ABSTRACT
resuJ'ts of R' AEkey' pePe! ls to l4rrove sctle purlose of the pre6ent P' Tursn on U' G!eD8'nder' G' Szeg6 an'l p. Erdlls, G. trteud, L' Ya' GeronlrEu6' eigThe
orthogonal Fburler serles'
Christoffe! functions' ortbogonal polynorolale' lntet?oLatlon' In ParticulsJ' Eatrlces and Lsgrange envalues of Toeplltz a'qy velgbt 1{ wltb be answere
wtll (posltively) lnte4rolattng polynofor each p ) 2 the T'a€raJrge that 6uch sr4tport co4tact f ? ror sme continuoue fl'nctlon cllverge fn { v to Blals correspondlng functlons arl'l thel! appllcatlons' tiealg wlth Christoffel Moet of the ?aper are found in the asBuslttlon for orthogonal polynorolals Ma&y llntt relatlons
I\]r,aors problem
the recurslon tbet the eoefflclente in
formuLa behave nlcely'
\1-A2o' U"A25'
\:'Kc' 4l-{Lo' ' classlflcatlo ns -E2A'2' 33A6'' 26N'' 26N8' )+o!2'' AMs(Mos) subJect \2A62' \2A56' \lA3), 4rA55' \LA6o' \2Aol+' 6',D3O' 658L',', 6>t3',' zge.a*, lolog, 3oAEr' 3OAB5' Fourler 6erle6' Inpolynonlals' Eradrature Processes' Key Worats
- Orthogonal
Toeptltz natrices' Chrigtoffel terpolatlon' Posltive operators' functlone' in Publication llata Library of Congress Cataloging
*v\4e5tr'
,."thi?l',,.1'
*';
-u-"'
n.r*.ie"f:,iq'.i . Mernoirs.o r t:" _*:\'":i ffi1T:'::1 i'" I ilrl.,''.,, "i no. 2I3 ISSI{^OOb)-Y/YU (
l
t'volume ro'
L' Ttlt:'-^tt' series: *:ii::' Y'nT?'i;ii.iH;{i,'Hiii' i' i! i'#'u..'o S'"'i3h't3;;iit*-" i1*i?til:pg:It lory'oroi"r".
-s
'-ilhin the cuidelines eetablished
*:J"";',:'":".*#"1111ffi :[il'$ifJ:6*dra'swithinthes'[ Malhemstical Soctely CopyriSht @ 19?9' Americsn
RePrinted 1987
1lhls work
is
dedicatetl
to Richald Askey
Table
of
Contents
1
Introcluctlon
1
D
Notetlons
3
Bsslc !bct6
3.1-. fihe GeneraLlzed
4.
Recurrence Fof!flr]-a
3.2.
l"lcal1fletl Que.alrature Processes
3.3.
Tbe $It't)ort
IO
of do .
20
Lrnrt Relatlons
'
l+.1. Polntwlee Llolts 1+.2.
WeaJr
'
L|nlts
Elgenvalues
of Toeplitz },1atrlces
49
,7
An Inter?ola,tlon
,7
hocess
6.2 A Sequence of Positlve Operators 6.3. cenerallzetl Chrlstoffel firnctlons 7.
9.
tn
:the Coefflclents
2, 39
Chrlstoffel F\.uctlons
6.L,
2'
ln the
Recu$ence
Fbmul-a
7l+
105
I2l+
Fourler Series '
147
Inequautles
L56
Iagrs,nge
Inter?ol-ation
References
L75 1BI+
2
'k
'zxt'
&\
&i
OBIAOGONAL POLYNOI{IAi€
1.
IntroAuction fhe purlrose of the present paper ls to fuqrrove
scrrne
results of R. Askey,
P. Rrtl8e, G. tlreutt, L. Ya. GeronlEus, U. GreEanale!, G. Szegd
8J1d
P'
Turan on
orthogonal polynoolels, chrigtoffet firnetlons, orthogonel Foufler series, elg-
of Toeplitz natrices ard Iegrange lnterpolatlon. In perticul-ar, TLEanrE probLem [f] wlfl be aJxswereal: ls there eny weiglt w with co!4)act sulrport such that for each p > 2 the Lagrange luterpolatlng po\moolals corenvalueg
to w dlverge ln { f"t sone continuous fbctlon f ? R' Askey [)-] conJectured. that the a:tlsfler lras ye8 and the solutlon was glven by the Follaczek welght because the logarlthm of the Polleczek welglrt ls not lntegra-
reqrontltng
ble.
