MATHÞMATICA,
_
REVUE D'ANAITYSE NUMÉRIgUE ET DE TIIÉORIE DE I]'APPROXIMATIONL'AN.{LYSE
NUMÉRIQUEET LA TIIEORIE DE L'APPROXIMATIOI{
Tome 12, ¡¡o
2,
1983,pp.
157-1G5I I
A CLASS OF JACKSON - TYPE OPERATORS
by
I{UCIANA I]UPA9 (Sibiu)
1. I,et C(K), K: [-1, l],
bethe
normed linear spaceof real
func-tions
defined. and. continuouson K; this
spaceis
consid.ered. normed,by
meansof the uniform norm.
W'eshall consider 8" the
setof all
poly- nomialswith
real coefficients,of the
degree4n, and
Po,P, ...,
Pn,...
the
sequenceof
I'egendre polynomials defined asP-(x\
:
'\ ' 2nl dxo' I'ikewise,
(1)
xo¡1 xu < ...
are
the roots of
Pra¡.If.
L,: C(K)
--- 9"is the
I,agrange interpolation operatorwith
respectto
(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
onefunction .fo, Íoe C(K),
sothat lim ll/, - L,Íoll +
O.'Ihe
aim ofthis
paperis to
,,mod.itv"iîä
operatorsL,, n : l, 2,
. . .so
that the
new sequenceof
operators converges pointwise,on the
wholespace C(K), to the identity-
operator.Using the
Christoffel-Darbouxformula ([5]), that
isK,(x, t) :+
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 provethat
thereaÏe
matficesM lor
which(3)
from
(2)we
get2j
+t
P,(x) P,(t),, m -l I P,*r(x) P,,(x¡n)
llnltt,
X¡r): -:-
On
the
otherhand (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
LagrangeoperatoÍ L,,..C(K) +gn
can be represented às(4) (L,Í)(x): É ce,!f)p,(x)
h:o
where the functionals
C¡,2C(K)---IR, / : 0, l, .,.,n, are
defined byc¡^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 oT,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)
giaenby (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) asa starting
point,we define the
operators AnzC(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 definedby
(S),while_M: llrnnnll,
mn,eR, is a triangaiar matrix still
undeter"imèá._
^Now
we give possibilities of choice ofItris
rrratrixM so that
llA,ll
<4
Co,n:7,2,..., and
rnorcoverLL^ llÍ - A,Íll - 0 for every Í e
C(K),rn this
casethe
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
sametime,
(6)and
(7)imply
(10), whilethe
equaliLies Po(ø):
1,Pr(x)
: x, Pr(x): |{t*, - 1) together with
(10)prove
(11)-1
160 LUCIANA LUPA$ 4
Let us denote by E the
setof all
sequences Q: Ø")î:, with
theproperties :
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 getD'(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 Alsoand
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
consideril;,at
nt¡n àre definedby
(12),i.e.
thematrix M is
generatedby an arbitrary
sequenceQ:
@,),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.fQ:
(q"),?=
E, tkenth,e þol,ynomiøl oþeratorsA,:C(K)-
*C(K), n: l, 2, ...,
d,eJined.by (13)
ørelineør
ønd, þositiue,uith
theþroþerties
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
.
Takinginto
accounta result
established.by
T. KooRNwTNDDR[], for (x, y) = (-1, 1) we
havePr\t) (l ç'@,
y,
t)l*) 'dt
D,,(7,
x¡,i M) : O#* ,
OIIence
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
coilsequencesof
(11)while the
monotony of.A,
THEOREM aswell
as3. the The fact that A^eo: conaergeil,ce
¿'o proveliers tfrai
nt,:
¡] A,,li¡ m,(Ç,,) generated.: l,
n- l,
2,.by
.'.ø sequerrce Q:
@,)e
E uerifytke ine
eslm,l<7, j:0, 1,...,n;
lmr - mn+tl <
(Ë+ 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
getlm,l: l(A"P,)(L) | < 1.
