^Recailing

^ttrc

expression of

(BÍ*r - ^{síi)ff; ø)} [16], f.om

^(2.13) (síí*r;.

-

sr,,¡)(_r;r)

- -*,., Er[,;,#+,u+l_](,,it)n

*

^{tlh i'1,}

^-,,4\

^-_ "c)1,,-!_.1 IltL+\, -a)

$u"s.,.,.1,i

-

^{ñf,7.. has.} exactncss_ degree

1.

1\ro'eovcr, âs noticecl before

rl ./ ls oolì\¡o\

'l- ^tilsb

otrler'^arrcl"¡..(01 l),. tlrerr alno

y is

conve-x

of

fir,s, orrlcr';., tlrc,r,cfole (ñÍ*,.,.

- ,\i,,¡.)(Íi,t)'+

_{o' V}

r e(o,

t _¡."

Then,

b.1,

a

blrco-

i'ctn

_suclr

of

_LhaL,

Popo'iciu Vre

^({t, _{115l, rve}

l),

r,'an"

tinit ilr'e<' loirìi*

^zi.,

^,;, lii.r ^",i,

Ë

tö,ïl

(SÍ*r,i -

^{Sä,¡)(l; a,)}

: (ñíí*r,¡.-

Síi,¡,)(eri

n)lÍt,

Í2,

Ís) ll,

^v

I

e C"(I)

lìccalling

(2.16), (S.1) fotlorvs. Lrrorn (8,1), rve oJ:rtain

BIANC-AMARIT\ DELLA VEC CIIIA D

À

i In

pa"r'iicular,

by

^{(2.8), j-ù follorvs}

Ili,i,(r1 ¡,, - " ;

t;) :

I t ¡, -,,I1' ^¡r,,,.,,,

)f

*,n,i,lli ¡,,!,,,.tt, ¡,

ylrq"

^J,f 0,,,¡,

is tire tliagonal matd,r

r.r'hose c'le¡ur¡nl,ri

ât'e

(L

-

^{^¡li,,r)^,}

i:712,...,\t.

The

procechue

given

above ca,n

l¡c

usecl

reall¡. to evaluate

Bä,¡

I.

For

example, r't'e have

(2.16)

^síi,t.k,;

n):

ⁿ²

| ^a(L _- ^,) _{[ ,å++]^,}

Jli,,¡(eri ¿)

- -

^BÍ.r((¿

_-;r'):; r)-

^,.t)(J

-,ùl-lyt1 t I ^{rr(r -j-} _")

^J

l^,

sÍ,,,("f;

fi):2ç1 -,9^X/o -2,Ltz-,L,fr)

^{,ur-F t(2$À}

- ^{Ðlo++(l _$^)./rrr+}

-l- ^(2Ð.À

- ^1)Ål r _ï,fn,

where

Ð'

-

^(2a

^{l7)(2a +}

^2)-1.

Iinallv l'e ^{notice that}

^síi.r, opera"tor (), e (0,

i )) is a palticular

_{case of}

the

operatol

sä,,,,ç, ù :

,,fr,, (-

^1¡,,+r+;

_{(1) tu¡' $;} ,a),

r¿

{

^À

{

^nt

|

¹

Ifo'wevel,^it is

ea,sy

to r.erify thai;,

V ira

)0,

^ßä,,,,,r.

I

^does

^not

^colÌ\relge

!o /.

^{S9, Sfi,7.}

^{is the onl¡' operator} of

t,ht¿ clasS '{8Í,u,,r},,eNo

corì\rer.é-

ing to

_./'.

3._The pro¡retl.ics

oI

rSä,¡

o¡rcra[ru

(À e

(0,1)). .Ihe follorving

pr,opo-

sitions holtl

:

PROPOSIif'IOì{ 3.1.

S'!,,¡ operrúor. (}. e

(0,1))

lrresetDes th,e cotlcu,-

ait'y (conoenity)

of

etter! order'oJ ihe ,fu,nctiort io'aqtpróuimøte.

