View of A Korovkin-type theorem for generalizations of boolean sum operators and approximation by trigonometric pseudopolynomials

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Torrrc 17, Nn

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_1t¡t.7-17

A I(OROVI(IN- TYI IE

_{TTIDO R}

Elvl FOÌì GIINEIù\ LIZ;\îfONS OF BOOLEAN SUM OPERÄTOIiS ¡-ND r\I'PROXII'IÂTION

BY'I.R I

^G

ONOME

TR I

C

^P SEU DOPOI-,,YNOIVII,A ^{I,, S} . C. ]]ÅDIJA, I. B¡\D]J,À, C. COTTIN

(llucu¡c;ti) (Cloiova) (l)uisbulg)

.

Abstt'act.

We _Jlrove a tr(orovkin-typc l,hcolcm on

approxitnation

lia

generaliz,ations

of

Boolean suln opetators

in a

space

of

-B-continuous funcl,ions u'hich are perioclic

i4

a certain sense. ¿\s a,n application rve treal,

the probleln of uniforrnly approximating

/J-continuous functions

by tri-

., gonometric

pscutlopol¡rnomials.

'1.

Intx¡duction. In

rece,nt Jrears

there has

been

some intelest in the

approxirnation

of bivariate

functions,

by lloolean

sums

of

pararnetric extensions of

univariate

approxirnation ope,ra,tors, especiallyin connection

with

problems

in the

field ,of Cornpul,cr

Airled

Geolnetric

Design;

cf. ^e.g-

the

correspondirrg secl,ions

irr _[8]

ancl

t]re

refcrcnces ^ci1,ecl there, as rvell

as tlìo

discussion

in the rccent paper

_[11.

If ^thc untlerlying

univariato

opcrators

clefine

(algctrraic or trigonornetric) polynomial

approximants ühe corresponding

bivariate

approximants are algebraic

or

trigonornetric pseud,oltolynornials,, i.e. of the form

f ^{n,ø1. ni +} f ^{u,@). ,,,}

i:0 ^{, j=o}

(1.1)

or _E

2nt A,(y)

i:o r¿(n)

t

j:o

_E

²ⁿ

^ll,(r)'

"¡('y),

. i+1

SIIì

-'2

.).

cos

--z

₂

,[:

if i is

ockl

1(z) :

if i ^is

^evcn

where Ar, B¡ are

udivariate

coefficienl, functions

which

shoulcl he periodic

in the

l,rigonomelric case.

In

most investigations

the appro

^to

be continuous. Ilorvever,

thc consiclerecl

^en

meaningful

for a

biggeri class of

'functi _ti-

nuous functions introduccd by

I{:

Bögel

in l4l

^{(cli. also}

[5],

[6]).

I

C, BADEA, I. BADEA arìd :C. COIT.IÀI.

Ä real

'aluect,

function / ^{on ÐclRz,is}

^calrecl B-comtinuous,

if for

every (n, y) e

D

^.rve have

. li*

^{4,,,,.( fr,}

^lll :

O, (r,vl+ F,r'l

-

Í(u'

Y\

* r(u' ul'

on

in tre

space e 12

u'ith

certain gue

for the

approximation _of certain

(1.2) _L*,,Í(n t ^2n,y *

²ⁿ⁾

: _Á,,,1(r,

y)

for

each

(r,

_!t), (ø, o) e [pz.

such functions _n'e carl B -

2rperiodic,_

and we' denote by Br, the

space

of atl

real-valued

B -

zn-ptr;-io¿ic

^"4 ãìo"iirroo.,*

_funcl,ions

on

IR2.

Before

b

rve

give some B" in

Section 3

sectio' 4 as jn _{section 2' rn}

eremenis

or'.Þ {"i¿,å:iäåT,rt"*

eriodie

funetions. It is clear

thab sual pointrvise

definition of

acklition

, ^{Brn is not}

^closed

under function

Br_

-functions

^neecl

^neil,her be

.B-

be

seen

by

consicler:ing functions

arc

a,ln'a)'s

in

Br*. Suclr frurclions

constant"

pseudopolynomials

with

/ir_.

