View of Approximation of continuous set-valued functions in Fréchet spaces I

l4 _{I{üscyin Ror 4} }TA1'ËIE\{À'I-ÌCI\ ^{--_ Tì} I]VT-IE I)' ANÄI,Y S1ì N TJM í'R I QUE

I'T DI''I'TII1ORIIj ì)lj I,'T\I'T'IìOXIùIÀ'TION

T,"I.NÀI,YSIì NUNIÉRI{)UIì

ET

^[,TI.

TIIEOITfiI ÐTì

I,IIPPROXTß'TATION Tome 20, No'

l-2'

^1Í)$1,

^pp. t5-23

tl- 1

ttt I

r-.1

x

:

O(1)

5

^lA(ag.)

^{I.X, -l}

^O(r) m,þ,,,,X,,

: 0(l

₎ l) ÂP,

J

1) 1

+

^O(1)

I,

^{I [3.*, I}

^N, -l ^û(I)

n,p,,,X,,,

: l)(1) as

?r¿

+

oo.

by virl,ne

of ,(1.õ), (2.1), (3.1),

and

(g.Z).

rt

_{_may} be re,markerl

thai from

ìhe.'hJ.potheses

of the

theor,crn, (À,,)

is l¡o'ncletl. 1'his .a' be

sho\\'n

rike ilris."Si"";.

ii"j-is"non-de""náoir.,g_,

Nu

>-N_0,

which is a positive

_constant.

rrence"i.rìq'i"rprio,. that

_1i,1

ìí

bounded. Thus

APPROXIMATION Otr CONTINUOUS SET-VAI,UED F'UNCTIONS IN FRÉCHET SPACES. I-

i\IICIII]LIÌ I)\ù{I) i'I'I

(llari ) D:1

X

,il

t'1,

I

In,rl" : Iì

^{I À,1 l?,,} lt-7t¿-tlt,l,,

:

O(t)

,Ð'

1r,,,Itt-1lt,,lr

ï\''e consiclor.scL-r'aluccl lunctions clcfinocl on a cornpacL Ilausdol'{f topological space and wìth conrpact colìvcti values in ¿r Fréchct spâce; \\¡c civc soltic conclilions rvhich ensttl'c th¿rt a sel

of sct-vaÌucd funcl"ions is a I{olovkil sc[ \'ith ^{rospcc[ [o} ccluicontinttous nets of nronotone Ì,iucar opelalors.

: o(1) i-ì' ^{l^r,,,12,} + ^o(t)

lÀ,,,

lx,,:t,(1) i.

^þilxn

^-l o(1)l),,,,1!*:o(l)

Ð,rs

rn

^--+ oo,

in vien of (f.B),

(1.6), (2.1), ancl (8.2)

'Iherefore, ^'rvc

get

i,hat

Xntrr¡duotion.

In

some r-ecenti pa,pers _[2

],

[3,], 14] ^ancl

[6] ^ihe

^interest

of l(orovkin-type approximation theory has been

extend.cd

to

^linear

operators on

sct-r'alüed.

function

cones, mol,ivatecl

by the variei,y

of circumstances

in li'hich

set-valuecl

functions

are

involved,

such as

opti- rnal contlol theory, mathematical

ecoDomics

and probability theory.

In a precursory study of Yitale _[6], the approximation of Ifaus-

cÌ.orff continuous sel-valued

functions rvil,h

compact convex values

in

a

finil,e

rlimensional normecl spa,ce has been cliscussecl

by

means

of

Bern-

"sl,ein polynomials

on

1,he oompact

rtal interval _l0r1l.

Àfterwarcls,

Keimel anil

trìoth have establishec-l

in _[3] a Korovkin-

h.1pe theorom

for

set-valuecl

llausclorff

conl,inuous

functions by

means of

Kotovkin positive

systems

for

single-valuecl

leal

functions

; ^the

^same

theorem ha,s been imprr-rved

in _[2] ]:¡. using suitable upper

^a,nc1 lo$'et' trnvelopos.

Finally, I(eimel

and.

Iloth in _[4]

^have developecl

an

al¡st;r'act

formulation of

some

locally

convcx topologies on ordered cones

by intro-

ducing

a notion of

(upper, lorver)

continuity which constitutes the

sub-

stitute of the

trIausclorff

continuity in

^{normed spaces}

; their

^{results gene-}

n-alize approximation ^processes

to

1,he case

of

infinite-d.imensional

locally

convex

topological

spaces.