IlJe
wlll
sholf that Askeyts conJecture 1s
ri@t but for dlfferent
In fact, there are uarryweights sol-vlng Turan's problem; lnteg:cable logarlthm, some of then dro not. Most
s@e
of then
reasons. do have
of thls paper deals wlth lnvestigatlon of christoffel functions ancl
1t8 geners-llzatlon. Ihe resujLts and the Eethods are stronger than those of the above authors. T'he chrlstoffel functions pl&y a very luportant role in the theorlr of orthogonal potynornlal.s.
Many
results ln olthogonal Fourier ser-
tes and interpolatlon are based on estitletes
aJtd
asyEptotics of Ctrrlstoffel
christoffel f\:nctlons can be applled in find_ing necesaary condltlons for weightecl mean convergence of orthogonal
fi:nctlons,
we re.llL show how successfuqr
Forrrler serles
axrd
Iagrange interpolation proces6e6. lntrotiuclng gene"alized
Chrlstoffel filllctions we shall flnd a corurectlon between different welghted LP noims of polynouials. Especial"ly lnteresting is the case when o < ! < bY Sponsoreat -
1) The Unitett States Arw uncler contract No. DAAG29-?t-c-0021+, e) Ttre Anerican l,tatheEstlcel Society, Fovltlence, RI Q.gLot,,3) fne l\Iattona-l Science Fountlation, under Grant No. Mcs75-056&1. Recelneal by the eclitors l,larch 5, I9l7 ancl ln revisetl form Januar:f zj-t LnB.
1'
PAUL G. NEVA]
Wewillinvestigateanevkind-ofquadratureprocesswhlchhelpstofindesyul}. outsLde the sulryort totics for dhristoffel f\nctlons and orthogonal polynoolals
oftheweightfunctlon.Takingtherecurslonforgr:la48&startlngpotntand. ln the recurslon forsuJ-a we w"lIL obassu.ning properties on the coefficients lle wl]-l be able to taln results on orthogonal foffnornfafl, fn-certain cases ln the vecurslon formufa' calcu.late the welght function uslng the coefficlents }le wlU a16o
flnd the (CrI) I{nlt of furan
tytr'e cleteroinants urder rather
veak conclitlons. has been
fteud rho I learned the theorT of orthogonal potynoEls'ls lYon G' yean' lrleny of the nethods I use 1n tbis supervlslng qf re8earch for several whlch ls a rldt forurd ln his book on ortbogonal polynoniats
paper can be source
of
method.s and unsorved research
lroblems. r wlgh to
erq)ress
4r
aeep
to R' Askey' L' Bers' M' cvlkeL' feeLlng of gratltucle to G' Fleud as weII e's the help of whlch thls paper J. IandJn, G.G. Iprentz and W' Pro:
Soclety, to the Natlonal Science Fouldation sponsorlng
and'
to the Unite'l States
Arrnv
for
qf research'
to solve problems mentioned Thls whole wo?k was born from the attenpts wlth R' Askey severa]- tines' ln R. Askey's paper [IJ' I tiscussedny results ilnmantuscrlPt aJId nade various swgestlons to He read a draft verslon of, the to Richard Askey because his prove the presentatlon' I detllcate thls work for me to carry out the research generous suptrrort 8nd help nade lt posslble $hich led to thls Peper'
for reacling the entlre FlnaLLy, I wouftt like to thank T' S' Chihara
rna^nu-
for pointlng out that sone of scrlpt, for naling ahost a huntbed' corrections' reflmown artd for sr4ryrlying ne with ad'dltional n6r resuJ-ts had prevlously been etence s.
\',
il lr
li
2.
Notatlons The
f\rnctlon 0! R + IR ls called. a weigbt functlon lf lt 1s
nontle-
ereaslng,lthasinflnlteJ-yoarrypolntsofincreasegJr{q,llthegonents
*2oao1*; (n =
\
or1r.
"
)
are f,lnlte*. For a glven welglt cr the eorrespontllng systern of ortbogonal po\ynonlals {nrr(ao))}o 1s deflned bv prr(dorx) = vrr(aa)xn + " ', Yn(tb) > a!al
J
o
lrr{oo,*) Pm(&,x) oa(x) = or* '
IfohappenstobeabsolutelycontinuousthenwewllJ.usrrallywritewand' general- case c no(vrx) lnstead' of q' a,nd prr(dorx) respectlvely' In the can be
wrltten ln the
forro
*=or"*ou*oj where oac
ls absolutely continuous, o" is slngu.}ar and oj ls a juqr func-
tLon.
of the baslc proltettles of a system of orthogonal polynonlafs tprr(do)l 18 that the polynoolals pn(dcY) satlsfV the three teru recurrence Gre
relatlon (do,x) = xo -n' (n = UrIr...