Simultáàóously(14)
PrU)_ Po+,(t):
(,ä+ 1)(l _
ùntn\v)where
by
¡f;'01we
d.enotethe
Jacobi polynomiar ofthe
degreeh,
normari- zedby the condition
R!",0)1t¡: t. lt
isknowl that for
æÞ þ, _ +
\ryehave lln["'P)ll : l;
therefore, (14) impliesDr(t)
: IPrØ - pr*r(t)l <
(A+ 1)(t _
er(t)).On the other
handlm,- 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,
762 F'rorn
| - rn,:
one find.s
LUCIANA LUPA9 o t7I JACKSON.TYPE OPERATORS
163
zr: zr(n) being the
greatestroot of the Jacobi
polynomialj-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
definel;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,
thenfor
eaeryc2"(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
rreansthat lim
rnr(q,): I implies lirn
ntr(q,): I
and.froin
lemma2 we
concto¿"*i#tin. llk -
(A,,ln;Qí'i-:O, for every polynomial
hof the
d.egreet*o. ,{&irding to
'r'.popovrcru (tal) and p. p.
Korlov-KrN (t2l) it follows that lim llf - Ø,.f ;
?)ll:0 for all
continuouschtt: (n!)'z(2h i 1)
(n-h)t(n+h+r)l
-Using
the
above equalities,we
haveø,,(x):Dr,r,n'#
fr,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
rreansof
lemma2
and theorem3
we obtain ll (1"Or;
?)ll <
2(1-
not\,)).But for f e C6) and
ò¡ 0, the
inequalitynl(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,
-tnwe give
(15)(A,f ; Q)(x): l;-)r, E
(l - *r,), þt ! x¡,) P l!xrr'x ri x*xm3. In
the following we shall stickto
some sequencesof linear
positive oper'.ators,illustrating the generality
ofthe
method presentedin thc
pre-vious
paragraph.For this, we
considerthe
sequencesof
polynornialsQ': þ,),
Q*: (bi'), Qr:
þ"") wherea,,(x):*; (L+'f
(17) b,,(*):æ0*x),ty:l
I Pn@n)l' (f (xn")
For
these operators| -
mr(a^): - ?-;
thereforen+2
,ti3 llf - Ø"Í;0J ll :0.
for
every/ which
belongsto
C(K).Since
(A^Q';
Q,)@):
,, *# * r)+ #*
r,764
\t¡e have
Thus
sequence
u,,, u,
=LUCIANA LUPA$
B
I
JACKSON-TYPE OPERATORS165 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
considerthe
sequence?* : (bî,)
definedby
(17).We
have mr(b,): zr(n)
andm,(b,)
- 1-
3(1-
z,(n))* ]I -
2,.(n)),* *.r(t _ zi@\.
I1 rr(n)
denotesthe
greatestroot of the
l,egendrepolynomial
p,,, then zt(n):
zr(s)for n eveí
and
the
equalityP"(") + P,+r(x): (t +
x)n!ort(x)furnishes us
zr(n)
{
zr(sf
1)for n
odd..i.e.,
A2n:fn(A",,Í;
Q"),*: l, 2, ..., are of
Jackson-type.In a
naturaTway, the following
problernof optimal
ápproximation may be considered.:to find in
Ea
seqtlenceQ- : 6i,) of poþornials
for whichnll'o)
(*): p,
@+
h.+
2)(n-
Þ+ g2J:!J er¡x¡
it is
easyto
seethat
rn r(c2,,)
By
rneansof
(16)we
conclude withllf - Ø*Í;
Qùll <
(1+ ./sl).
(19) A(?*) :
min Q=EA(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 exislsa
sequenceQ*
which satisfies (1g), and rtroreover .thatQ*:
Q*:
(bi,), bT, being definedin
(17).It may
Ëe'rrotedthat the linear
operators -4, furnishesus
somesummâbility
methods for I,agrangeinterpolation
(see [6]).zr(n)
1zr(s f t), s: I +Í!l
ø.- 1 ,
l2l' "-t't'""
Bnt
(see[5],
Theorem 6.21.1and
pag.IBg) zr(n) Þ
rr(s),n:2m I l,
and
/r(s*
1)<
cos".,
This
enables usto
assert thabthere
exists a I5
^12),
suchthat
^,ll-tn@:1, n:),g,...
From-(l5)
we co'cludethat
there existsa
positive constantc, c <
(2+ +
11Ð" suchthat
ItÍ-Ø,Í;0*) il < r"Vt)), Í ec(K).
This_ proves