In this

paper we are interestetl

in

^'bhe

Korovkin-type

approximation

of

set-valued" funct,ioirs

with

compact convex values

in n'réchei

spaces ;

Lhis

setting is not tho most

general

as

consid.ered.

in _{lal but}

^provides

u.s some tools

of

selecbion

theory which will

^play a crucial rôle

in

achiev-

ing the main result.

Moreover,

the tr(orovkin-type

theorem obtained ^is rrob expressecl

in terms

of

l(olovkin

systems

for

single-valued.

real

^funo-

fions

as

in

_[a].

,i:',r+l'Jtn.,l":0

^{(1) as ??¿ -+ oo,}

^ror

^t'

:

L¡2.

This

cornplel,os

the ploof of the

theorem

tìtìtltiÌìIìN0ljS

l' ^{.ìl I c L} t, 1'. i\I., Ott. rttt e .¡:l¿ttsion. oI nbsolulc stuttmabiliLg atrcl sr.lnc l]tcrtrents sf l,iltle¿:onl anrl PaIc¡¡, PIoc. Lo¡<ì. IIaLtr. Soc., Z (1957), 1.Ill-j.41.

2. lí o gl¡ c tl i irntT', Ìì., Sr.rl lcs sét'ies ul¡s¡tlumcnl sotiunablcs ptrr Ia mé.llrcrlc rles ntoqennr:s

aritlttnétirlttes, llull. Sci. l\,Iath., ^/a$ (1925), 234 - ^256.

3'l\lislrra' Ii.N. ¿ìrÌdSrivasl-Íì\¡Ír,.|.ì.S.L.,Onrúsolule Oesùroiutnntcthility _fru:laLs

of ittfirtitc scrrcs, Portuealiao ]\Iath., 4È (1983-.,1g84), 53- 61.

lìcccivcrì _15.IX.1990

l)eparlntcnl ol X,I0IlrcntaIit:s, Iìrcigcs U niaeNil!1, Iloyseri 38039,'l'Lu kcg

I Worli pcllortnctl ^under the auspiccs of thc G'N.A.F.r\' (C'N.R.) and M.IJ.R'S'T' t60o/.) and supportecl ìry I.N.d..A.nI.

1ß Michelc Campiti

In the

second

part of this

paper, N'e

shall

consider singlevalued continuous functions

rvith

valuos

in a Fréchet

space artd

\re

shall stud5r

the

convergerìce

of

equicontinuous

nets of

opera,l,ors

satisf¡ri¡g

suitable

condil,iorx; as a

consequence, u'e

obtain a natulal

generalization oT the rvell-l<nou,n case

of

real

valued

functions.

l.

Prelirninulitt¡; ^¿tutl

notirtiqrn. fn this

^Secl,ion

lve recall sone

preli-

minary definitions

ancl some classical

lesults on

selc¡ction theor'¡'

l'hich v'ill be

usecl

in the

sequel.

AII the vectol

spa,ces

tnclel

consic-leral,ion

halre to

^.lte consiclererl

over

1,he

lieid lR of real

numl¡els.

\\re

Ì:egin

liy

fi.xing a,

locally

con\¡cx

Ilausclorff

topological space -Ð

ancl

a

base

E of

cotLt'ex open neighborhoocls

of 0 in Z ;

rnoleovet', \\'e denote

by 6ont'(D) the cone of all non

ernpl,y convex subsel,s

of

-Ð, endon'ecl

rvith

1,he

natural

arlclil,ion ancl

rnultiplication by positive

scalars ancl

l¡v

(6Grnu(1tr)

the

^se1,

of all tron ernpty

cornpact

convex

subsets ot E.

IrcL

'Y

^ìte

a

topological sp¿ìce;

if

_,f

:X

-Ør,,¿,,(-ãl)

is a

set-r.aluccl

function on

,Y, rve

recall thi¡i, a

(continuous) selection of

,/ ^{is a}

^(con-

tinuous) function

<2:X

-,14

sal,isf¡'ing ^p(âr)

el'(r) for all r eX.