Y.(ds)
Yn-r(do)
, P,,*1(tu,x) + o,r(da) nn(do'x) - -t;-dr
""-F ,,
where P-r=Or
pn-t(&I'x)
Y-t=O artl e-
on(do) = S_-
tn'"(*,',
dcx(t) '
functlon tf lt ls s.bsolute\y continuous. Obherwlse o ls a tlistrlbutton function, neasure or integra'l weight fimction'bJrWea wil_l use. the same temLnolory for Loth cases. If the weigbt ls denoted.
@t
greek
letter then
we rea^n
tirat it ls actual'ly a dlstrlbutlon' latln letters
iean that we are clealing w'lth sbsolutely continuous welgltts'
PAUL G. NEVAI
polnts of lnterest' wiLL be one of our Batn Thls resurrence rel-atlon
if a systen of polynonlals tlo(x))|6 farous resu-lt of J. Favartt
W
a
satlsf,Les
the recufr€nce fo:guLa
Y:er.(x) =
fr
nn*tt*'
-.- /--\ +*tt-to + onPn'rx/
l;
.(x) vn-l\-/
Po=Y6r Yo>o 8lrd' 0n€lR wlth P-1=or Y-t=o' wblch na'Ir not vlth resPect*to sooe welgbt c tben tlr,(x)) rs ortt'ogonl:tlea'l In th18 pa?er rle are golng to (see trYeucl' $rf'r)"' ileteroinecl unlquelybe then o ls and [v"-'/vo] are bounded and qtf t both vhen caseE vlth zuctr
for u=orrr,..
rurtquelY tleflnecl' Ttre
real zerts of 1ln(do) ' whlch are
a'n
distinct' will be denoted by
x-(ao): x-(do) > xrrr(do) definetl W corres;rontllng to cr ls t"t*l (do.z) = ,otn r.rx.
\-
fi € Fo_lr__
ln(t) l2*(t)
rr(z) = r
of lPn ls the set of polimonlaLs for z € 0r o= L'2"" vhere to see that rcst n . It ls rather easy ,,2 . n-I rrr(oo,z)-r = to lPk(do'z)l-'
degree at
usuaL\r de-
and' are are calLed Ctrrlstoffet numbers nrmbels In(do'xkn(ds)) nunresults lnvotving Chrlstoffel fhere are tl{o lEportant noted' by \m(do) ' the Gauss-Jecobi lech&nlcThe flrst of them is
Tbe
bers whlch
trill often be used'
el qua'lrature for:rula; (
--[
J-6
*
i. lt*'too) "(x*'(ao)) n(t) do(t) = .r{=r
books llste
referred by !0entlon1n the references wlIL be
ORTTOGONAI POLYNOMIALS
for each n € F2rr-l , the nhlcb
can
be expressed
is the Markov-Stle1tJes lnequallties
seconcl one
as
n (e) < ;frn(e)tu(t) nt rk(e) s r L !
k=1+1h-J---k=l*
the set of polnts (! = Lrzr...rn) . ['be sr4rport of c , that ls suprp(dcr) ' ls oflncreeaeofg.Ifswp(dc)lgborrnd'edA(e)wlutlenotetheanall-eet
closetllntervalcontalnl.t8gr+rp(do).fhesymbo}sAanitf!I1}lal$ays A ls denoted by f end' the Ean cloBed lntervala, tbe lnterlor part of velglt correspondlng to length of A by lAl . trbr a glven r the Chebysbev $e $rlte v lDBtead of vf ' t rdlJ. be vrltten eg v, . If r = [-1rI] then then
Tf t= [a-bra+b]
v-(x)=tt2-(*-"\21-U2 'I
For f € tL sum
. partLal (euSp(da) ls bountted) Sn(do'f) clenotes the nth
of the orthogonal Fourler serlee of f ' s-(ao,r,x) = u
l|here
Hence
for x € R
\J-6 f(t)x.(
rL(do,x,t) = r pn(tb,x) n*(oa,t) -' ll=U
or by the Cbrlstoffel-Darborx for:ru'la
t*rlt) Pn-I(do,t) Iq(tb,x,t)=Ttoot
pn(tb,x) - pn-r('r}'x) Pn(db't)
x-t
.
lnter?ol'etl-on polynonlal Ln(do'f) co!*"t*e polynomlal of ilegree at most responillng to cr le deflned to be tbe unlgue (k = 1'2""'B) ' rf we n - t vhlch a€rees tt'1tb f at the nocles xkn(do) the firntla'nental polynonlal8 of lt'grange lnte4rolatlon denote by
For a glven fiurctlon ,
'n"
4Kn(dd)
L*(dorf)
can be
nrltten
as
Ln(tbrfrx)
=
E
E=L
f(xkn(dd)) .c,*(oo,x)
.
11_ i ll i
?AITL G. NEVAI
It vil1 be usefu.1 to
remember
v
,,kn(do,x) (xo, =x*r(a))
.
that
l.(darx)
(dcr)
l*,(dCI) lrr-rido'xor) fr6
.