COTìOLLA ll

\.

8.3

oriler, then,, V

r

^e (0, 1)

l f ^the

Ju,nctioru

J is

conaer (concrtae)

oJ

_,fi,rst ,Sí*r,¡(,/;

r)

1Ir S,,i¡,Ç;

r).

oo)^anrl

if I

^e _-(0,

1), Llx'n, lr¡, (2.t)

o,ì

urt. tao.o vkin's tlteorem ancl.there-

ltrovecl l¡J' l\[oldovan in _114],

rve

COIìOIrl,Älì\i

8.4.

I;f f is cr I

cmd

if

rtrc s.eqtr,ettce..[Si!.1,(f.;

r)]."

^(À e

(0,1)) i

reusittg¡,"

tlen

llte J:rtncliott,

f ^is

no,t-cottraú (itcti-cot

_."^.

9"lgl1-1..]

3.*, togethcr s'ith the previous Corollary B.B,

gir.es a clralact,r'rization of functions conr¡e-\ (concave) of

first

orcler, on ^'1.

"

No\\' \'e

clenote

b¡'

r-,ip,r¡r

the

class

ôf Hrilcler-continrous

func- l,iorLs, i.e.

Lip¡¿t¿

:

^t,f

.

C'Q)I _lJ'@)

-/(y)l ^S trln ^_ ylt').

'The follon'ing proposition

_holcls :

Pro preserve funsl,ion

o/.

fncleecl,.

x'e

lec_all

[11] tìrat

Sf!, ancl

thelefore its

iter.ates, 1,he

concavity

or:

the convcrity of

evcl.)r ol'clel

s

^{(s e} tY)

of

1,lié

/ ^to

approximate.

Þn,

from

(11.14),-if 7r e

(0,1),

also

the function

y is concave

(co'vex)

of olclcr

s, ancl then, b5' (2.l.9)1ÁY,r/ is concave (corwex) ojl

order

s.

I

120 BIANCAMARIA DELLA VECCIIIA o

PÌìOPOSITION

3.5.

IÍ

^a

:

a"

: o(l)

^{(ru -->} æJ, th,en,

(3.2) _/e

^I-,ip¿¡r

*

Si,t,l e

lip¡a¡r, YruT-t ancl'

V Àe (0,11.

Proof. Í'et' f be a function el-,ip¡lt, and ø

^a'nd' 11 anJ¡

two points on

-Z-.

Then, by a theorem

provecl

in f3l,

^also

^tho j-tìr iteratc of

Sl,ancu

operator (Vj

^e ;Y) e l-,ip¡¡y., i.c.

(e,3)

l/(ø) -l@l ^<

^14@

-y ^{l'' =}

^(Bíl)i

^U;n) - ^{(sí)i (l;n)l < J¿lc} -yl*

T'helefore, from the clefinition of ñÍ,r, operator,

by

^(3.3)

it

follorvs

lsÍ,,(/ ) n) -sÍ,¡(/;y)t: ìË,t- :J" ^t)'(1) \J/ ^{ftun' U;Ð} - (sÍ)'(/;y)ìl

^I <

,A NI'W CLÂSS OF LINEI\II POS]TIVß OP}ìRATORS 72I

< Q^ -1)l\n - ^{yl* ç ,rlllr} -

^Ul"

Wenotice l'hat, iÎ

^o,

:

Q'n

:

0(1), (tt, ^--+ æ) ancl 7' e (0, 1], then, as otrselyectr -before,

Sf,¡f

converges

unifortnl¡' to ,/ ^on I ;

^so,

ⁱⁿ

1,hcsc hypothcses,

1,he converse

of

^(3.2) holds

too. r\

simple expression of

thc

rernainclel J?f,7f can also be given.