Nloreovor,

the Tollol'ing notations

r.r'ill be

ust¡ful in the

^sequel ;

G(X, n)

denot;es l,he space

of all

corrl,inuous

functions on -T l'ith

values

in

-/J enrlou'ed

u,ith the

topologv

of the unifolm

con\¡ergence;

if ./r,...,1,, are set-r'alued Íunctions on N,

u,e consicler

the set-valuecl

^functjon co(f

,...,f,) clefinetlb¡.puttins, fol

eß1rr ü

e-T,

co

(lr,..r,f ,)(t) :

co (lr(a),

"...,,f"(u)), I'here

ao(,ft(u),

...r1,@))

clenotes

the corn'exhulloli/t(ø)

u

...

...uÍ"(r);

rnoreorler,

if

^cp

e(€(X,r), [ç]

^{clenotes the} set-valuecl tunction defirred.bvputl,ing,

foleach

u

eX, _{q}(r) : _{ç(r)}

ancl

finall¡,,il|qy...¡9,, arein Ø(X)n)j

^u'e sùnìrh'rvrit,e

co(9r,...,?,)

insteacl

of

co

({9r],...,[q,]).

the

follorving

plelirninary result is rvell-l<nol'n

ancl c¿rn

be

easily derit.ecl

from

lS, Ierlrnlzt

4.1 and lheoleur

3.2" ].

llr¿olost:r'rox 1.1.

If X is

^(, palu,conlpad [fa,usd,ortf slta,ce

antl

IX

is

^a

Frëchet sp&ce1 tlten, eu,cl¡ lozoer se'nt,icotttittttous set-au[,uedJu,n,ctiott ccchnits u co'ntinuous ^s election,,

Moreouer,

if

_.f

:X

^--+ GØr,ttu(14)'is ø lo¿oet' senticontitt,u,otts set-aa'lued Jwnction and

if A is a

^closed subset

of X and 7:tl

^--+

ll ^is

q, cott.tinuous ol _,f

ø,

^th,en,

7

^{can be eutend,ed Lo ct,} cotttinu,ous selectíon o,f

f ;

^in,

', i."Í'noeX

eutcl

lloel@ò,

^tltere

^{erists ø}

continuous selection of

f

es l,he ao,lu,e yo

at no.

ffil

LeL

X

lre a palelcornpact Haustlor'llf space anû

Ð a Ilrí¡chct

space.

ff f ^{:X --}

G6c,¡¿t,(D)

is a

lorn'er semicontinuous set-v¿r,luecl

function, ìn

the sequel

s'e

shall denote

by 9t/(f) the

^(non

emptv)

convex set of

all

con-

tinuous

selections

of /; ^{bv viri,uo of the}

^preceding

Proposition 1.1

^'rr'e have,

for

each a

eX,

(1.1)

_T@)

-.¡¿",,{,v(n)}

^.

Jlloreover,

the follorving

lernrna

ivill be useful in the following

Sections.

,4,pproxiìììation oI colrtint-tor¡s set-valtrecl frrncljons. I t7

þ'récl¡et sp&ce.

r.f

_{.f ,tt}

:x -

8(€o)ruetr)

aie

loruer sem,icäinfilntol.ts set-,.ccr,l,ued

J'unct'iort

nnrl

), e [R,

il

^resu,Its

(1.2) _Yef(l -l tJ):9¿/(h

1-

e,,/(o),

(1.3) ?cr(xJ) : t,r"t$).

_

^l]roo,[. Vtrehave,onl.ytoshorvtheinclusion.g"f _(.f

-le)cgat(f)lgcf(cr), tht¡ otlrel

bein.g

trir.ial, 'ro this

^enc-[,

fìx

₉ e s,,cî(j'¡'¡1i'ancl let'(v',,¡,,.^'tr"q1

a

sequerce

of

convex closerl sets

in È l'hich

is a'úasé'of neighbòr,hó;ìi*

"i 0

"satisfS'ing trr,t,t

c 2-"V, fcir

e¿lch ra e

[\. ^I3y incluction,'1,e

shorv 1,he

existence^of a, se,quenoe

(ù,)"u^

oll coni,inuous selections of

,/ ^ând

a fìoq¡erlce (X¿),€sr ol' .onLi nuorrs scleclions

of

r¡ suclr Lhn,l

, ^frr,

^{r,acli .c e .t',}

(1.4) _{,n,(ø)

e ^tf _,,(r.')