The Chebyshev
polynonlals cosnO (x = cos B)
w1Ll- alweys
be
by
'lenoted
function of tB ls l.ls rrr(x) . lbr a glven set Is the characteristic
g(.) (e > o)
Eeans tbe e-nelghborhood
nr,. *d
of E'
Pr
an'i
denote polynon-
denote the set of natural lnlals belonging to lPn ' The Letters N' lR and" O R- is the set of Bostegers, real ntnbers and corylex numbers respectlve\r'
ltlve real nlebers. ror o(p(-,
11
11","- isdeflnedbY IlrtB^
(0f course, for
O
Sonetimes we
-
=
\ lr(t)l-
do(t)
< p < 1 this ls not a norn'
)
(E.€. wtIL omit unnecessary para'neters in the foruu.las'
x*=x*(da).) We assume
thet the
read'er
Is fanlllar with
methods
in
the theory
of
r
one-
sided approxlnetlon and positlve operators'
Fortheconvenlenceofthereaderweglvea:rinclexwheretoflndthecle-
ftnition of l+
ln
slmbo]-s used
frequently' D'3'I'4
below roesns
Ctrapter 1.1
n0 o0 u:I-t n' Ay-t ^o d.ft "a 0.
s
"r,
PI
snk\ v
/
^&rb/a.,\ "k\\g/ D(dorz)
- D.7.6, P. I29 - D.6.2.37 ' p, 9t+ - 0.6.1.3, p. ,9 - D.6.2,\2, p. )2 - D.\.2,L1, p. I+t - D.3.1,[r p. Io - o.6.L.L6, p. 67
that
see Definition
ORTHOGONAL POLYNOMIAI,S
b. I
f(s) JS
r n'(d0.p,x) l n'(do.x) M(arb)
Pollaczek ueigbt p("
)
D
(a.b
un'(x) n
)
- D.?.11+, p. 13f - D.9.28, p. L5) - D.4.2.4, p. kI - D.]0.I7, p. 181 - D.6.3.1, p. 106 - ForEul-a 4.f(1 ), p. eB - D.3.L.5t p. l0 - D.5.2.J2, p. B0 - D.\.1.8, p. 30 - D.4.2.1, P. 39 - D.5.2.7, p. 79 -p.8. - D.6.3,\, p. ro? - D.9,28, p. L69 - lJ.o.J.Jr
p. rul
Atlthoseslrnbolswhlchuerenotlisted'herewerelntroducedlnthlschap-
3.
Baslc Facts The Generallzetl Rectrrence ForIilI]-a
3.f.
I€t Un(x)
denote the Chebysbev polynonlal
un(x)
(o:
O
=t1S#
'
of
second'
klnd, that
1s
x=cosB
the recufence forS,r , -l S x S 1) . lbe polynonlels urr(x) satlsfl
nul6
axuo-r(x)=un(x)+Uo-r(x),
(r)
vtrere U_r(x) =O
a^ntt Uo(x)
Theorenl. I€t OSkSn.
n=L,2,."
=r. Toranarbitraryl'elght o,
Pn(darx) canbe
expressed ae
(2)
rrr(ctcrrx) = urr-*(x) pk(do,x)
- ur,-*-t(x) l*-r(da,x)
+ Rrr*(da,x)
lthere
(j)
n,-(ao,x) hrE'
2 I u.-.,(x) =J=k+I r1-.1 ttr - +# rj\q/
-
zd, ,(do) pr_r(o,x) + J-r J-r
J n-',(ch,x) o
Y, -(d0)
tr - 2 #fuT rJ_]\*/
-
J n.,-r(0o,x)).
k = n then (e) ena (3) elat pn(dorx) = pn(oorx) . If n > f anci k + L = n then (2) a'nd (3) coinclile vlth the recurrence for:rnula. Now flx n and let n - I ) k : 0' Suptrtose that (2) and (3) hold If we replace thelr k by k + I , that ls
Foof.
We
wl1l prove (2) by lnduction' If
Pn = un-k-r Pk+l
- un-k-2 Pk * Rnrk*1 '
App\ylns (2) and (3) to the case I = k + I > I we obtain Pk*r. = !,i
f.i !:r
i'.
Il
urlk - uo !k-t * \*r,u
'
ORTHOCONAT POLYI{OMIAIS
Ibus
W (1) Po = (utuo-k-t -
uo
un-x+)l* - un-k-r pk-1 + Rn,k*l * un-k-t L+rrr
= un-k !k - un-k-I Pk-L + Rnrk '
that ls (2) ReEark
e.rrd
(3) hold algo for k
2. Rrttlng k = o anal rb = chebvshev welght ne obrtaln fron To(x) =
Fo! k=1
(2)and(3)gfvetbesaneasfor
fo!"lauta na,y' easlly be checked
geeterl
k=0.
