View of On some properties of Stancu operator

REWE Ð'ANALYSE NUMÉRIQUE ET DE THÉORIE DE L'APPROXIMATION Tome XXVIÍ, No

l,

^{199g, pp.} 99_106

ON SOME PROPERTIES OF STANCU _OPERATOR

ZOLTÁN FINTA

I.INTRODUCTION The positive linear polynomial operator defined by

(l) P:

(.f

,")= É f ftln)wn,t(x,a), neN, xe

_[0,1],

a>0,

k=0

where/is a real function on [0,

l]

^and

k-l n-k-l

w,,t(x,*)=f,1 _{\k) (l+cr)} ,'! ^t'..*' (t+2o")...(t+ ^,--4 lt ,"-'^' (n_t)a)'

was introduced by stancu [6], who studied, ^among other properties, the conver- gence

of

Pf

f ^{tofas n-)}

^{@ and 0 <}

^s

= c¿(z) _+ 0.

In the case cx, =

0,

pro is the Bemstein operator

B,

^given by

(2) B,(î ,')

=

É ^r&

^t

^{n)(:).r (r- *),-o}

^,

where

f ^eCl\,ll.

^Lorentz

[4,p.|}2]proved

^rhat for

f ^eCl},ll:

(3)

_lB,(-f

,x)--f(x)l=M,'0

_n

^*)

^ifandonty

if a2çf ,h)=o(hr).

Here r¡ ²

(f

,

t)

is the classical modulus of smoothness defined by

(4)

úJ'

(f

_{,r) =}

o:ï!,

lllr'r.f @)llrp,,t ^, where

nr.f (*) -{r ^rr

⁺

^h)-2f

^(x)+

^f

^{(x -}

^h), if

^{x+ h}

^e[o,r];

¡. 0,

^otherwise.

AMS Subject Classification: 41436.

100 Zoltán Finta

In

this ^paper, we pfovide further approximation properties

for P,". Let

^rp denote

the following function:

^q(.¡;)=

{rO-ù,xe[0,1]' ^For a

^function

ge c[0,1] we

denote

the

uniform norfn

on a

subinterval

[a,blc[o,\]

^by

llgll,,t",ol ^{= sup {} lg(¡) l: x efa, b)]'

2. MAIN RESULTS

3. LEMMAS

For proving Theorem

I

we need the following ^lemmas:

Lsvnr,le

|

(the localization theorem).

If f ^ec[0,1]

^vanishes

^{on a}

^sub-

interval

la,blç10,1)

^and

a(n)=o(n't),

^then

(7)

^P:

^{(f,x)= o(n-t),}

^x

^e(a,b)'

Proof'

suppose that

f

has second derivative

f ^{,,(x) for}

some x

e[0,l]

Then, by lheorem _{7.1 [6,}

p. ll92]

and a(n) ₌ o(n_t), _{we have}

(B)

]Y*"tP;

^{(-f , x)}

^-

^{.f (x)1 =}

{l jr"f ^- _'t

^"(x),

From Lemma 5.2

[1,p.

^ßa]we ger (7), Lnn¿¡¿e 2.

If f

^{e C[0,}

^l],

^u(n) =

o(n-t)

_and

tï* {r[p,"

^(.f , x)

- f ^(x)]]

^>

^0,

x e(a,å)

c

_[0,

_l],

The theorems in ^{question can be stated as} follows:

Tneonpu

I'

Let u(n) = o(n-t _),

a(n)'

n 3 r,

for

^{n =}

l'2' "'

^and

f

^e

^c[0'

^1]'

Then

for

^{each M > 0,}

(s)

lP:

(f ,x)-,f(x)l< ,ÉP,x

^e

[0, rf, n=t,2'.'.

holds exactly when a2

(Í

_{,h) <} lvIh2 , h> 0'

Trmon¡v

^2. Let O<o(n)

-+0 (n->ø)

and

.f .40,1)

with oz("f 'h)<l'¡hþ,

h>0,and0<P<2,Then

lp.

