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MATHÞMATICA,

_

REVUE D'ANAITYSE NUMÉRIgUE ET DE TIIÉORIE DE I]'APPROXIMATION

L'AN.{LYSE

NUMÉRIQUE

ET LA TIIEORIE DE L'APPROXIMATIOI{

Tome 12, ¡¡o

2,

1983,

pp.

157-1G5

I I

A CLASS OF JACKSON - TYPE OPERATORS

by

I{UCIANA I]UPA9 (Sibiu)

1. I,et C(K), K: [-1, l],

be

the

normed linear space

of real

func-

tions

defined. and. continuous

on K; this

space

is

consid.ered. normed,

by

means

of the uniform norm.

W'e

shall consider 8" the

set

of all

poly- nomials

with

real coefficients,

of the

degree

4n, and

Po,

P, ...,

Pn,

...

the

sequence

of

I'egendre polynomials defined as

P-(x\

:

'\ ' 2nl dxo' I'ikewise,

(1)

xo¡

1 xu < ...

are

the roots of

Pra¡.

If.

L,: C(K)

--- 9"

is the

I,agrange interpolation operator

with

respect

to

(1), i.e.

(2) (L,fl(x) - f ¡po,)

¿=o- ' (x - x¡n) P'+'(r)Pi¡r(t¿n)

then ll¿"ll , #r, n: l, 2, ... (t3l). This means that there is at least

one

function .fo, Íoe C(K),

so

that lim ll/, - L,Íoll +

O.

'Ihe

aim of

this

paper

is to

,,mod.itv"

iîä

operators

L,, n : l, 2,

. . .

so

that the

new sequence

of

operators converges pointwise,

on the

whole

space C(K), to the identity-

operator.

Using the

Christoffel-Darboux

formula ([5]), that

is

K,(x, t) :+

(2)

158

where

(5)

From

(4)

K,(x, t) :f,

i:o

LUCIANA LUPA$

2

I J JACI(SON-TYPE OPERATORS 159

In the

following we shall prove

that

there

aÏe

matfices

M lor

which

(3)

from

(2)

we

get

2j

+t

P,(x) P,(t),

, m -l I P,*r(x) P,,(x¡n)

llnltt,

X¡r)

: -:-

On

the

other

hand (I -

xz)Pi,¡{x)

:

(n

* t)lp,(x) - xp,¡lx)l

and, the-

refore

Pn+r(x) _ 2(t - tln) Ko@, xrn) _ Rn(x, xø) (x - xn) Pí+r(xr) (n ! l)zlpn@¡,)12 Rn(t¡,, *¡,)

ft follows that

Lagrange

operatoÍ L,,..C(K) +gn

can be represented às

(4) (L,Í)(x): É ce,!f)p,(x)

h:o

where the functionals

C¡,2

C(K)---IR, / : 0, l, .,.,n, are

defined by

c¡^lfl:##,f Í@u^)--Ë##

co,(f):"+ t p,(x) (L,f)(x)

d.x,

¡Í - A.f ¡ < c^ (r; !), lL:1,2,...,f eC(K),

where C

is

a constant independ.ent o

T,EMMA

l. If A^:C(K)-"C(K) is

d'efined,

by (7)'

then

(9) (A,,f)(*): i:

D,,(x,

x¡n; M)f(xn)

þ-O wkere

D,,(x,

xp,; M) : +tr#

(B)

ui.th K,(x, t)

giaen

by (3)

and'

ft.,¡*, t; M) :

that

is

(6)

Co,(p,)

:'+

I

\ er¡*¡n¡*)d.x : 8p¡, 0 <i, h <

n.

-1

2."Taking

into

consideration.the representation (4) as

a starting

point,

we define the

operators Anz

C(I{)--*T(K), n: l, 2,.

. .,

by o

L

----'

(7) Ø,f)(x) :i *o^c¡,(f) pr\x)

A:0

where the functionals C¡\ P:0, 1,...,%,

had been defined

by

(S),

while_M: llrnnnll,

mn,

eR, is a triangaiar matrix still

undeter"imèá.