- r*-r(x)
When
uo-*-r(x)
2Sk(n
b€R+,
.
oS' kSr.
pk(dd,x) -
pn(do,x) = un-k(+)
vhere
Then
u!-k-r(T)
pk-r(ds,x) * {'f,{ao,*)
(do)
Y, . p"(dd,x) 1 ,, ,'x-a\. i tr - ^ i" *-i=r I "n-J' {,i(oo,*)= b ' rJr*r I n -'J=i+r
(5)
o
p"."f.
Lt
[a -
cr-r(e)J nr-r(ao,x) * [r
o* be defined ty c*(t) = o(tt + a)
J
|fvrr-r(o)/Yn(do)l l|ie
will
cal_l_
No$
+
lr-r(ao,*)l
rhen prr(dorx) =
prr(dd*,--e) , orr{ao*) = fton(ao) - aJ ana Yrr-r(da*)/vrr(ao*; =
weget
(e) ald (3).
llheorem3. Iet a€IR,
(lr)
I
xuo-r(x) - un-r(x) = tr(x) uo-r(x) - ro(x) uo-r(x) '
ro(x) = r*(x) uo-*(x)
fhls
lbeorem
=
alEly Theores I to o* and then return to o '
(4) and (t) the generallzed. lecurrence forsula. It will help
ustoproveuarqrpropertiesoforthogonstpo\moroielslncasethecoefficlents
of the recurrence foraul-a are
eonvergent.
PAIII,
a€lRrb>0.
Deflnltlonl+. kt
G. NEVAI
Then
c;'b(dCI) = l"*(o"l -
Yk(da)
v. ,(d0) br I -;l . liFd' - ;lb' . ltff "l
corollary S. I€t a €R, b €R+, 0 S k < o .
Eben
pn(&,x) = un-k(#) pk(&,x) - uo-ir-r(\3) r*-.(oo,x)
(6)
+ o(r)
for x€(a-b,a+b)
4 JiTGT
I
1=k-r
+
.l'of*llr.(oc,*)l
'
n
{here lo(r)lcz'
Definitlon5. I€t a€nrb>0.
o€M(a,b) 1f
Then
urcf,'b{oo)=0. --
k5
Renark
?.
when consl
of
Forrlf o€M(a'b) or e'=Orb=O' eralitythatelther a=Orb=l rf o€M(a,o) w'tth b>o then o*€tnllorr; wh"re o*(t)=o(lt+a) then o# € M(oro) vhere o#(t) = o(t + a) TheoremB. I€t a€R, b €n+, o( 6(r,
lu2
- (* - a;21 r'n*.{oo,*;
pf,1ao,*;
s
x € [a-b,e+b]
'
6(*. ",r-",,?r.,rtcl'b{oo)l2t
for n = Lr?r,-. Proof.
we obtaln
Then
fron (6) that for [(1 - e)n] + l"( k(
*
"r=,,i",o,ic')2
n
{l't*)tt/n
'
gen-
ORTHOGOML POLYNO}.{IALS
that ls [r2
-
(*-ay21 plqxl
l
io'rt*i*) * ru2 pfl_r(x) * o^nlr(*)r=rrj.lnt(.r)t.
-
(* "l2t
fhus
;
u[(r-e)n]+I
[t2
p2n(*)
< et2r;]rt*t
2
/n \a
J=[(1-e)n]\ The theoren follows
'
F
1
J' x=t(r-i)nl*r-
'
fron this lnequality.
fheoreng' I€t a€l,t(arb) wlth b>0. **
fhen
fot - (* - ")21 \+l(dd,x) n',,{uo,*) = o
unlfo:m\y for x € [a - bra + bJ
Proof. qf
. utr]rt*l
Itreorem B we heve
to
.
shov
that
we can choose o = €n €
(0r1)
so
that
l3'rot
+
2)t# *
..=*t " "
ir--.,,)l.r.,,tcl'b{ao)l2l
=
o'
s',+) ([c.,J2+ r-zrr-l-/z l>_t/z
'I
Then
"r,1i. nto * =|
Thus
I
(l-eo)nlJ jn
[c.]2 + J S -'en
-
-
(e. n+1-) s'rP [crJ2 5. J/n/z u
s''rp [[c,]2 + f2ll/2 + sr4, [c.J2 + o . t7o1" d n-' JZn/z u
10. It ls usefu.l to renenber that lf "trpp(oo) = [-lrl] vlogo' € Ll then cr € M(OrI) (see e.g. I"euil). Renark
It nlght be lnterestlng to coryare lheorem p nlth (vealer) Geronlmrs (see Chapter III ln hts book).