(f ,x)-,r(x)l

₌

r(Él-)u'' ,

^{x e}

[0,1], n=1,2,'..

T¡ggREM 3.

Let

^cr =cr(n) ^{= O(n-1 ), o?., =}

(l+nu)l(n(l+o)), n=1,2,,,.

and

f

^{e C[0,}

^ll.

^{Then there exists M > 0 such that}

(6)

llP:

f - f

llrro,r¡l M'arfl

(f ,a,)r¡o,,¡,

where cll{.f

,/)cro,rt

=otïp=,llI'rr<,tÍ(*)ll.¡0,'¡

to the Ditzian-Totik modulus

of

smoothness

l3f.

Tireonnu 4, Lei u=u(n) =o(nt), ^and 0<þ<2' i¡ f ^eCi1,il

^anci

ai _U, ^h)

cp.tt = o(hþ

),

^{then ll P:}

f ^-,f

ll"ro,, r =

o(n-þ''

_)'

In order to prove theorems, we need some lemmas

first'

ⁱ

2 3

On Some Properties

10r

thenf _{is convex} on[0,11.

Proof'

_{suppose that}

f ^is

^not convex._Then, using the parabora _technique

[1,p.

^124], there exist xo

e(a,ó),ô>0

_and

e@)="ri +d**r,".0

_{such that}

"f(xò=e(xo)

^and

f(xo+t)<eeo+/) _{for allt,}

ltl<ô. wecanexrend ftoa

fturction _{x e}

[0,l]. F eC[0,1)

^{By Lemma} 4.1.16,p,

_so that F(x)=f ilg4l (x),

x e [xo _ô,xo

+ð], F(x)<ee),

(e)

so we obtain

pno

(u', r)

=

*

Prt

(r,

x) _{= x}

fx(l-x) I

L , ^+x{x+c)j

Pn" (1, x) = ^1,

and

n[Pf

(F,

x)

^-

F(xo)]t

n[p," (e, _xo) ^_

eeù]

But Lemma

I

shows that

=c.xo(t-,ïo)

11'?g<0

n[p,"

U, x)

^_ _-f

(xù]

< _c. xo(l ^_ xo

).

^{t,* no +}

o(l),

which contradicts _the assumption

(9),

^{I + ct}

In order to prove Theorem

2le

need some other lemmas:

Lewre

3. For

.f

^{eCIO,1] we} have _llp,ï

f

llrto,,l s l]./ll.ro,,i ^.

Proof. Since

.i ^wn,t(x,u)

⁼

^I

^{[6, p. l lg4], we have}

k =0

I P," (.f

,")

I

=

å

^{w n. t ( x, a) I "f Ø t n) I <}

^ll.f

^{llqo,,t .}

5 On Solne Properties

103

v/here G(v) ₌ ^{(p 2} (y)

f

^,,

^(v).

^{By Lemma 5} we obrain

(16) _{llpi Gr(f}

,.,x),x)llc¡et,.t_Atnt<

,r(F#,ìll.,^,.,.^,,

llM(G,x)llcro, _rl

and (17) Since

we have (18)

ll M (G,x) _{llcro,,t slle'}

Í,,

llrrc,,t.

P," ((u

- ,)' ^.g-, _e),

x)

= _n(l+o)'

^{1: no}

ll

p:

_{( R, (}

f

^,.,

^x),

^xllc¡ e r u. t _,r r,1 s

ffi ^{,r, f}

^{,, llr¡0,}

^t:,

using (16) and (17). By Taylor's formula:

u

I

^{@) =}

f

^{(x) +}

î,(x) ^{(u_,) +}

_J @_ ^v)

f

^{,,(v) dv.}

Then

P,ï

(f

^(.), x) ₌

f

^{(x) +}

f

^,(x)

^p:

^{((. _ x), x) + p,"} (Rr(-f _,., x), x).

Hence, by (18), we obtain (15).