*

^l/

u,

x,. t(n)

e T,,eì)+ll

^,, and

(l.ri)

rp(a)

* þ,(n)

^y-,,(n)

^ê

^{l; ,,.}

th ^'l {,'"V,lr'

pr

_exist

Q.,, e

an

llorvs the

hen the funct,ions

^tJ,,o :

-ll * il

ancl

üo(ø)

:=,fi,

1,,{")

,1,,{r), n@) :

,1,,,,r,(*)

^x,{o)

fol

each n e

x,

are conl,jnuous selections of _,l' ancl r,espectivel¡, g. rlroleover

r{.,0

and

_1e

satisfl

conilition (1.5).

ditions (l .,t) anrt (l

.i)

anrl con$idcr.

ing argunr.ent

to

shorv

the

existence aÛd _Xn.t,t

of 9,+t satisf¡'ing ç(n) -

tT; ä{*l'n" derinitions

or

J"'

^anct

_.By

^(1.4-),

the

sequerrces

(ü,),,rm ancl

(2,),,e

or satisfy the

Cauchy eondil,ion

rvith

lespect

to

the _r.rnifbrm

topology'il

^zaçx,

n¡ änd

ilrerefore

us

funcl,ions ^qr: .Y

-- D

and, respec-

d _{X are}

conl,inuous selections of

/

(1.5)).

shall

consicler. set-r.aluecl fnncl,ions

us an(l we shall indicate

these set-

18 Micl.rele Campiti

-valued functions simply as

continuous funcl,iolìs.

llhe

cone

of all

con-

tinnons

set-valued

funciions from a topological

space

X in €G"nu(fr) rvill

be clenoted

by G(X,

Øßouu(H)).

the

cone

G(X,

6(6r,t,u(D))

is

cquippecl

with the

l,opology

of

^the

ulliform

convergence; namely we shall say

that a net (/,)Í.t of

set-valued lunctions

from,X

^t'o G€onu(D) converges

to

a seb-valuecl

function /:X -'

-GGortr.,(H), if, for

eâch

I/ eE, there

exists øe

I

such

that

Observe

that if

l\foreoverr an

operator

'rt

:6 --

^re¡,r,

gre'i."(t;jjîcallecr

*on',rr,nrìr, ;¡:

(2.1) _{1,g e'€, ,f <}

_{U =>}

lt(fl.<

?,(t)

(cf.

⁽ 1.7)),

, - ^Fina]j¡r,

^{\\'c s¿lv}

that ?

^is conl,inuous

if it

is oontinuous r.vith r,éìs'ìer,û

to

¡,he unit'olrrL syrrirrre.ûric toporogy irrduced

;,; Ø; ìïì;;, ii,:'åi,,r,"ìriii

thcrc

e.rists U e öt

*uclr tttat,'it"f,,"!.G;ti*ii, ^" '

^u¡¡(¡

.l'(n)

-

s@)

I ^U, s(u) c f(n) |

^U

for

each ,r

e,Y

(i.e.

,f < tl ï

^(1,

^g </+ U), ilren ,I(h

@)

-

^T(s)

^(r) I

tr,, ,I(s) (¿)

-

"(,fX

a)

a-

^V.

for each-¿.e-{ (i.e. ff

¹⁽ is ¿ù rìor) errrpl,¡,

?(/) _<

conrþacl ^,.f\g)

* conv,ri ^1,,

^,f,(g)

subst,['ôf'lr, <

1'(./,) F

lr).

_n,n rlerrol,c l¡r, ^1,,..

Lhc

crnstanb

sef;-r,atuerr

furctior _{on .r'} ,f .";ri*¿'i*rul i?';il :rìrjíï

laJ'

that

a subcone'z,of

Ø(x,l?Gont(E))

_contains

iü"

"å".t*nt

sct_v¿rl*ecl

ftrnctions _lhe ij.ft,

follorÀ'ins

e.e

Lernma

for

each

rvi[ '.o'

be

"rrìpty

irsörr¿

"orr,pu"t in^the ;;;.;;; ,ãq"ãi.

subset

K

^{or H.}

Lnr'r'ri^ _2.1.