8nd
resuJ-ts of
PAI'L
L1
G. NEVAI
llheoreml-1. I€t a€M(arb) fl-Ith b>0' 2n
Iln
t
suP
!l +o
IBt
Sultltosethet
1cl,b1ao;2 <
-
.
are equlvalent' frc (a-bra+b) ' fhen the fol1owlag tlro staterents
s} =
D .(1) tn2rrtar*)t ls rurlforrly boundecl for x €I$ ' x €E ' (11) tn-1*r(ddrx)] ls unlformly bountletl for -
proof. (1) e (11): {(n + r)-1 ll}rtoc'*)l
is the arltbnetlcal
nean
of
trzotoo,*)t. (11) a (r):
Use Theorem 8'
I€t us lelosrk thst if or€ltr
cl'b1o; = O(l-r) then the condltlons of the the-
po\ynonlal's' are satlsfieal. Exasple: Jecobi and Follaczek
Eheorem
12. I€t a € M(arb) wlth b > o @
a'nd
let
o1.
I cl'-(do)<-' u
{-A
If
A
ls unlfom'ly
c (a - b,a + b) tt€n tbe aelluence tlpo(a'x)[]
bounclecl
for x €4.
Foof. I€t, for stqrllcltY, 0€M(orl), A= l-ere], such that ao,r16) = c. Iet us flx k = k(e) J.,l
1:: i1.":
Tf
2 i t.'|' u{lT l=x-r n>
k
1n-1x) *LZ ,i,tr
o(6(1'
then bY corollenr
I
s
t+*1l JL-E
5
iJtlr*(x) I * tn*_r(*)
|.
i=,_i*,
., lnr(*)l
't?
ORrI{OCONAL ?OLYIIOMIALS
I{ence
(x)l .-, b '-ltr' '-
- .'
^A
-
*
* |Jtlry(")l
+ lu.x !n
lp*-r(x)lJ * * k+t:iJ
lp,
(*)l
thusfor n>k
I r+-,llib*(x)l* .k+l.ry-_ln,(*)l (m(n */f_e =
t+*
lpr-r(*)ll * l **rH.o o*r?l.jpl(*)l=
|Jt[n*1x)l
"A-e-
* lpr-r(*)lt * ] *.r5.o b,(r)l
,
ln partleuter, for n > k
bo(x)lS
f+
+
rJtlr*(x)l + lp*-r(x)ll '
^h-"' Now reuember
that k
Sonetfuoes
does
not
depentl
on n .
lasteatl of fireorem J we w111 use the foAlorhg generallzatlon
of the recurence foruula. theorcro13, I€t e €& b €A+, o(n(k, pn(do,x) =
Ihen
q.-"(T) pk(e,x) - \-n-r(#)
ni*r(ao,x) *
ll,ltao,*)
where
{;lt*'*r . ttr - 3Cffi, +
pr""f. Ihe theorem
tr - fr
il ",-"t+' '
rr(ao,x) * fit" - o1*1(do)J p3*1(&,*) *
*ffi,
can be provetl
fbeorem 3.
CorollarJr14. I€t k>0.
=
lfhen
P3*2(do,x))
w lnaiuction 1n exactly the
sane rra{r as
PAI'L
un-r(x) p**r(do,x) =
*
k-t llo
G. NEVAI
\(x) 4(x)
ut(*) ttr -
2
+
v. (do)
nr(ao,x)
ffit
-
eor*r(ao) !;a1(do,*)
Y. - (da)
+ [1 - 2 .,:;;151f pJ*2(do,x)] - vo(ao) ' TheorenIS. I€t 0=9.+10^
w"ith g^
nn(orx) (x = cos 0) can be represented
Thenfor 1=orl...1
as
(?) slnoprr(ctt,cose) = l,gro(dor"t')l 'slnf(n + t)e - arg vrrr(aor"i0)1 when 0r=O (s)
antl
2t sin oprr(tb,cos I ) - ul(n+r)g ,qn{0a,"-ie) - "-i(n+})e
otherw'lse. Here
n
Prn(oa,ele) ar(do,x) = lL
v,
tlo
, (oo)
- 2.,t61-f
,p2n(do,eie)
tt(*,"o" e )"uo ,
pJ(do,x)
-
roJ-rP3-1(ctr,x)
+
p* + [r - 2 5:?9r v3-1t.gr J'c^(&,x) consequent\y, 92n(e) lE a pol-vnonlal of degree at most
(J = or1r...)
wlth grrr(ctrro) = e-nyrr(oo) and qarr(do,r-L) = qlooTT Foof. l,et pn(
Thus (7
)
us
ir
l"l = r '
write (7) in the forn
) = Re Err{oo,ulo) un(x) -
1-mprrr(do,ei0)
trr*r(x) 1t ' "2;r/z
neans that
pn(dcr,x) = Rr,-1(tu,x) = un(x) po(do,x) + R.,o(dd,x)
vhlch is equl\ral-ent to (2) applied with k = 0 (7
).