Turning to the

absolute value, (consider.

an

obvious

modification

(I2)-case

in _[1]), nor

B-2æ_per.io_

ith

(2'l) _l@, y): (n -

^{2tc, æ) sin}

^y l-

Ztc",tt, rvhere

Ic,,eZ

such

that relàloø,Z(lc, j-1)æ1.

_{We have}

.

^.

^ ^{Í(q,,!t) :,.fQ,.a)} : (n -

2/c,n)(sin

9 -

^{sin o)}

:

: (æj-2n-z(k,-lt)¡c)

_{(sin y}

-

^{sin ø)}

^:'i(ø í

^{zn, ù1}

: _¡e* |

^{2r,, u)}

rot valiable !2,,u),.VtÐ" it

^{follorvs 1p2..,} thab

^since / ^is 2r-periocfic rvith

r.cspecb bo brre seconcr

for ari (r,

y),

(*, r)2.;r:.n'

!t)

: L''' l(n i

^2n'

v t ²ⁿ⁾

Ilurtherrnore

' lù,,,,f(n,A)l: _ln -u,

^¡2(1c,,

-Ir,)r[ ^{.lsiny _}

^sinol__+

0 ^for

(u, a) ^---+

(t,

_{E) ;}

i.e. / ^is B-continuous.

(One

can

even checlc

thal, lfl ^is

B-continuous.) Thns

/

^{€ Bzn.}

^But

^{we have}

^for example

_i

ir(' ^{", ;)i -,n-r,o)r} :,' .

ancl.

lr(", ;)

^I

^-

^r-r(,,,

^o)r :',

I

2 \ KOROVKTN-'T¡¿PE TIIEORE.NI

from which it

follows

that

A,,o l-/l

( 'Ð

^7l

* ^L',,olll

^Ic)

!!

o

i.e. l/l ^{is not} Br^-periodic. I

The space Crn,rnof

allbivariate

continuous and rea,l-valued functions

which

are Zn-peiodic

with

respect

to both

variables is a proper sutrspace

o1

Brn as ca,n

be

shown again

via the

example of B-constants or

via

^the

preceecling example

; A lìr^-function in

general

neither

satisfies

the

con-

tinuity

nér

the

pèriocticity properties

of

Crn,rn-finctions.

It

^even nèed

not t¡e

bouncled.

The situation

is different for

the

associated difference function

A,,i

w_ith fixecl (u, a) e IR2. tr'or

I

^e

^Br,

^this

^function

^{is boundecl} and. 2æ=pgriodic

wìth

,respeót

to trotn

va¡'iables as

the

follorving

two

lemrnas

will

show,

f,nuru¡.

2.2.

Il I

^i,s

^ß -

2n-period,ic, then

(2,2)

^L u+2tnp+th¡l@

*

^Zmtrr31

t

^Znn)

:

Lu,u f ^{(n, A)}

tor

^eoery (n, y), (u, a)

e

^p.2, wkere ^lt,, h, rm, and,

n

are integers-

Proof. There

are

six

steps

in the

proof.

(a) We prove (2.2) when lt :

Ic

:

O,

ttt: ^{1r ând ø is a}

^positivo

integer. This

is

done

try

mathern-at'ical ind'uction.

rr ⁿ :

'' ""'o).]i.t";::i';aï 1il,;;,ïÏ" ^'oi''

for every

(nrU), (ura) _{e [p2-} Moreover,

:-

^'lve have

'

A,u,r+znnl(u

] ^{2n'E l2næ} |-2,t):

^Lu,v+z,,n

^l@ry | 2nn):¡,

^:

anct aclding these

two

equalities we find'

''

L,,,j(n i

^{2n, A}

* Qn *

²⁾ⁿ⁾

:

^Lu,u _1(n, U),

i.e.

the

desired result.

(b) In a similar manner we

can prover

using step (a)t t'l.at

(2'2) is

true ^for

^h

: k :

Ot and posil,ive integers zr, ^ø.

(c)

Now we

prove

(2.2) when ^l¿

:

lt,

:

At

flL:

0, and

n

is a positive integer.

t0 C. B.ADE^, T. BÁDEI\JAITCI C. 'COfTIN ^5: ,,;

^,I(OROVI<üNÊTIIpE,'IÈtÞOREIvtr t7i

4

' _,We:harne I ,; ::

^,

Lu,l(x,q *Znn): Lu.,l@,y\ I L,,,,f(x,y !2,nn).