In

fact,, becausc

Sí,r

(À e

(0,1l) is a ^positive

operat'ol

with

exactness clegree

1, then, by a theolern

plor.ecl

in _l8], tve

have

(,3.4)

tli!,,i.$;

r)),

-t, ,rä,r,(rr;

u)Ï"(E) - - _[*,,n #]^/"ttt

Y

f

^e C'z(l) ancl

rvith I ^a

^suitat¡le

^{point in (0,1).}

Now,

let,ting

Ð,9(o):

_{lg@ -F ø)}

- ^{g(t)la-1, Dlf} : Ð,,(Di-'), (Di,!@):

^g(n,)),

y

n ^e

tr,

V u e

R+

anc-[ V r7 defined on

1, the

follou'ing

proposition

holcls :

PIìOPOSITIOT\ 3.6.'Ilte

^seqnen,ce

3.5)

rrDi'S",t(Í

) ^n)j:

n, ⁰ { ^ma 11.,

^1t1,

< t?j

^{À e (0,}

7l

ancl n e

I

a ø'i.f i ^e

s

^th, e f oll owt¡t,g nton o [,orr,icity pr o1t erties ^:

t) Jor

^na

--

⁷

a) if

on, th,e

interon

lo,7)

me Írør"tion,

f

^is conr)en (concaue) of

first

and, second, orcler)

tlten tha

^seçruence

(3.5) is

clecrea,sing (increasing)

o* lo'=tl'

b) if on

the interaal,

l=, Lz t -

*1tlte/urtcliora ^I

f

^{is concat:e (cotmten) of}

Jirst

order ancl conaen of

fi,rst

ordet' ancl conuen ^(concaae)

of

second,

oriler, tlten tl¿e se(fuetLae (B.b) is decrea,sirtg (inct'eositr,g)

*l+ _,t -.f,

c) i.f on

th,e 'í,nterl)a¿

tl - ^a, If

^th,e Jun,ction

J is

^{concaae (conaen). of}

first

and, secoùd ordei^, th,en, th,e sequ,eizce (3.5)

is

d'euea,sing (increa'sittg)

on lI - ^e,

^1');

il\formÞ2

a) i! on ^ttrc

^í,nteroat

10, ) ^u,,

¡urrrtíon,

f is

^cünc(tl)c

(conren)

of

ord,er ^n¿,

- I LI

^{a'ttil eonaen} (conotr,ue)

of

order

nt

aratl nt'

!

^L,

thetr tl¡c rcqtleruac (3.5)

is

deurrusittg (itrr:reasittg)

on.lOr\-9-)!l

^"

L 2 I'

h) if

on tlte intet'aal

i L ,- ¹ -

^(2nt'

-

^1)u

^{l, -}

urr]'thn funcLior

f ^is

^concanse

4. ûn

[he eorn,erctcnee of rSÍ,r. Assume

that

L e

(0,1)

and, o,

:

0,,,

: ':

0(1) ^(ra -+ oo). Thcu, as noLicetl

be[orc,

rSÍ,r

verifit's the

hypothoscs o[

l(olovl<in's blteolem on 1 and

iSÍ.r

/

^{convergcs 1,o}

I unifot'ml¡' otl

¹¹

Vf e

O'(1).

Ì\[oreover,

following _[12],

we

obtain the

evaluations :

(41) l/- ^sÍ,¡/l ₌ ^-î ^F,l#+]'J, yreo"e)

(+

2) ilt ^{BÍ,^/il_} _{å |} ;¡lii]^,).(r, ,l#r+ïl^,' ),

^y

Ï

^{e c"Q)}

(conaea) of order

nt,-7

cttucl nt, u,ncl contretr (concer,øe)of ord,ernt,flrtlueru tlre sequencc(3.-o)'ts tlccrausittg (inccttsin(J)

oul LJ _, t -,r*f

^;

t) if on tlte intertal lI - ^nta,L')

^th,c frtnction,

I ^is

^colLca,De (cort'ueu) o,f

ot"iler nt,

- I, ^tn

^{a¡¿al ilt,}

I

^{L,, th,en, th,o} serlu"ence

(3.5) is

^decreasing

(increa,sing) ota

l7 -

^ntcr',\).