(8) obvlously foll"ows fron
ORTHOGOML POLYNOMIA],S
15, i€t k be a notuegative integer
CoroUary
for J>k.
and. l-et
fhenforeech n)k
sln gprr(&rcos
e
) = l9r*(ao,ele)l srn;(n + r)e - arg,pr*1ds,"1€)J (g € R),
that ls pn(do,x) = urr_*(x) 11(do,x) - urr_*_r(x) r5_r(do,x)
Iet us note that the contlitions of corollary supp(dq)
= [-1,1J
15 are satisfied.
and
o(*)=(*ff n\ J-1
u'l
u.
(-r<x<1)
where n is a potynonlal which is posl-tlve on [-1r1]
2r 1€.r2 ; l92ktdflre )l = tr(cos (e €rR)
if
(see szegd, Chapter
II.)
E
J
In thls
cese
PAT'L G. NEVAI
IO
i[ocllfled
3.2.
Qua'IratrT
e BoceB8es
Iet u beanollnegativeinteger' It nay be wrltten ln the forro
b>o' I"€Erpat. IFt o€M(arb) vlth n > rn
(r)
- t then xn Po-t(ttz,x) ,.-. --r * nfi"t(x) ab pn-r(dd'x) & -& t * xE pn-r(do,x) = \e1,n(x) ln(dc'x) "fft,t *r&
*o no)zr^ are lol'ynoolals of degree
$here \]f..,
spective\Y.ta nff1,rr=o lf
(2)
n=O
r0
- I and n - 2 re-
l\:rther
= |S "f,,, # f,: "
vt{il#
u' '
Ibr m= O and- n- I the Proof. I€t, for 814)Ucity, cr € M(o'I) ' n > I we bave ls certalnly true' Sup?ose that for n-I &h pk(.I],x) xt-l ?rr-t(ttr,*) = nflr,n(") rn(ao'x) + : 1]fi_r wlth
ex18t1ng
(k = n-n' n-B+r' "'
un {;:t n* $hlch
alepends
r n-l)
of the particular only on M(O't) and ls lndependent gee that the recurslon fornu1a we
o € M(or1) Uslng
xn p,,-r(do,x)
= t*
ffr,,,(*) .
. #il'-'''"-'(uo)J
* t'flil'-r
"-,s **,,#iL, W-, {;1, .
r"fl;1,-,
on-,(cb)
.fiLr+J#J
CIk(dCI)
* .fli|,^-r-
. .fJ,-r
H#
+
Pn-'(ckr'x)
#ii'-,
*$#'
r,,(cb'x)
P,'-'-r(tb'x)
+
pk(e'x)
#ft'
pn-E(tl,'x)
+
+
Lt
ORTHOGONAL POLYNOMIATS
fhls foruula proves (1)
anal thows
To colpute (2) we Put
M(0,1)
that
? 5-l
,G
*
we have
u. = ?
=
j, *,",
=
'l-r,, J, *,",
I€@a
=
lll
exists and alePeuds orily bave
ln
ttr16
for n+l(2n
j,
\*(v) tr - 'f,t"r:
ir"t
it"t
=
=
rf,-r(*,,L,,(.,)) *
"rr-r,
We
on
fhus by tbe Geu6s-Jacobl uectra.nlcal quail-
'
n2o-r(.,,**o(.'))
'ni-r,r{1or{"))
e&. n-t.
h (I) o = Chebyshev velgbt.
= r - rrfut"l "es" Il2n-r(v,x*o(v))
rature fortu1a antl by (1)
l1ra
n*
J, \"rt"l rr'-t(v,xor(v)
)
^ Ihen [a-b,a+b] c
2. Let o € M(erb) w1th b > o
A(do)
Proof.rtfol}owsroml.e@althEtifflscontlnuousonF'8ld.ha6coEpact st4tpo"t tben
rin.l-
-' n* k=} \n(dCI) If n
[a-b,a+bi
r(*u,) pfl-r(e,xo,)
=
#
5"-:
I A(e) , then we can choose f
r(t)
^rFG
so that
u' '
r(*ut) - o for every
and k = L'zt... rn slld
r'-t d*b iJa-b f(t)^h"-(t-8)-dt
whlch contredtcts the above
llult reletlon'
I€tusnotethatl€@e2fo].lowsfronresuftsofo.Btumenthalt!+]. Theorem3.I€to€M(8rb)arithb>o'I€tfbeacmplexvaluecl'bountled. frmctlon A(dd) . If f is Rienarur integrab)'e on [a-bra+b]
(3)
lT
j,
lkn(dd) r(*n,) p2o-r(do,*ko) =
# SII t,t,
then
?AIIL
1B
G. ITEVAI
proof. If f ls a polynonlal then the theoren foLlows {mpdiatelJr flosr I€ma ]". Otherrrrlse we write
f = Re(f) tA + Re(f) ll(ao)\a + i l-n(f) 1o + i ln(f) la(ao)\a n2o-r(aorx*)' where a = [a-b,a+bJ . l€t, fo! s14r]Icltv, A.(s) = I ioet"o) k=l * '^
i'fx
e
)
O
.