But,

using ag&La (a), rve mâ,y \vrite

L,,ofin,E J_2nn): ^Lu,rl(n -

²ⁿ

{Zn,ll -f2nn): L,'l@ _2n,?Ð:0, (d) Similarly, rve can piove (2.2) for

l¿

: lç: 0,

-¿

: 0, aîd.

nx

a

^positirre integer.

Ffenge,

in

these

four

steps rve pr.ovetl (2.2)

for

h,

:

k

: 0,

antd m, n non-negative integers.

(c)

Now .*o provc

(2.2) rvrrcn. rt,,rr,,,z,m ùt'e n.n-neg*Live i'tr:gurs.

Indecrl

,

^using

thc abovc

sl,c¡ls, rvc

haíc

Lu ¡2tn,v1-2kn

l@ +

^2mr,

^y | ^2nnl :

Lut-znn'+zn,

f(n,

^y)

4

L,t,u

lþt f

^Zlm,:r

t

^zkæl

: L,' _l(u,

o)

: L,,,1(n,,1.

,.

(f

) the equality

(2.2)

is

arso

true if

somc

of the

intcgers rr, k, rn, n,

m

(e) because

the

last

îqualíties

are he periodicil,.y ploperû

y of

A,u.u,f

the

B(-r'¿)-ca,se we

formulate in

^{ich rvas given} irr f3' rremmal

ioí

,

Lnuua

_{?._8..^Il}

I

^{i.s an elemerú of ttte space I3rn,} then there

is u

^positioe

nunzber

M : ll[(l)'suct¡

th,at

for

^euery

^(x,y')

^and,

^(i,'t) ltoi,i n,

^we

^tíaae-

^.

IL,.'l@,a)l <

^M.

Proof.

Any

real number

u

^{cùn be}

written in

the

form

,tt

:,ttrt !

^2æu2

where urel0,2æ[,

and

u, is an integcr(tnu _iotugral

_parb

_of aì

\"2")

this

decomposition

lor

^m, y,

s,

and.

t,

^and. I¡emma 2.2 we obtain

{Ising

(2.3)

L,.r

l(r,

^U)

:

L",+"nsz,t.Å-2ñ,Ãl(ür

| ^2na, _h *

^2nUz)

:

A,,,r, l@t,

llrl.

Norv, Lemma 2.3 follows

from

(2.8) and

flom _lB, Lemal; I

..4. Eagþ

I

=

BZ

may be

wrítten

d,s (r sunn

of a

bound,ed

rJ-co ion

whicrt,

is

Zn-períod,ic witt¿ respect

to

eäcr¡ uariable

anil

^r

Proof. T[e may write :

^,

l(n, y) : L

_o,yo,l(üt

y) *

f(n9t a)

,l ^l@t

^t/s)

-

^{l@r, !to)}

with

fixecL (xo,yol

e

^p.r.

,; i

A.ccording.to Lernrna

2.2

artd,Lemma

,2.3;,the lunction A"^" f

^is

2æ.periodic r,viúl,resþect to each r.ariable ancl boundecl. 4..,r,

/

^{is also} -B"-dónü-

4uous sinee 4,,,,;(A ,;,n"

l)@; y) :

L,,,,

l(û, lJ)..|

ⁱ

flema,rk. It is not true that I ^e

^{Brn can}

^bc rvr{tten as a

sum of'

a' crn.rn-fttnction anrl a B-constant. _compare the example of a B-continuous

function l'bich

is no1, continuous

uþ

Lo a -B-eonstant

in i6]. (

^.

The preceecling consiclerations enable us

to

show

-

^Lnnrm

t' 2,5.

Bs,clt

' .f ^e-

Ilv^ is

uniforntl,gl ' B-continuous o,n, .

Rr;

^i.ê.

for_each

e )0

^the,e

i,s ø

ò(e)

tO

su,c'h',tltat ,for: eaery

(nrù,(ura)'eTlz with _lu - ^øl (

3(e) anil,

IlJ , al

<. à(e) ^zoe,h,aue

(2.4) _l\*,f(u,

^y)

_{I <}

Proof.