Let G

be u' suttcou,e

of G(x,

GG,,t,u(rx)) cortt*i,rt.i'tt¡1 il.e cct't¿stttrtt t¡tonotot¿e set-aa[,u,eù, Iin.eur oncÌ'rL!,.ort.J't.,ont .fu,nct,ions cut,il ,6 _in,6(,1' ^Let

(?,)X.

be ct,n equ,ico,u,ti,ttrtotts _t¿el, of

,

Çrß,.,,r,1b¡¡.'fnir,,,¡:oi:

,rrit,

i:'årl¿',

lltcre

erists

ü

e

S

sucl¿ lhî|,

l, I ^{eG,.f 4}

^g

j-

U

.- I,ff) _< ?',(t) |

^V

.for

^{eacl¿ u}

eI.

Proof

. l..'l, lI

^e

tl _;

since

(?,)L, _is

_ecf

icontinurus at 0, ilre'e

exisl;s {.I,

eE

srrötr btrat, r'nr

å*"t-f ¿}r:'.'''

^'

./ <

^At

= I,(Í) ^{ç Ì/ fol}

^{each ¿ e 1.}

+ i":åì'1?'.r

u-

co of Z(cf.,

e.g. _{l_}

1' _ant

_set-valùed

..

In l4l,I(eirnel

^{and Roth have al ro} inl,roclucetl

the

class

of uniformly

continuous operators; zr,ncl opelatot,

r'

: G

-G(x,GGnn"(n¡;r

^{called uni"-}

l'ormly_gontinuous

(or _briefly t¿-continuous)'rf, for u,ùrr" z et],

the;re

exists I/

^e

E

such

that, for

eâch _.f

,

^{g er€,}

f ^<

^s

* u- ^{?(/) <} 'r(s)-lv.

4

al A¡rploxinratiol.r of contirtuous set-valnr:rl functious. I II

J'@)

-

"f,(c) -F

V,l,@) - l@) a ì;'fol

^{each n}

^e,\l

^{ancl ¿}

^)ø.

fn

[4,],

I(cinrel

and.

Iloth have

inl,roclucecl 1,he symrnetlic topology

on

(o,rru(fi))

by

considering

the farnily

({1)

eqo,',,@)

_I

B - ^A + ^{lr, A} c B -f

^tr/})r,Es

irs

a

neiglrbolhood base

of an arbitlary

element A. eGonu(E).

Sinc,e rve

shall restrici, our attention to

set-valuecl Íunctic¡ns wil,h r'àluesr

in

l,he subcone 6Goatu(El of Øoatu(E),

i! ihis

casetheconl,inuous set-valued

functions

coincide

rvith the

set-valuecl functions

rvhich

are continuous

rvith

respect

to the

symtnetric topologv

(cf., e.g.,

_11,

Corollaly 1, p.

6?_l)

;

consequently, the

notation G('Yr66onu1l)) is

consistentl,vith

that

one rLsed by

Keimel

ancl

Roth and. a

set-valued

function f ^{:X --}

-->G6onz,(11)is continuous

if

ancl

onlyif, foreacn'

V

eE

anclcne

X,

there

cxists a

neighborhood rY

of

øo suoh

that

(1.6) l@) -

l@o)

I Y, l@ù . _f@) t ^v

for

each

neN.

We

conclucte

this

Section

by lecalling thab in Ø(X, G€onu(El)

is definecl

the

follorving order

relation

(1.?) l<go,f(r)c:g(n)To,-eact-ue'Y

for

eaclr set-r'alued. functions

l,g ^X:,

^{--+ îîØonu(Ð).}

We shall

^{also use}

the

nol,al,ion

.l ^<

^g

+

^V

^(1,

^g

^e(E(X,

^V6o*u(E))

and V

eE) to

indicate

l@) -

^g(n)

-l V ^lor

^{each u}

^eX.

2. tIp¡rroxirnation q¡f eontiruous sct-vniuetl funetions. fn this

Section rve

fix a

compact

llausdorff

topological space

X and a

Fréchet space

Il

1 we shall consider a sutrcon e Ø of

6(X,

G'€ ,rou(Ð)) cont'uining the single-valuerl

functions (i.e. {p} eç for

eah

9eØ((X, E))anct ^'we

^shall stucly

the

convergence

of

equicontinuous nets

of

monotone

linear

opera- {,ors

from 6 it G(X,

GGonø{E)).

First

of all, ^rve

recall

Lhat an

operator I

^:Ø

_-,

Ø(Xr(66oaau(Ð)) fuorn

â, suboone G of

G(X,€6r,nu(E)linØ(X,Ø(€ontt(D))

is callerl linear

if r(f _{+ s)} : !I(l) I ^{r(s), ?(À/)} :

À?(/)

fcrr cach set-valuetl

functions l, I ^{eG and}

^À

>

^0.