Proof

. Proposition 3.6 follorvs florn (2.13) ancl frorn a

^theolem

provccl

in

_[2].

f,lamu,ùt. We, nol,ico

that, for a:A, Proposition 3.6

gives

a

^nerv monotonicil,y resu-ll,

for the

sequenccì

!r(8".¡J)1"'t(r)),,

_{7, e}

(0,11,

^171,

_<

îL ancl ø e

f

Thjs rcsult is

moro general 1,han

that onc cl¡taineil by

Stancu

in

_[21]

for thc

scçlncìnce \(13,J)t"')(a)1,,.

Iìinally

1re proyo

PROI'OSITIOII 3.î. Let !(a) ^{be a}

^piecewise

^linear

J'tutction ^toit'Ír,

at

most u,

-1-

^clmnges

of

^slope, wlt'iclt, c(t'n occLll'

only at

tlte

ltointsJ-,

'i:7t2, ,..r'tt' -7.

^Tlten,

^{for al|}

^natura,l nnt"mbets nt,, Sl,m+t,x(J;

n) is

nof degree

nm

a,nil, nt'ot'aol¡ef

,

,SÍ,,,r,(,f ; n)

:

SÍ,,+r,¡,

U;

^n).

Proof

.

Proposition 3.? follorvs from (1.1) ancl

flom a theolern

provetl

by

Fleechnan

antl

^]?assow

in

_[5].

122

ancl

thc

^puncl,ual estirna't,es, V

r

^e

^¡,

I3IANC.\\]TAÌìIA DELL1\ VECCI'II1\

n(L

-., h*-j;]'), otec"(r)

l'rc¡ttr

^(2.1 l)

j ^{lccaJling that,} tlicr

eigenYalucs "ií!,,,,

'i :7 ,2,

' ' ',11, ^are

uot gleatel than 1, I'e

have

tirrr

SÍ

r,(/i ¡) :

4-',r,-

)f,-Lr,,,,tl'k,,,,,,1'À-,¿ll-L,1 ,,,tt,¡Llr,,.l'(0)

: : å

^{r rr,}

r'!i,

^{I t' ¡'}

_-

^{urlr ¡",,, n('l|'} ¡,,,,"1¡,") - ltr'¡Àf)

"/( ^{0 )}

On

^1,ho

ottt". tral¿,

ft'orn

(2.7)t itfollorvs

It'¡,,.,n\Ir¡,,,,,,,(Ì1i,,,,,o1'k,o)-r

-

lt't,-ullrk,r,o(::Yu,r,u'ufIIt',,,,")-r

:

-'h, ^{t,, - o Il' r, r, nl|' n,|,,,, Ð k - j, -,,Ð ;1 r,t,,,} I ;,!,,

and

letting

f¡, : ^(r1r, l-;2, ..,r'iltt),

ii,

lesults

lt,¡,-n.Il''¡,,,,n(Iì¡¡,,,,,uf¡,,,)-t-,YxDllt,y,,1-1.å:lt'¡,,r¡rl;,|,

!

So rve c¿r,ir srlit,e

lirrr,9ji,7.(./;,i,)

- f,

n^iü.,,'.''", ^1, r,,,,,1,,;,lilil-\f7,,/(0)

7.-æ "ô l; ^t

-- f

_t4" ^,-/11- r,{r',t/")¡t,/(0)

1.. I

that is

^(4.5).

^ Nlì\\¡ CL¡\SS OF LINIiAR POSI'l'lVll OI'l:llìÄTOl-.S

tìlì1ìrìtìlìNcr'ls

10

il1 123

l,f(n,)

- ^{Si,t.U; ø)l (}

^2co

^{(t rll}

lÍ@)

^{Si,,t.ff; er)}

^l<2 n(r- r,)f t"'i

¹

L

^r¿(1

f

^ø)

^{]^.' (r''}

tr,u

! t

n(r-n) I

rr,(1

f

^ø) I

Y

,f'

^{e C"(I)}

\4¡c notice

tìrat, for

^¿¿

:

0

in

^(a,1) zi,nct (+1.2), rve fincl the

lclations

obtairr-

ed :åi;""'"rllT i:" f*1,"n^i,li'ü;- ^srudietl

^b):

l\Ia in

_[101.