Because

/ ^is

B-continuous

on

IRz

ii '. follows

Lhat

!

is'also B-conti- nuous

on

_10, 3nl2

:

_10, Sztl X [0, 3æ], ancl then by, [6, ^Satz

l]

the function

/ ^{is uniformly} 3-continuous on

_10,3æ12.

Thus

1,here e-ris1s

a

function

àrf "")

ì ^{0, r} ì

0, such thaL (2.+) is satisfied

for

every (u, y)s(u,

a) e l},Bnlz

witlr

_{vy'e ln}

- norv define ^{ul <}

^àr(e)

^and the function

^137-ol

^<

^òr('e).ò(e)

frorn the definition of

uniform

-B-continuity of .f ^on R' _{bV ì(e)} : min{n,8r(e)}, and

sho.w

that

^(2.4)

h_olds for-every

(t,y),(u, o)e,tR, rvith lø'?¿l <

^{ò(e) and} ¡ly

- ^a¡ ç

^Aç"¡.

Because

ò(.) <

^{æ, \rye can choose}

h,leV

such

that

n:ntlZknrøre[0,3æ]; i : . :

i

lL

:

n4

|

^{2kn, øt1q}

[0,32];

,

U

:

_!/t

|

^{Zln, y, e 10,}

3rl;

'u:Dtl2tnrr:rel0,3rl. i:':

^:

Then

l*r- url:

^ln

- ^{ul <} ò(") ('ò1(e) and

_l,y,

- ^orl : _ly - ^ol

^<

< ^ò(.) (

à'(e).

Because

(uu!lt),(uu,ur)e

[0,3æ]2 anrt / ^{is uniformly} B-continuous

on lOr Sn)z we

obtain

lLu,,u,l(nr., gr)

I <

^e. Using this inecluality and,Ilemma 2.2

we

get

lL,,, l@r

Y)i : lL,,',,,t(ruyr)l ( t Now the proof is

complete.

I

3. A

Korovkin-type theoreinrlor ãpproximation'in B,n

rvith

_$dnerali-' zations

of

Boolean

sum

operators. TTaving suppliecl

the

necessaiy

auxi liary

results

ig

secti'on 2 we

may

no'rv prõceeâ*similarly _as

in ttä pf*t,

of the Korovkin-t¡le

theorem

in _l2l to

obtain

our main

assertion.

.

^We

^first

^prove

^the

^following

Lprvirvra

3.1.

^L9t _.d ^e

.Brr, y

chosery.

Ior'

_{, eaerl¡} positiae ,

number ^{e l,here} are toòo

pbsitiad -:

A(",rfl

ctnd,

B(e)"i Éþ,Í)t

such that Jor ^en:erE @, E)r.(s,1,) e

l;,,,l@,y)l < å*,4(e)*i** ^B(") *?-

⁾

A KOROVÍ{IN-TYI'A'TI-IEORI'M ¹³

t2 C. BÁ'DE.A, l. B.4,DEA aùct C. COT.ITN 6

P.roof:

r¡et

e be-a gtven positive _real number. rJecause

/ ^{is uniformly}

B-continuous on.[pz

by

Lemma 2.5

there

exists

a ù(e)e]0irl such thai for ever¡' (nu!lt),(nz,yr)epr lvith _lø, 'nzl < à(.) anù

_ly:,

- ^lal (

ù1e¡

rve have

(3.1)

lÂ",,r,

l@r,&.r)l < :.

Given (n, y)r@, f) e _fR2 we choose /c, I e

:

(n

+

^2krç¡

!! !

Zttt)there

holds z

'o"ri ^{that for kí}

'

!J') :

lu'-sl ( r, Iy'-tl (

^æ.

IMe disúinguish

the

follon'ing

four

situations

(i) _ln' - ^{sf < 3("), Iy'} - ^{tl <}

^S(");

(ii) I*'-!l ^>8(.), Iy'-tl ^<

^8(u);

(i,ii) ln'- ^sf ç ^8("), ly'-

^f

| >

^à(e);

(ia) _ln: -sl >à("), ly'-fl ^>à(e).

ln

case (i), using (3.1), we have

(3.2) l\,,,l(r',y')l ^< +.