_

^Now

we give possibilities of choice of

Itris

rrratrix

M so that

ll

A,ll

<

4

Co,

n:7,2,..., and

rnorcover

LL^ llÍ - A,Íll - 0 for every Í e

C(K),

rn this

case

the

numbers fir.¡n àr'. called ,,convergence multipliers".

D

j:o

*0,'t#

P,@) P,(t) Moreouer

(10)

(A,,P,)(x)

: m¡,P/x), i :0, l,

. . ., %,

ønd using the

notation erþ)

: tr ae

høue

(A,eo)@)

- rusnetilx), (A,er)(x):

rnlileL(x),

(11)

(A, er)(x)

:

1/t'2¡ er(x)

*

mou

:

!Lz'. '

Proof.

From

(7)

and

(5)

(A,,Í)

(*) -- þ,n,,,"# r,l*¡ f;, -#A

f @ o,)

:

: Ð 4;i*:,';#Í@u*):Ð,o'o' x¡*i

M)

¡(xe')'

h:0

-A.t

the

same

time,

(6)

and

(7)

imply

(10), while

the

equaliLies Po(ø)

:

1,

Pr(x)

: x, Pr(x): |{t*, - 1) together with

(10)

prove

(11)

-1

(3)

160 LUCIANA LUPA$ 4

Let us denote by E the

set

of all

sequences Q

: Ø")î:, with

the

properties :

l) q, e 8,;

where

e(%,

y,4:

(1

-

*r)'(1

- !), -

(t

-

xy),

anð.

lzl+:'lø. rc M: llmr@,,)ll, then for x = (-1,

1) we get

D'(x' r¡'; M) :

- -: - i

g,,(t)(lq(x,

**, t)lr) ï dt >

0.

n Ìtnv;*n, ft*ì J -l

JACKSON-TYPE OPERATORS

161

dt q";lt)

2)

q"

I

3) f

Þ0onK;

I

r1.

q, e8

and q^lx)

:ä**'+P*(ø)

then Also

and

D,(- L,

xnn]

M) : #F,_,j;,

O

(12)

ftti^

:

m,(q,):

:

J q"tt)P,(t)dt, :i

:0, l, ...,ni il,: l, 2, ...,

-l aud

molq^)

: l.

In the following we

consider

il;,at

nt¡n àre defined

by

(12),

i.e.

the

matrix M is

generated

by an arbitrary

sequence

Q:

@,),

Q e 6. In

ord.er

to underline the

above mentioned. d.epend.ence,

we

denote

(13) (A,f)(x) : (A,Í;

Q)@) :

:f *r(q,)

Ce,(f) Pr@)

where

mr(Ç")

is given by

(L2).

T,EMMA

2.

I.f

Q:

(q"),

?=

E, tkenth,e þol,ynomiøl oþerators

A,:C(K)-

*C(K), n: l, 2, ...,

d,eJined.

by (13)

øre

lineør

ønd, þositiue,

uith

the

þroþerties

l) A,eo:

as,

Anet:

rnt!Q,)

at, A,ez:

mz(I,)

", * 4

3 ;

2) llA"ll : l, n: l, 2, ...;

3) (A,Ar)@) :

lmr(q,)

-

2rnrlq,)

| ller(x) + t+

ukere

Qr,(t,

x) :

(t

-

x)2.

Proof

.

Taking

into

account

a result

established.

by

T. KooRNwTNDDR

[], for (x, y) = (-1, 1) we

have

Pr\t) (l ç'@,

y,

t)

l*) 'dt

D,,(7,

x¡,i M) : O#* ,

O

IIence

D^(x, x¡n; M) > 0 for every x = L- I, Il, h:0,

1,

...,,n.

nv P), Í > 0 implies (A,,Í;

Q)

Þ 0 on K.

-

.The equalities l).and,?)-

gi

coilsequences

of

(11)

while the

monotony of.

A,

THEOREM as

well

as

3. the The fact that A^eo: conaergeil,ce

¿'o prove

liers tfrai

nt,

:

¡] A,,li¡ m,(Ç,,) generated.

: l,

n

- l,

2,.

by

.'.ø sequerrce Q

:

@,)

e

E uerify

tke ine

es

lm,l<7, j:0, 1,...,n;

lmr - mn+tl <

+ l)(1 - mr), /t,: 0, t, ...,n _ li

r-m,<il+)fl-nt,).