So

assutne

thal,'), :i + ^{8, u'ith}

^r)

.biggcst

integt'r'

nurn̡cr < ), anrl

^{ò e} (0,

1). then, frorn

^(3.4),

it

follorvs ^:

ltri,ti; r)l

ltÌ,:, ò(Itä.,U;

n))l < j.tttry.,11,,,,,[ # i;]

On the

othcrl

hand,

because [101

llQti;,of)ø

ll (

c(ru

- ^2)-'

^{ll./(') 11,,, Y.f e c2t+2 1r¡}

rvltclc

ll/llr, : rìì'àx

_ll/(j)ll

ancl

C is a constant

depending

on i antl intlcperrdent of /

^a,nc1

^n,

^u'e

ol¡t¿lin

COIìO],LAIì}:4.2. Let ),:i+ 8,'toitlt ieN

^an,d

^òe

^(0,

1).

'l'hen, V

Í

^e C2i+2(1.),

iI

^resuLts

ll

^t¿ä,,.Ill

(

^o, ll /,,) ll,,l

_L ry:J _{I +r,}

^lo,,

^{-'-u, n )}

²

J Q,ultcre o¡

is rt

cr¡nstant itttlependen¿

ol I ^tutrl

^n.

Filrallt'

1\'e plo\¡o

'I'FIIlOlìlllL

^4.3.

I'et

_{.f e}

C'(I). 'Ihen,

^th,e Jollotuittg reLtr,tiort' holds:

(4.ri) lirn

SÍi,r(/ ₎

u) : L,(l;

ⁿ⁾

wtr,ere

L,(J;

a,)

fs

.Løgt'nttqe polynont'iu,l' interpolcttittg [lt'e Ju,nctiott'

J

^{on' tlt,e}

\çvt,o[s

!-, i:0,

1, . . .¡,tt,.

,IL

Proof

. Ililsi, of all, l'e notice that, flotn

(2.12),

it,

follon's

lirn

Á"ji,7(/ ) n)

:

Éo^rÍí,^ ]|-h,,,-,,11'r,n,ul

lim

.Fr,n,¡l'Ilr;,];,otr¡,Ltl¡,,

j. ilrrsliaìiorr, \'. r\., -,lrr ctctnt¡tle of a scrltLcncc of littctu ¡tosíliuc operalors itt lltc s¡:trce

ol' t:ottlinurnrs Iuttt:!iotrs, l)okl. Àltacl. Nauk., Iì:l ^{(1 957), '249-'2õ1} '

Z. lj ctlrr yccchia 8,, On ntottolonictly ol'sonte littcar posilinc o¡tcralors, t6 Ìppt'al ^in.

..1>rococdilìgs oli [he'I'hil,<ì Co11lcÌcltcc o11 NunrcÌical ]tcthocls arrtl Appt'oxitrration'l'ltcety", Nio:, Yugoslaria ^(1987),

B. J)clla'i¡ccchil )'l,, OtrlhepreserotrlionofLipscltil:conslrutlslorsotttelittctto¡terllors, stìl)nliLLed Lo .'llollctl-iuo _dcÌla l-lltionc ìIatcrnnLica ltaliaua" 11987).

4. Iì a v I l rl .I., ^S¿¡r' l¿s ttutlli¡tlicaleut s rl'inlct ¡toLutior¿, ,I. ltaLìr. PuÌ'cs '\ì)p1., 9:l (1944), 219-',247.

5, lìr'ecrlnran I). anrl Passorv Iì., I)crlurcrtLle lieuttLcitt. Pol¡¡nontÌtLl.s,.I..\ppt'ox.