₃

ln

case

(ii),

_there holcts

. 8(e) . u'-s

Sln-'-(s1¡-'

So

we

havo

sin2 (3.3) 2

à(")

o

)-l

sin2

3.6)

(Consecprentl¡',employing(3'2),(3'4)'(3'5)'ancL(3'6)wehavethefollowing inequality

arrd

(3.7)

M

_sln¿

. ^fi' -s ---

^stn'"

2

, .fr' -

^g

srn4

--

_o ^.

lL,,,l(r',Y')l < _.,T(4

sina 2

I A",,

"f(r',

7')l ^< å ^{* {} t

., 4'-,ç

\'sinz-+

)' + ltt

-.. ^"¡1u

'*ttt"

sln"

---

o₄

Using

I-,emma

2.3 and

^(3.3)

we fincl that

(3.4) lLo,,l(r',8') I < ^{tu-- . *io,} '' _-- ^t.

g¡12

o(e/

₂

2

In the

oase

(iü)

_ancl

(iv)

we

obtain in a similar

manner

(3.5)

_I L,.,

:-, l(r',y')

I

< {,=, *io,of) ^.

^{sin U' 2}

^{l- t}

^,

In

consideration

of

l,,emma 2,2

alld of tl]e

pcrr.iodicit¡1

propertics ot !h.

sine functior.

*," r'.ryãoù*ìiì"tu ^n'hy

^{n and}

y'by

^{y in (3.7) which cornpletes}

the proof. I

Now

^{rv-e are readY}

to

^show

TlruoRnm 3.2.

Let (L,,.,),Ïn,ne[N,

^bÚ

^ã'

^{seqy'ence o'f} positiue.linear onø,ato,s transt'orm,tli" 6'rlrl,i:aák 'Br*-J|,nctions totiich' u're 2ß-pati'oili'c ^witl¿

i.iieïål'ti

øoln"i'*¿outt't ^into

functiois

^ó¡

pn'

and' satisJvins

(i\ L,,

o(e;

n,A) : L,

tulrcre ^e(s,

t) :

1' For ,f

e

Brn 'untl

(r,

_U) e Q2 ^let

U'n'nf(n,y): : L","(l(''U) -l l(n'*) - l(''*); ^n'

^ll)'

If

^th,e cond'it'ions

(ii) L,,,n(¡ti

nt U)

:

sin

n I

u,,,,"(n, _Y), (í,i,i) L,^,,(qzi

n,!Ð:

^sin ?l

*

u,n,n(ü,A),

(ia)

L,,,,(þt

i

^{n, U)}

:

cos

n I

^{t^,n(u, Y),}

('r,)

Lo,,,(Qzi

ûtU):

^cos

ll t

w,,,u(n,Y),

:

sin ^Ú, _Ùr(s, ^Ú)

: :

cos

s, _tr(s' f): :

^cos

^t' t,,

ott'Derga' t'i-øero^ uní'Jormly om fR2 üs-m'

?

ta, ^tùenior

^euery

^f

^e

B*

ttt'e seçIuetuca

(A"'"fl

rln

Proof. ^I-,et'

f ^eBrn arLd(n,Y)ep2'

^Because

of

conclition

(i) we

mey

write

(3's)

^U*'n

l@' y) : _l@'

_U)

- L^'"(L''u f

; n' _Y)'

Bvtheresultsotsection2(cf.especiallyoorollary2,4ald,theproof

;'t'"tö ;;ä tú;tt; ^{; t-}

^a ùer-tteÎinecl linear operator

on B'n'

1,4 C. E4DP4,,1.. R,q:rìEA and C.. COT,'TIN

B

Furthermore,

taking into

account

the positivitS,sl

I,,,,,,, Lernrna'S.|, implies

: r,

^,

ll@,y) t U,,.uÍ(n,y)I :

max

{L^,,(L,t,rl)o,!l)tL,,,u(- L,,rif;n,y)} |

^t,

t. -

4 L,,.uf'å+.4(e) '\¡ si¡23--'* z B(r)*i,,, ll* z _{;,r,y\ )}

för' 'éach 'pcisitive

,r.,*[.r' ,.;

^:;

i i' ,

ⁱ

t

^{Í '}

. . _^

carr,ying'

thro*gh

sgae operations ancr a,ppryin_g the

relatio's _(i)

_tol (v)

from thc

statement o.f

thç

thcorernrlvc

â,rlivä*i ttiu

estimate

ì i ll@ryi"-:

[J,,.t,1@,y)l

e-l':

* , * i"t¿ ^{'{2-c,os r} 'L,,,,,,(úti

n,!J,,),- sin

u.L,,,n(pr) 4!Ð _

/r.