4. PROOFS

ProofofTheorem

I.If

lP:

U,x)-.f(x)l< M.'P'..(*),

_{x e[0,}

t], n=r,2,3,.,.

n

is satisfìed, _rye pur g(u) ₌

f

^{(u) + M}

^tf

^/

²

^{and obtain}

' n[P,"(g,'x)-g(x)lr r'x(l-x)

⁽ⁿ

-l)htn

2 l+a

Then, by Lemma 2, g is convex on [0,

l]

and, therefore, g(x + h) _2g(x) ₊ g(x ^_ h)>0.

Hence

f (x+h)-2f (x)+/(x-h)>-Mh2.

Analogously,for

g(u)=_f (u)+

Mu212

Zoltan Finta ⁴

102

(14) So

llP:

f

^{ll.¡0,,1 < ll"/ ll.ro;'r'}

Lnwa

^4.

^For

^{some constant}

^{C>0 andall} f aCl},ll'

(10) lP: (f _,x)- lQ)l<Ca'(f ,qç¡tJi),

^{x e[0,1]'}

Proof.LeT.

f

eC2¡0,1.1. By Taylor's formula:

(1,) _t(+)

⁼

_t G).(+- ò,' ^{ur.(:- ò'lir}

^"

^u,. r(+-i]'

where

s0)

⁼

s,o)-) ⁰

^{as y}

)

^{0.This shows that ths last term in}

^{(l l)}

^{does not}

exceed ll,f " ll.ro,

¡(k

^{I n}

^-

^{Ð2 12. Then}

(r2) _{llp. Í-}

lll"¡0,,1<ll./"ll.ro,,r

4*

#, ¡

^ec2¡0,t1.

Using Lemma ^3, we obtain the boundedness of the operator

Pf

^{and, by (12) and}

Theorem 5.3 _12,p. 2181, we obtain (10)'

Remark.

For

c¿=0 the inequality (10) ^reduces to the known inequality

of

Popoviciu [5] conesponding to ^the Bernstein polynomial ^(2):

(13) lB,(f ,x)- f (x)l<Ca'(f _,q(r) l^ñ),

^{x e} [0,1]' Theorem 4 requires the following ^lemmas

LEÀ/ß44 5.

For

Rr(f _{,u, x)}

= ! ^f"-v)

^{,f " (v) dv we have}

,T

lu-

xl

lRr(f

,u,

x)l< +

Q- (x) 92(v) ^|

f "(v)ldv

J

for

^{x, zr e} [0, ^1].

Proof. Lemma 5 is a particular case of Lemma

9'6'I

13, p. 1aOl'

LEMMA 6.

For

A >

0

a fixed ^{number and}

f

^a dffirentiable function on 10,

l)

with

f ^'

^locally absolutely continuous in f0,

ll

^{we have}

(ls)

llP.

l -îllrrn,,.,-n,,t ffi

^ff

^s'"f

"11"10.,7'

Proof. We consider the maximal function ^of Hardy-Littlewood

M(G,x)=supl lr'l t'

IC1u;Arl,

u lx-u "

^I

(20)

llo's','llr¡o.t¡ 32n

Kz,eç

,n-1)cto.tt.

By Kz,r(f

't')cro.¡1 ^we denote the K-ñrnctionar of the pair of spaces

c[0,1]

_and

a coresponding _weighted Sobolev space with weight frrnction g2 given _by

Kr.r(-f

_,t2)qo,,r =ir"tr

{llf

-gllc¡0,r1

+tt _llq,

g,,llc1o,r1

ig,e

A.c.,o"}, where

g'

e A.C.,o" means that g is differentiabl

e

and

g,

is absolutely continuous m every

Í

⁼ _"f

- g,

_closed

*g,,

intervar we estimate

[a,b]c-[',1].

using Lemma 3 and Lemma 6 and writing

(21)

and

llP:

(l _{- s,) -} (Í-&

_{)llc¡0, tt< C ll.f}

_-

_B,llc¡o,ry

(22)

_llp,i

s, - Inllc¡an,t_,u,.1s#ft.ller8,,ll.¡0.,;

.