:20 lrzfichele Carnpiti

¡tpro_rinratiotr of corr[ìnrrous s¡i-i,¿lr¡ccl lil nctiotrs _TI 2l

IL is

¿leal

that

an

uniformly

conl,inuous op€ìÌ,atot,is

both

continuous

rì,1)r[ rnonoto]re

;

^{as a} consequence of Lremma 2.11 a,lso

the

conyerse ^is true.

Cottc't,r,.\rr.r 2.2.

Let (l

_bc

_{a s,tr}

_on,e

_{of ß(X,} G$ooru)(Il))

cojttu,inirlfl

't,l¿e

r:on,sttntt,

set- I,

1ø\X ,'Ø(i ..tn,(

1tr)\'\

^,â1X

,

^{ØØ o r,,L}

(tt))

bc.t1

lincu,t

_,,1Сi

_{iit ilset/.}

^'

Ihen,,

tlte

are'erly,i.ualent:'

a) T is

tr,rt,í.formlt1 con,tinu,ou,s..

b) ? fs

continu,ous and ,nt,onotoirc.

"i f

^,l,s contLnnrous

at 0

^ctnd, n¿otltotone.

.

.. ^r?roof

. ilhe implications a) > ìr) anil b) ='c)

^¿ùre

trivial ancl

the

iurplicrbion

_{c) =+}

a)

f'ollorvs fr,om' fremma

2.1.

_H!.

ing Coroliaty

2.2,

il G

conlains bhe

tucly of the

con-r,ergenoe

of nets

of

from

⁽⁶

in 6(,1,

(6oan,(IX))

is

equi- ce

of nets of unilolnrly

cclnl,inuous

Iìr

1,he_ secluol,

rve-shall

considel subcones

.d _{of G(x,}

.6Øotru(Il)) co-ntaining

the

single-valued

functions

_ancl monotone contiriuous lineai trpclator:s 1' : (€ -n

6(x, Glîo^u(E))

satisfying

the folloriing

concutiorÌs:

(2.2) for

cach ^cg

eG(X,Il),

^,I({91¡) is single-valuecl;

(2'3) _for

_each

f ^eG

^antl

^{n €X?}

r(l)

@)

-,.f,JrTQeÌ)

^(ø).

|"v ^(1.1)' the identity

gperator satisfies concritions

(2.2)

uotd (z.s).

In the

second

palt

_we shall see

that, in

1,he case

lt - R, moo"t"íe

continuous

linear

operators

ftom G(x, Rj in itself

generate

ii a

^nal,ur,al

lYay monotone continuo[s linear 'opô"*tors

fuon"G(X,

,ArAoorrlþ¡l in

it;se

,lefi i'ä.hï :,'å!i,iiîå"1i,îî"î1Jïxiîifå.i*

fol

lls¡rilrrroN

2.5.

Let 6

ba ct

sub

cctntøini,ng the

single-auluecl

Jt nd,l

ntonotone con_

Iinuotts

l,iu,eor

sati

lf II

^,is

ct, ,6, _t:kin

_set

_in,

_Ç

if _(€(X,

_,,fo, eaclr equicontinuous nct

(r,)f.¡

o.f nronotonelinear operators _J'r.om

6

^in,

6Gztzu(11))

suclt iltat ilte

n,et

(T,(lt))Í.t

"orrr"rgri to

T(h,)

for

^eaclt,

h

eÍIr"ue

aL.so ltatse.tltø.í the rtet

(T,(Jl,<.¡

co??,Dlt.!cs

to I([) _,for

eaen¡

f eß.

-. -rl ^{'r is tltc}

iden.ti'ty.^operurorr""a"ìt-T(orouhír, _set

ir"rh"*ni ir'{¿*piir

cal[.ed, u,

](ot'otl;ùt

sc[ ^'i,t¿ Ø.

As

observed

in _{l4l, if} H

conLajns

the

constant sei,-valrred funotions, we carn

omit the

ecluicontinuity of the nel,

(2,)å, of

continuous monol,one

linear

operators

in Definition'2.8.