Approxim¿rtion

with trigolometrie

pscudopol.,r,nom-ials.

\\rhile it

^is

elear

bhat-^f.unctions

in

C2-.2, óan

be appiorimated arbitralil¡' l'ell by

funcl,io¡s of the

form

(1.1)

iri

ttre uniform

noÌln,

this facüis

not

so ,obvious

for the largel

sp&ce ^B2E.

'

IIog,ever,

it

can be oJ¡tained aS,a conscqrrence oli 'I'lrcorem 3:2. ll'he id.ea

is to takó the

^{,Boolean sutn oT} par.ameti'iclextensions

of univariate

pOlynoinial

approrirnation

operatot's as pseudopolSrlgmial;approximation opeì'ators.

lhis:hinal

of'procedure ^\\'ârs also used

in _[2]

to

llrove

^ân _-appro-

xirnation

theorern

for'

3-conti¡r¡crus

functions by

aìgpbraic pseudopoly- nomials.

While thc

unclerlying approxirnation probleirn' rvas

not netv in the

algebraic case

(cf. the

refel'ences

in _i2]), to oul'

klorvledge \\¡e al'e

the firút to consitlu the

corrcspnntling trigonometric problem

In the trigonometric

case

the

^te

polynornial appräximation

op.erators

^o-

lean

suur u'ould.

only.be

ctetinecl

on

^so

use suitable discretelv tlefineil

opelat

^a,-

luecl furrctions.

Before qqngider'ing,any special sittrations. rve looh

lat

^{l,'lvo a,tbitrary}

cliscretely definecl ancl ìonstant-r'eprod'ucing positiyeiline¿ìt operators

z-

atnd

Ln, i.e.

operators gir.en b"v

"' e

konovr<rNlfYFu tnaöhtn¡'

' _1tl

L^(l; d -rE l@')'Pu,¡ (r),

i:o

r,ø

;

ù : f

_l:0

^s(vil'

^{4,,,, (y),'}

, |

^..

n)

: _D

q,,¡

(y): _!,for: ^all

^{m, ø}

^{and all r,}

^{y e}

[p,

^and

j:o

' ì :r"

p,,t(n) ) ^{0 lor 0} ç

^rl

ç m

and

all

æe [P,

l

q",i@)

) ^{0 for 0 (,J} ( n and'all, 7e[P,

$ ^{1 l'r}

:l

where c(e)

:

rnax{.4(e), 13(e)}.

Letting nt

and. r¿ tend 1,o

infinil,y

yietdsrthe _desired

result. I '

_{Reniarks 3.3}

(c) If equalit¡' (i) of

t,he

lhcor is not true. ff

one replaces

(i) by

(r

then the above rnethocl of

_lproq

we have

pointwise convcrgence to converges

unifolmly to

zero à,s.tm) n

ing the main result in

_[2]).

(d) Note

t,hat

rvith the

damc.argunrcnt _as

in the prôof of

theorem [1],and

[2] it

woutd be sufficient B-conlinuous l'uncbions.

:

^ì

all

-B-continuous funciions, oundedncss,

is

somewhat,

dircclly

be àpplietL

to

such, operators

sincc a

-B-conti_

that it

^is,

operatorg -functions ^j

es to

the,

i,--Now the tensor

ploduct L,,,,: Lh'l!,,

^{acts on'thg}

^bivariate

funcl,ion

l(n,y) is it

_E

is a

flxect

par ₌

^{Lu acts on}

l@rU)

as

if r

^iö

^{itilea, is a positive} linear

ep_rodì+cing

-o-perator definecl

for

any

Îunctioi /: ^fR2*

^tp. The_operalor

a*.n

de^fined in _J!e^o1ep 3.2

in this

casö equals thó Boolean sum

L'ot :

Lk

a

^{Íi'@T' R'gmarh 3'3(b))'}

and

the

conclitiois

(ii)-(v) iü

Theorem 3'.2 lrray be replaced

lry the

con- ditions

(iit) L,,(9iu):sinrf u,*(u), :; ì ''i itri¡ ^{z,,,ir'i !t;} :