By Theorem 7.2.1 and, Theorem 7.3.1 [3,

p.79

and,p, g4, respectively], _{\,].e can} choose

g' _{to be}

{",

^the

^{u*st tJãi-¡å}

^degree

ór;";;

approximario' in C[0,1]. We have

(23) _{ilPi Prt;t-}

PtJ;tlic¡',1¡ s

Mltpi \.t;t-

prJ;tllr¡n,,,,_n,n1,

using Theorem g.4.g [3,

p.

10g] translatcd from [_1,

l]

to _{[0, 1] and}

n to [\Ç].

But, in view of Theorem2.l.l _{[3, p. I 1], there}

exists

M>

'such

(24) M-'rtr(.f ,o-,,r)"to,,l I Kr,r(f ,n,l)cro,rt<

Ma2r(-f ,n,,,r).,r0,,r.that

Then. by (23), (24)^ (.22¡ vn¿120). we obtain

(2s) _llP"

_Pr^t,t- PrJ;tLlc¡0,,11

" ñft'2n.Kr.*(f ,r-')"ro,,r r

<

M' 'ff '

'l{'f 'n-"')

_',.,,t'

On the other hand, by (21),( l9) and (24), we have

(26)

llp: (f _- \r;t)-U -{ø¡)ll.¡0.,1<

<2CK,.e(f

_,

n-')rto,,l s

M,, ^.

rrrÇ,r_"r).t0,,1.

7 On Some Properties

t05 and

104 Zoltrán Finta

we have f (x+h)-ZlG)+ Í(x-h)<

^Mh2

, ^so l\'ol(Ðl< ^Mhz.

^Therefore

a'(.f

,h) < lrlht

,

h> ^o.

Conversely,

let f

^eC[g,

1] with o'

^(-f , h) < lrlh'

.

^h>

0'

Then

f

^belongs

to the Lipschitz space

Lip(2,C[0,1]),

which implies,

in view,of

^{Theorem 9'3} 12,p.53f, that there exists

/'

^{on [0, 1.1 and}

/' ^is

absolutely continuous with

lf "(x)l< M

^{a.e. .r e[0,1].}

Let

x e[0, 1] ^{be a} fixed point. For the linear function

l(y)

=

f

^{(x) +}

f

^'

^(x)'

'

(, - y),

by Taylor's formula, ^we have _l-f

0) - ^t(y)l<

^{M (y}

-

^{x¡2 I}

^2'

^{Since P,"}

preserues linear frurctions, we obtain

lP: (f,x)- f(x)l=lP: (Í -t,x)l<

Y

^{p,: ((.}

- x)z;x)= M. t(l_-t)

+!Y

<-2 2n

^{1+ G}

By the hypothesis

n'a(n) (

_{1, for}

_n=!,7,...

_{we obtain}

lP: (f,x)- _f(x)l< M q2(x), _xe[o,i],

n

Proof

of

Theorem 2. Lemma 4 and ol2

(f,h)<

^Mhþ

' 0<P<2,

^{imply the}

inequality

lP: (f

, x)

- I Q)l<

^M

(ç'

(x) I n)þtz _' Proof of Theorem ^{3 .} We have

Pf(l,x¡=1, ^Pi@,x)=x

and

[_ . I lxfl_x)+r("+o)-].

'ì

^(u"

^,x)

⁼

_I

_r.ü

L---.,, ^j

Then

Pi

^(@

-

^x)2

,x)

⁼

x(l - Ð

^'

+:+=

_{n(t +}

_ü)