We

^have

the

following

rnain

theorem,.

ruú ilcn

't'ot,,(D

'J'r;.1;:,f;:rr,l;r,i

ct,

,uì,ono

linctí,t oqrrr*iiì,

'tt,¡¡ .the,f

otknuing contl,í,t,iott

(2'4)

J',ot' eucrt .[

e'6,

no e

x

cøtcr

)/

^e

E,

th,crc erists rt, e

rr

sttcr¡, ,¿ut

/ <

^/¿

antl

rl,(tt) (øo)

.

,11(f)(no¡

¡

^¡r,

l,l¿ut,

I{

^,is a, ,!t_I(o¡,oul;in, set ,in, G.

IiLroo'f

'

^LeL

^(!t,)i'.¡

be an ecluicontin'ons

..ct

of

rnonot'.e

_{rjnea,r, ope'a-}

l?iä.,ï1î '!ìiïì',,iÌt¿,^K;:Ií?;

^ancr

^{s'ppose ilrar} nie ner (,r,,(tt))Íet

_{con_}

,'," ,ri:','iìf,:-i',iìit:i:T; lî*:'"

r;h¿t tr'¡e ncr,

(",(/))[¡

cor]\¡crs(,s

to

^,1,(f),

3Il, ::¿i,ir

_,sjuste ^v¿rtuect.

^r,et

^{V e}

E anct

consicler Ifl e

E s'ctr _tirat ,f

and cron

.(z +l)¡.

rL.v ^y:iií ir!åT\i,TÍff1"1,,Æ#fl

hood

N(r¡) ,Ãìji;;: irr(i.tt.t +

2t4, *.r,en_

.i.¡or

earit teI, I,is

monotone ancl

thercjlore I,(Í) _< ,j,,(h); on

1,he

<¡LltcÌ li¿nd' tlre

nct

(^r',(tt'))Íçr

"ooìru.g-à* ^,Lo .,r(tr,)¿o,r tLìï.orore 1,hcl0 cxisr,s o:(n) e

r

suctr

trrat, ior'òáäì',;;;ì'3'

_o(ro),

(2.b)

_?,(l¿)

_<

_T(tt)

+Iv,

"(/i) ç ^T,(h)1_

fir.

) _3tr1'.antl for. eaclt n e

N(ns), I,(fl

(n)

c

e. Ilet ZeE

ancl

consider WeE

Let

îcne Z : try (2.4), thercexilf,s

/r,,1{¡rrctr,tat {.ç

t_anrt T(tr)

(rn)

c.

'T(J)

^@o)

^1-

^Tiz';

moieoíã.,r,rrusriäir]i w¡ ^is

^cornpact andtherefo.c _{,here

erist

_!/t¡. . . ,y,, e

T(fl (n) such thaL

^1,

each

i!' ^1,'.

.'.,;,;|.1=,"ií,J,*

"i_' _:I..,;"?

i: !,.

^.

^.,n

^{antl filrûhei,}

,.I(f\(r^\c

vrrtue of thc continüíty"'of

Michele Catnpiti I Approxirnal ion of ctontitruon:;

REFI]I.ìENCIìS

St:t-y¿l¡p¡¡ lunctions I ¿,t

T({pr\),...,f({q"}), T(Í) tì,rd

af(/¿)

-\\¡e nra)¡

fiutl tttr open

neighbor'}roorLì rV(*,') oT *ro such l,hat,,fol e\¡ery ⁿ⁾

eN(nù,

(2.6) 2(l

)

(n) - ^I(,f)lr) +

^{2IV 1}

^rU)

@)

c

co

(Ir({erl'.

^. .,Í({ç,,1 _Xrx) ^J,-2['i¿.

At

1,his poinl,, lx'e obscìr've

that

_{cp¿} ^is single-r.alucti

fol

ea,ch ¿

:

1

t.

^{. . t tt} antt therefole

the net (f',([çù)),1,

conrrerges

lo

"([9¡]); moleovet',

sirlt:e h,

eII

^ther

^net

('1',(lt,))Íer cotl\¡et'ges

to

ft'; hence,

thele exists

^ø.(a-;o)

e1

.qtlclh

tlrat, for

^{t¡a,ch ¿}

eI, t)

ø"(ero)

ilntl j:7r...ttìt

(2.1) f,([ç,]¡ ç

?({<ai})

+ lt'', ^{?([q,]) <} fi,({qil)-f ll/'

?,(/,) <'I(h)+W) fØ) <

",(l¿)+

^l,l'.