^sin

_s I

^n:,,(s)¡,

,(ia') L^(þin): cosrf

^tr-(a),

(o')

^T'"(Q

;9) :

cos E

+

^u"(U),

rvhero

q(z): .- _sina,

_{,'(a):

:

^cos ø-, ?î!'ttr^,tt-tunt'tlJn ^converge

to

zero

uni- formly

^'on

n

^as

m

and

n

approach

infinity'

(4.1)

rq.here _i:()

l,

^{p,"l (}

16 C. q-ADEA, l. IIADEA ancl C. COT,TIN

1

The

preceecling cliscussion yields

(L,,),,eryt

(I,),.n

ng¡Lrg+g¡

,,t C01?S¿,ì,U(jl,ed _O?L Derges teniforntly

To obtain the result

tì,nnoutìceal ,at

thc

beginning

of this

scction ..re ca,n nolv use

for

ins[ance ¿ù s(ì(lrrercc

of

opcr,atoi,i

tir,ioi-tl,äi",ä"'"" ""

I(,(J

_;

g - -J,

^1ril=I-2

"*ij'.f(,^,,,).

k:l ^en(tr.,

-

ⁿ⁾

'where

1¡,,,

2hn

N, + ,' lü: L,2, ..., itn * ^{2, àntl}

^<Þ,,

is a

notnogatirrc cosine

polyrronrirl of

^1,lre form

11 .q' KOROVKIN-TYPE TIIEOREM I7

rrou'everr

in vie¡r' of corollary

^2.+

it actually

suffices

to

consider the approximation problem

for

l:ounded Br.-functions

:

trI¡e

first

appro- ximate

the

bounded

part

tant

^ud pseudopolynomial ancl then

adcl to

it the

^/J-cons obviously a

foo.

ff

n'e call the

sun

of and a B-constant

a

lj-IIarch,aud, pseudoptohlnomia'L

we

have

2 - ^c.2789

Conou,'\ny

4.3_. Each,

I ^e-Br,

^{ma,¡¡ be} uni,Jorm,lS¡ cr,lrytroním,ated, bE u seEuence

of

B- Mcn'chaud, pseucloltolynontials.

I

Note that

as

an

extension

of the

resrùts

in

_12, section

4l a

similar assertion can also ì:e .¡rrovetl

in t]re

algebraic case.

[1].Baclca, ^C., Barìca, I., Cottìn,C., Gonska, FI.FI., No/cs on llte degree of approtimaliotr of B-conlitrttorts ctttrl B-differcnliable funcliorts.'[o appcar in: j. Ãpprox.

1'heory Appl.

[2].Barlea, c., Badca, L, (ìonska, ìI. kr.,Aleslfttncliottlheoremancltrpprorimation by pseudopolynontials. Ilull. Austral. Math. Soc., 3¿ (1986), 53-64.

[3]. ^{B a <l c} a I. øle În sens Bögel çi ^unelc aplicr(ii ln aproæimarca prinlr-un opercdor Be Babeç-Bol¡'ai, Ser. luâth.-l,Iech. tb (1978), 6$-?3.

[4]. ^{B ö g-} el, I( I)ifferenlialion oon ltunltlionen meh¡eter ierüntlerlicher.

J. Reine Angeu'. Ì\,Iath., 170 (7534), 797-277.

[5]. ^{B ö g c} l, K., Ùber tlie ntehrdimettsionale DifferenlÌaliott, Inlcl,rutlion untl beschriÌnhle Yarialion. J. lìeinc r\ngevr'. 5), 5-29.

[6]. ^{B ö g} el, I{.., Über ^tli.e ¡neh Tfferentiation. Jber. DMV 65 (1962), 45_71.

[7]. Boj anic, R., shisha, ion of continuous, periodic funciions'bg ^discrete

--- ^posiLiue linear operulors. J. f f ^(1924), 2gt-23b.

[8]. G ^o n s k a, ÉL H., Quanlttcrtiue Approrimalion ìn C(*). Habititationsschrift, Urìiversit¿¡t Duisburg 1985.

[9]. Ichim, I., La co¡tslru<:lion cl'utt opéralet,.r linéairc posilif. Re'r,. Roumaine Math. pures Appt. 30 (1985), 659-665.

tlOl.Kis, O', Szá¡a¿os, J,, ^{On some} tleIaYaltée _Po¿¿ssin \ype iliscrelelinear op"erators.