,

,Pro.o{.. S:+çç.

lP,(t)l

<. 1,

.¿(1) :.1, t = l-1,

11,

from

(10)

we

get

lm,l: l(A"P,)(L) | < 1.

Simultáàóously

(14)

PrU)

_ Po+,(t):

(,ä

+ 1)(l _

ùntn\v)

where

by

¡f;'01

we

d.enote

the

Jacobi polynomiar of

the

degree

h,

normari- zed

by the condition

R!",0)1t¡

: t. lt

is

knowl that for

æ

Þ þ, _ +

\rye

have lln["'P)ll : l;

therefore, (14) implies

Dr(t)

: IPrØ - pr*r(t)l <

(A

+ 1)(t _

er(t)).

On the other

hand

lm,- me+tl:l(A"P,,)(l) - (A,po+r)(l)l< Ø,Dr)l):

(h

+ t)(t _

rnr).

$ - L'analyse Dumérique et la théorie de l,approximation Tome 12, \r. 2, lgl3

Pr@)Pr(y):

+ j,

(4)

762 F'rorn

| - rn,:

one find.s

LUCIANA LUPA9 o t7I JACKSON.TYPE OPERATORS

163

zr: zr(n) being the

greatest

root of the Jacobi

polynomial

j-r j-1 j -1

D@"-tixvt-) < Ðt//l,-,llly¡tl <

(1

-mt) D(r+

t)

v:0 v:0 v:0

R(o,d)

and s:s(ø):1+

I,ikewise, we

define

l;l,d':d(n)-,-'É]

r-*,<i+1)0-*,)

'(ryr

Sirnilarly it is proved that

'r'HEoREM

4. If Q e

E,

.f ,

.f = C(K), ue

høue.

I

a

'mrl

{

1.

:

(q,,) ond

,}8

rnlÇ,)

: I,

then

for

eaery

c2"(x)

:

2(n2

l3n l

3)

I

Rll'ot lx¡ 1,.

According

to a formula given by Bateman

(see

[5]), if

,111

llÍ -

Ø,,"f

;

?)

ll :

o.

llÍ-(A,,"f;

?)

ll < Q+^lt),ù(¡;^lT-m,@i¡

l+)":f'

co'Prþ)

Moreouer (15)

then (18) where

I cn,Pr(x)Pr(y): (ryi' ,,|ffi¡,

Proof.

We

have mo(q)

:

1,

0 <

1

-

mr(Ç,,)

<

3

(l -

nør(tl,)).

this

rreans

that lim

rnr(q,)

: I implies lirn

ntr(q,)

: I

and.

froin

lemma

2 we

concto¿"

*i#tin. llk -

(A,,ln;

Qí'i-:O, for every polynomial

h

of the

d.egree

t*o. ,{&irding to

'r'.

popovrcru (tal) and p. p.

Korlov-

KrN (t2l) it follows that lim llf - Ø,.f ;

?)

ll:0 for all

continuous

chtt: (n!)'z(2h i 1)

(n-h)t(n+h+r)l

-

Using

the

above equalities,

we

have

ø,,(x):Dr,r,n'#

f

r,t l, ffi¡: m,(ø,):

Also, from (18) with M: llm,(ø,)ll, we find

fr^;¡*, t ;

M) : tlJ

lt"a i' o"l In

conclusion,

from

(9)

functions

f, f eC(I{).

By

rreans

of

lemma

2

and theorem

3

we obtain ll (1"

Or;

?)

ll <

2(1

-

not\,)).

But for f e C6) and

ò

¡ 0, the

inequality

nl(n

t

l) |

(n+h+l)t(n-h)t

(16)

ltf -

(A,,f

;

?)

ll < (t n *

ll

(A,Q,;

Q)

lf ).Lr;

ò)

,#)

is verificd. For

ò'

: ./ I-m,

-tn

we give

(15)

(A,f ; Q)(x): l;-)r, E

(l - *r,), þt ! x¡,) P l!xrr'x ri x*xm

3. In

the following we shall stick

to

some sequences

of linear

positive oper'.ators,

illustrating the generality

of

the

method presented

in thc

pre-

vious

paragraph.