^{q2 1x¡ 'crl '}

fot

n=1,2,...So,

^{our statement is a direct} consequence of Theorem 1 [7' p. ^165]' Proof of

fneorei

^{+. We can choose}

g, eCf0,1]

such that ^{<p2 g'n'e} C[0,

l]

and g', is locally absolutely continuous in [0,

l],

which satisfies

(19)

ll"f

-B,llcr.l,'l <2Kz,q(f ,r-t)rro.,t

6

n =

lr2,

MEAN- VALUE FOR'MULAE FOR INTEGRALS INVOLVING

GENERALIZED ORTHOGONAL POLYNOMIALS

REVUE D'ANALYSE NUMERIQUE ET DE THÉORIE DE L'APPROXIMATION Tome XXVII, No 1,199g, pp. f07_ll5

LATIRA _GORI, D. D. STANCU

l. It

is known that

if

we have a se

associated with a nonnegative _measure

d

^ul polynomiaß

(p,), if

we consider a f,inctio

n -f el,,,(o,,

^tlrn

'jlti:JffT

value formula, _of N, Cioranescu

p],

_for

(1) _[

b

_tra p,G)da(_r)

₌

f

^(')

^(E)

I _", p,(x) do(x),

a <E <b.

.

^In this paper we-give several

.*t.nriolr.

_{of.it to some crasses}

of nonclassical orthogonal polynomiars, incruding

th;;;;r-"rrhogonal

porynoiiur, correspond_

ing to a measure of the form oldä,

*rrir.

given real

rrtt¡t. ,"ro*,

^{cÙ ls} a nonnegative polynomial _having

, ^t'

^is remarkable the speciar _case of s-orthogonal polynomials

p^,",

_foïwhich

I ,Ì::'

(x) _{da(.r) =}

ñh,

when we obtain the exrension

Ø _[

b

_rø ^p];i' _u)dcr(x)

₌

f ^(2G)

I

*^ o:,,,-,1.r¡ dcrlx¡,

n"r"il||,1lces ^to

^formula

⁽¹⁾

^{when s =}

⁰

^{(the case}

^of

^{ordinary ofihogonar}

2'

we

start from-

a

"method

of

parameters,,. (see [20])

for

constructing _a

ffiii,r"J,:ï:T:i:rrer

quadratu'"

roilîu

uy u,i,,g

,nu;tî oî.u,,,*ed

nodes

n., o,Ll;.jffj};l;llå:

^innnitery many poinrs or increase and

rhat

da(x)

AMS Subjecr Classification; ZStZ+, _SZCZS, 4]ASS,65D32.

Consequently, from (25) ^and (26),we obtain

llP,ï

f -./ llcro,,,t(*, # ^{* r,,). r'*{r, r-''' ),p.,t.}

tJsingthehypotheses

a=o(n-t) ^{and of (f}

,'n-''')cto,rì ⁼

O(hþ)' 0<B< 2'

we obtain llP:

f ^- f

^{llrw,tt= O(n-þt2 ). Thus the theolem i5 proved} completely'

REFERENCES

l. R. A. Devore, The Approximation ^of Continuous Functions ^by Positive Linear operators, Lecture Notes in Mathematics, 293, ^Springer Yerlag,7972'

2. R. A. DeVore and G. G. Lorentz, Constructive Approxímation, Springer Verlag, Berlin-Heidel- berg-New York-London, ¹ 993'

3, Z. Ditziw.tand V. Totik, Moduli ^of Smoothenes.s, Springer Verlag, Berlin-Heidelberg-New York- London, 1987.

4, G. G. Loreutz, Approximation of Functíons, Athena Series' Holt' Rinehart and Winston' New York' 1966'

'^-)-'nti¡ø ^{ztoo rnn¡tinns} rn lathematicacluj"l0 5.T.Popovicìu,Srtl'approximaliondesfonclionsconvexesd'ordresuptirieur'lt

(1935), 49- s4.

6.D. D. ^Starrcu, Approxintation ^of functions by a netv ^class of linear polynomíal operators,Rev' Routnaine Math. Pures Appl. XIII, 8 (1968), 1173-1194'

7. V. Totik, LIníþrn approximation by posítive oPerotofs on infnile intervals' Aural' Math' 10 (1984), 163-182.

Received May 10, 1997 Depaitmenl of Mathematics

" Bab eS-Bolyai " Universi IY

3400 Cluj-Napoca Rl\mania

Zoltán Finta ⁸

106