Àcta rllath. Flung., à7 (l-2) (1986), 239-260.

[11].Ma1chaud,A., Diffërencesetdétiuéesd'unefonclionr]ed.euæuariables.C.R.Acad.Sci.

r7B (7924), 746?-t47O.

[12]. Marchaud, A', Sr¿¡'l¿s dériuées el sur les rlifférences des fonctions cle uaüablcs réelles.

J. Math. Purcs etr\ppl. G (1927),337-42a.

[13]']'Iorozov, E. ^N.,. Conuergence of a set1ttence of positiue linear operalors inlhe space of conlinuous 2n-periotlic futrclions of lao aarktbles (Russian), Kalinin Gos. ped. Inst. Uëen.

zap. 26 (7s58), 12s-142.

[14]. P ^o p o v i c itt, T', Surl¿s solalions bornéesetles solulions mesural¡Ies d.ecertaines equatíons fonctionelles. Mathematica (Cluj) f4 (1998), 4Z-106.

Received 28.XIL1987

7^ 2 s Uniuersilg U-níuersitg Uniuersitg of of of Bucharest,Fctcultç Duisburg, Cruioua, Dept. Dept. , 70109 Bucharesl, 1700 4700 Duísbuig Cruioùa, ROMANIAROMANIA1, FRG

5.

Concludinq

lìclnrìrl{. -{ ^natural

continuation of

the

considera,tions

of this

papeT consists

of quantitat ve

versions

of rheorem 8.2,

similar

tg-the

c¡uan_tjtative version

of its

algebraic analogon

in [1].

statemenús of

this type'lvill

be contained

in the

doctoral dissertàtion oT Che thircl author.

'Ihe

^authors

^aÌe

^.i¡ery

^grateful to Prof. Dr. H. Ff.

Gonska

for

his

kinrl

help during the preparation

of this

article.

REFl]RENCES

10

@"(n)

Nn

I

^{p",, ' cos v#,}

^lirn

^pr.,,

:

1

V:l ,L+æ

E-vampres

of

such operators cân be

fo.nd

e.g.

in

_[?]. As other

rathel new

references

for disclete lincar polynorniai nirp"oiiirrrtion op"iuiãis

LItaL

K,,

lepr.ocluces cotìstants, anct

that

z¿ tenrls to

infinitv

for,

all/in

túe space.;f

iodic functions, i.e. espäciaily ið;"-ih;

functions

sin

#,

cos ^ßr.

Now

consicler

the

Boorean

s'm operatoÌ'wnt*

crefinect

by

W,,,u

f(r, ^y) :

(14;, @ ^Itfl)

f(a,

^y)

4 ,Vil)-2 Nn+2

: (il,

-l-

2)(rl,

r_

1 't er ^h\:1 .I.

^hz:l

^.1,.

ll@,tr",u)

I

l(tr,.n,, !t)

-

l(th,,n,,tr,,u)l

. 4),o(t,,,,,u

-

^{#) .}

^e,(t¡,,, -

^U).

The clefinition of J(,,and. the results citecl above shorv

that w,o*Í is atri-

gonometric pseudopolynomial â'ncl

tliat _T{.,,

_satisfies

_the

assuiùbtions of Theorem 3.2. Thus

we

have

Conor,r,

^{th,en (W,o',}

_J)

conaerg.e^s

uniformlg to f

^on

e2,

^i.e. Jor.

e

^{sts ø sequence}

of

uniformtE" øppr:o*t*hh',g

rígonomelric I

\Me are now interestecl

in thc

questi

atlditional information on the

coeffi pseudopolynomials

in

Corollary 4.2.

I

continuous coetficie,nt fttnctions

if t

nuous. Moieover,

if /

is bounded, also ness

of

a pseud.opolynomial is equival cient functions.

For

the trigonometric as ^'was done

for the

algebraic ^case

ip

Bounded

pseudopolynomials Ttolynomials a,fter.

A.

Marchaud. ^rvho

Since

the limit of

a

uniforml¡.

conver

is still

bound.ed_

it is

clear

tìr4t a

^gen

by (trigonometric) Marchautl

pscutl

:tl