For this, we

consider

the

sequences

of

polynornials

Q': þ,),

Q*

: (bi'), Qr:

þ"") where

a,,(x):*; (L+'f

(17) b,,(*):æ0*x),ty:l

I Pn@n)l' (f (xn")

For

these operators

| -

mr(a^)

: - ?-;

therefore

n+2

,ti3 llf - Ø"Í;0J ll :0.

for

every

/ which

belongs

to

C(K).

Since

(A^Q';

Q,)@)

:

,, *# * r)+ #*

r,

(5)

764

\t¡e have

Thus

sequence

u,,, u,

=

LUCIANA LUPA$

B

I

JACKSON-TYPE OPERATORS

165 Since

ll(A,Q,; 0J ll < 3,

so that from (16) one

obtains

|"f _(A"Í; ?Jil< 0+,Jr)^V,#), .f ec(K).

I,et us

consider

the

sequence

?* : (bî,)

defined

by

(17).

We

have mr(b,)

: zr(n)

and

m,(b,)

- 1-

3(1

-

z,(n))

* ]I -

2,.(n)),

* *.r(t _ zi@\.

I1 rr(n)

denotes

the

greatest

root of the

l,egendre

polynomial

p,,, then zt(n)

:

zr(s)

for n eveí

and

the

equality

P"(") + P,+r(x): (t +

x)n!ort(x)

furnishes us

zr(n)

{

zr(s

f

1)

for n

odd..

i.e.,

A2n:f

n(A",,Í;

Q"),

*: l, 2, ..., are of

Jackson-type.

In a

naturaT

way, the following

problern

of optimal

ápproximation may be considered.:

to find in

E

a

seqtlence

Q- : 6i,) of poþornials

for which

nll'o)

(*): p,

@

+

h.

+

2)(n

-

Þ

+ g2J:!J er¡x¡

it is

easy

to

see

that

rn r(c2,,)

By

rneans

of

(16)

we

conclude with

llf - Ø*Í;

ll <

(1

+ ./sl).

(19) A(?*) :

min Q=E

A(Q) where

L(Q)

: V1 -

mr(q,)

REFERENCÞS

[1]Koornwinder, T., Jacobi þotrynornials (Ir). An ønalytic þroof of the þroduct formu!,a.

SIAM J. Math. Analysis, 5 t2S-132, (1974).

[2] Korovkiu, P. P., Littear oþerators and, øþþr:oximalion theory. Gorð-on ancl Breach, New York, 1960.

[3] Natanson, L P., Constructi,aefunction theory. Vol. III, Fr. Ungar publ. Co., Newyork,

1965.

[4] Popoviciu, T., Asuþra d,emonstualiei te

crärile sesiu

[5] Szeg6, olynontials.

[6] Varma 'Û.\f.,On

ximation Theory, 9, ¿ 349-956, (1929).

Received l0.III.l982.

Facultatea de tnecanicã,

Str. I. Røli.w ny.7

2400 SIBIU, R, S. România

(¡,))'

we

have shown that^there exisls

a

sequence

Q*

which satisfies (1g), and rtroreover .that

Q*:

Q*

:

(bi,), bT, being defined

in

(17).

It may

Ëe'rroted

that the linear

operators -4, furnishes

us

some

summâbility

methods for I,agrange

interpolation

(see [6]).

zr(n)

1zr(s f t), s: I +Í!l

ø.

- 1 ,

l2l' "-t't'""

Bnt

(see

[5],

Theorem 6.21.1

and

pag.

IBg) zr(n) Þ

rr(s),

n:2m I l,

and

/r(s

*

1)

<

cos

".,

This

enables us

to

assert thab

there

exists a I

5

^12),

such

that

^,ll-tn@:1, n:),g,...

From-(l5)

we co'clude

that

there exists

a

positive constant

c, c <

(2

+ +

11Ð" such

that

ItÍ-Ø,Í;0*) il < r"Vt)), Í ec(K).

This_ proves

that the polynomial

operators /---+

(A,f ; Q\

are operators

of the

Jackson type.

rt remains to estimate the order of approximatio,

giverr

by

the

operators Azn

generated

by the

sequence

Q-r':

(c"*).

Referințe

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