'lìheor¡', _3t) (198:l), 89--92.

6. ITc¡i.rran,ì ^'Jl ., On l)asltrtlirtu-lypcopcrelots, AcLallalh. Sci. IJung-ar., Jt) (3-4) ^(1978), 307 ^{31 6.}

7. .ll ot o r, á L, ,l ^nolc on lhc scquutcc fornretl Lu lhc [irst ^ordct dcrinaliucs ol ^llrc Szttsz

ù[ira]i¡¡tot op(t Qlot s, IIaLhctu., 24 (47) (1982), 49-ã2'

g. l!.I¿rsII'oia¡¡i G., Sui rcstidialcunc lornrc Liuctu'idittpprossÌntr:iotrc,C'alco)o. llt (1978),li43-368.

S. ì.Ias'ír'oianni G,. Su untt ^t:lasse di t'tpercrlori li¡tccui c posilioi, ltrrltl' _'\ct:acl' ^Sc' l\{.F.N. Sclic I\:, XLVIII ^{(1980), 217-235.}

10. ì,I¿rsl"r.oia*'i (ì. o,rit Occot'sio lI. l'.., ^UtttL ylctrcralizztt.ziottc dc:I L'tt¡tet uloLc rLi

Slcotcrt, lìcntl. ^Accatl. Sc. II.lr.N. ^Selio I\r, XLV ⁽¹ 978), l5l-109'

1!24 BIANC^MARIA DELL¡\ VÌìCCIIIA

11.tr,fasl.roianni G. and Occorsio l\,I. R., SuIIerlcrit¡a[edei ¡:olitronidiStancu, Rend. Àccad. Sc. M.F.N. Ser., f,V, (1,978), 273-281.

'12. }{astroianniG. aud Occorsio lVI. R., Attewoperatorof Bernsteinlype, Mathem.

Rev. Anal. Nurn. 'Iheo. Äpprox., 16 (1987), 1, 55-63.

13. Mirakyala G., Appt'oximaLion cles fonctÌons contittues au tnolJeil tle pol¡¡norttes de Ia forne ... , ^Dol<I. Akad. Nauk. SSSR, 31 (1947), 207-205.

X4. Moìrlovan 7!., Obserualio¡ss¿¿r'Ict suil.erlespolynomes d¿ S. N. Brrns{eittd'unefortctiott contintte, l\{ath., ^zr (97) (1962), 289-292.

15. Popoviciu T., SurleresledanscerlainesforntuLeslinëairestl'approritnationdel'atrcLyse n,Iath., I (24) (1959), 95-742.

16. Starrcn D. \)., Approrcintalionof funcliorLsb¡1 ct ttctuclass oflinccu'polgnoutielopetetots) Rev. Roum. Math. Pu¡cs Appl., 13 (1968), 1173-1194.

17, S t a n c u l). D., ^Use of probabilístic nrcllntls in tlrc theoru of unifornt approrcintalion of conlinttous funclions, lìer'. Iìoum. Pures Appl., 16 (5) (1969), 673-691 .

18. S t a rÌ o u I). IJ., Approtímalion properties of a class oÍ' linenr positiue o¡tet'aIors, MaLb,, 3 (1970), 33-38.

19, Stancu I). D., Ot llrcte¡na[ndcrof a¡;proxintation.of fu,ttcliorLsby ntectnsof aparanrcter- tle¡tcntlent linear polgnontial operator, lVIaLh.. 5 (1971), 59-6ã.

20. Starrcn I). f)., Approximation of _funclions by ntcans of ^sontc rtea classcs of positiuel linettr opcrulor'.s, in "Proc. Conf. l{atìr. lìcs. Inst. Circlrvollach", ccl. by L. Collatz ancl G. l,Ieinarclus (1972).

21. Starrcu I). 1)., tlppticalion of diuklett cliffercnces to tlrc sttttly of trtottotorticily of lhe clctiurt!.iucs ol'11rc scqucnces oI Bunslcin ¡tolgtrcntials, Calcolo, iC (1979), 43I--445.