Ìly

(2.6) an<ì (2.?) u'o ob1,airt, Tol eaoh ,' e

f _,

¿

2

^a(';c,),

r

e,\r(øo) a'rtil .i

:

L,...trL¡

?({q,}) (r) c: I},([pi])

^(¿r)

* ^fl'' c

'1',(.f) (n\

)- ll'

ilncl hcnce, sinco

?,(/)

^(e:)

is

cottvex,

Q.9)

^T U) ^(ar)

c ge(1.r(tp,ì,...,f'({ç,ì)

_(,r)

* ^2II; c

"'(.f) (r) -l 3'l'l'c:

'x,(h (r) ï ^lr

^;

on the other

^h¿r,ncl

(2.10) rl|)

_@) c:'t-"(h ) (ø)

-

^{'t'(h) (n)}

^{-l- Ìl¡ c}

^{'x(,1') @)}

+

3Ì4/ c. T(l')(')i)

+v

^-

Är'grring on

the

compastness of ,U a,s

ilr the first

case,

þ¡'

^{(2.9) a,nct}

(2.10)'n'e tleduce

tlre

exislence

of ø.e1

such i,hal,,

lot'each

_'. ^e

l,.t )

^u-,

ftl) < ^f,(J) + lt, I,(,Ð.<

t'("f)

-¡

^{T'- zlrtd}

^tìris

c-omp1-ei,es-the

plooi.

In tllb

palticular; case u.here

the tipelrtor' ?

^is

the

^identitJ¡ opet'¿tto,'r

n'e

obtain the

liollorving Corollar'¡'.

Oor¡oi,t,¡l¿l: 2.6. f'et

X

^be u' cornpnr:t ÍIctustl,or.f,f I'o1tolol¡ical s'pcr,ccr.Il-o

Frócltet sqtctce cund

G

a, stiltcott,e

of G(X,

(6Gottu(Il)) c:ottttrí,nin'gl'he si,n,gle- -ualu,erl _Ju,rt ctions.

If" II is

u ^subset

oîG

sutisJyitt'(t tl¿e Jollotui'ng cott'd'il,ion

Q.E) for

^{eaclr' J e} '6,

s;teX

a,t¡d

lr

^{e1{3-^,} tltere enists lt

eII

suclt thttt

I ^{< l,}

ctnd tt'(n)

c f(r) ¡

^V

_'

tl¿en,

If ^is

^u, K,oroul¡:in set in, ^((,1

.

m

lìnlu.r,r

2.?

Ilntler

tlre hypotheses of

theolem

2.4,

if the suÌ;conc'l

also coltl;ains

the

constant set-valuetl l'unctions, conclitjorl ^(2.4) ma,l' ^'l¡t¿

replacetl

by tÌre

^follorving

(2.L2) fol

each _.:[

e'€,

aoë,X.arLcl

I/

^e

^E,

there cxists ^/¿e

]f

such

that

I ^<

^h

f ^{-ll and 'I(tt) (nfi} - 'I(l)

^(#o)

+

^Y.

'l'he proof is similar to

insteacl of the monotonioity o is applied

to

an elernent Û e 6(ú't

nu(fr))

ancl

if / <

^g

+

^{L/, bhe}

lVlor,eover,, ii' rve consirler, 1,lre

11 r'oplacerl

by thc

following.

(3.1-3)

fol

cach

Í ^{e16, noeX-attl}

^V

/ ^<

^/¿

^{_f Z}

^{ancl tt(nn)} _{c. l@o)}

+

^V.

(p

_-L. e 'lB

:

ir,(ø), rvhile, for each 'llS"c/(øo) -F

(y(r)

-l-

") ^.

[B:

å(n,);

e ' Its,

thc ploof

isl

cornplete. g

i a I i nc I u _ç iott s, Glunrìlchlcn der l¡ a Lhcnì atisclÌcn for II,29__35.scl_ualuecl Ilausclorff conlinuotts furtcíions t lype approximdlion lltcorcü.r for sel_ucrluccl ftutc_

8t 9-.s2Íi.

a t u.l. ct p p r o.t i t nallon, p r.c¡r r.in t .I,ecìruischc I lrr chs.

- -s9.

n. t,Ia[lr., 6J (1956), 2, 961 _982.

ualued Iuttt:liotrs, J. Approx. ^.I,hcoly, ¿0 (1g?g)

Iìecoivcd 1.IX.19gO

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