22. Stanctr F.. ^AsupLcL t'cstului în fornLtlcle rle nprorintare prin operalorií ltti X'Iiralcian tle tutttçi tlouà uariebile, Änal. Stud. Univ. ,,4t. l. Crr,a",1/-r(1968), 415-422.

23. S t a n c u ll., Asnpru u¡trot,inú.rii furtcliilor ^{de ttna} ;i d.ouä uericrbíIe ut a.iulorttl opere- Ic¡rilor lui Bctslcùtoo, Sttd. Celc. Celc. NIaL., 22(79?0), 531-542. |

24. S z a s z O., Gc¡tcralizctlion of S. Ilcrnstcín's polgnomÌals to llrc ìnfinile ínterual, J. fìes' Nat. Stantl., ^¿r5 (1950), 239-245.

Rcccivecl 15.III.19B8

IsIilu[o po Applicazioni deila nlalcnm(.ica

C.N.lì, Napoli, uict P. Caslclli¡to, 111-8013f

1\IATI'IE]\I..\.IìcA

_

_IìEvUI], ì),ANI\I,YS]J NUÀIÉIì I QUI]

]]T D}J '|IIÉOI'\IE DI] L'APPIìOXI\{ATION

,I,,aNÀLYSE NUI,IEIìI0UE ET

X,A THÉOFåíE

ilE

I-"{PI¡IùOXIRTATION

ïomo

17,

N" 2,

198{i,

pp.

125-13e

RDPRESENf|ATION OX' OON1INUOIIS TTINEAIì

FUNCTIONAI-,S ON SiUOOTI{ NO}ìMED I-IINEAR SPACES

SEVER SILVESI'}ìU I)IìAGOMIR (Timiçoara)

al¡struct.

rn

this paper we shrùll give some i,heorems

of

r.eplesentation

for the _lineär

_spaces

_by

use

of 1' I ^{and Tâpia} [5j,

and

the

_{iements oif tinãa,ï}

subspaces.

pp.

3Bg). I:,ct

X be a

^rcal

XxX'K(IR, _e) is

called or' ¿-semi-inner prorluct,

12

Introduetion. DTJI¡IìI{ITIOT\

1 (l3l,

_[1]

or

complex lineat, _space.

A mapping

( _{, )"}

I

,semi-inner

plod.uct in the

sense

of

^I_.,umer

;for short,

if the following

condil,ions holcl:

(i) (n*y,ø)":(n,ø)x*k/,ø)¡., _ü,ytãeNi t(ii)

(on,

!l), :

a(n,

U)¡¡

^{a e}

I{,

Ø, lJ e

X

_;

(iii) (n,n)")0 if r*0;

(iv)

_{I@, E)"1,}

( (r, r)r(y, y)r,

û,

y

e

X

_i

(v) (n,)r?it:\(r,A)r, ì,eK, n,y €X.

tr'or

^t_he proporties of l¡-¡lerni-inner product, we ßend 1,o

tll

^{pp. gq6}

_

389,

or l2l

^where

^further

^{refe cnces a,re given.}

DEIìINITION 2 ([5], _t1l pp.

BBg). LeL

(X,

_ll

.ll) _{be a real}

_{nor._}

,merl'linear

_spacc

andl:X

^--+

IR,,f(ø):f

^ff

^{nlp. neX.}

2tt""tt t we<.

'Ihen the

^{lna,pping :}

(n, g)o

:

(v+

fl(g)

^{. æ}

: tilg t@Jtvl,

_$,

lt

^€

x

_i

tJ,o t

{e called semi.inner product

in

the sense of

rapia or

?-gemi-inrìer product,

for

short.

n'or the

usua,l_properties oT ?-se¡ni-inner

product, wo

send. 1,o

[1]

Pp. 389-393 or _[2]

where

furthor

rcferencos a-re giveá.

E-¡, 2fZ(