We
'rr(x)
for x€A((b)
end lr,
constnrct two polynonlals r,
l
ne(r)(x) ro(x)
!
such that
'rr(x)
enal
4(
trt'Jo(oo)
tr^(t)-n,(t)ltn<e.
thls because Re(f) lo ls Rlerarur lntegrable on A(do) (see e'g''
We eari
do
Szegd,
1.t).
Hence
rin An(ne(r) to)
=
#
su_o
'-_---------6- (t - a)' *utt){t)^,!'
dt
l{e have, firrther,
l^n(Re(r) and ve can
to(*)\o)l
flnd a po\momlal n
=
,
ifr*,
lr(t) I n,(la(oo)\a)
such that
(x € a(ao))
ro,*y1o(x) S 'r(x) and
^ +\
-a+b
rb- Ja-b
n(t)dt <
e
Thus
rre An(Re(f) to{*;10) = o . ns ttrs f{n'll of Arr(In(f)) can be found ln the saee }ray' Theorem
4. I€t a ( lR, b > o . If for every polynornial
p2n-r(a,xo,) "(**,,) 1T,i. '-' n* k=I ^o,(dcx)
=
#
S,:"
n
'(t)vfGF
u'
'to
OFf HOGONAL POLYNOMIALS
then o € M(arb) Froof.
We
have by the recursion formula
I \n(dCI) *rr' p'n_r(e,*tr) = cn_r(e)
k=1
and.
nDDuY;-r(da) \sr(do) { n'"_r(*,"*) .r_ k=r--
D
= oi-r(oo)
- f2;* Y;_r(dd)
Ihe theoren follows lrmedlately fron the above ldentltles'
-
PAIJL G. NEVAI
20
3.3
Ttre SuPPorb
of
d0
properbies of Eupp(O) In thls Bectlon ue are going to prove several heavlly on those pmp€rpap€r for o € t"t(a,b). Ma.ny result8 ln thls 'lepentt results of thls eectlon are lt€II tleg. I€t us note that practlcally a]-l the to B1urnenthal [1]+]' Chlhera [15]' knorm. Foz' eltelnele proofs, lte refer
Kreln [I3l
s,nit sher@n [15] '
elvays closed' that 1s the aet of polnts of lncrease of d' 1s bounded' Ilence supp(o) le compact lff lt 1e Supp(dcx),
followingthreestatenentsareequlva.Ient.(1)srnp(ao)ls -' l < - . (lil) -tY-coq)act. (Ii) Br4) |x.,..(oo) " k€N{'ota"l' -t
l€me 1.
lfhe
odr
'. rn e (il)' p"*L, Easy coq)utatlon' Let us prove e'g'' (111) k=1r.
l.te have
the fol-
nechenical quadrature formula: lowlng i.qtortant l'tentity by the Gauss-Jacobl
n-
(r)
xkn(dCI)
D.
= \,,'(dCI) \_- * tfn(*,*,x*)cc(x)
n-]
t oj (do) p: ( do,xo,) + = \Ktr(d0) {-n
r*(
ao,
:i, tiSI
llhus
l"o,(o)l
number
zero in
I = N(erx) [x
o.rH_,
swp(do) is
I€ma 2. If a,
S
-
erx + eJ
such
=
lcrr(oo)l *
t
Pl-r( tu'xr*,') r, ( ao'xo,)'
yr_r (do)
,=Hn-,
t@t-
x € s':pp(do) s'nd- e ) O then there exists one that for every n ) N ' pn(do't) has at reast
conrSract'
, ln Particular' A(dCI)
= ;rrn x*(cru)'
rtun
x-(do)J '
q ls constant on an lnterval A, then for every n , Pn(dort) A' haB no nore than one zero ln zurther, lf
2T
ORTHOGONAL POLYNOI{IAIS
Proof.
See Szeg8,
$5.1, ana l"eual, $I'2
c(x) + d(-x) = const , a(x) = const on (-1rI) thetr for every n , p2rr*1(doro) = o but p2n(dort) bas no zero t'n (-1r1) ' Note, that lf
I€@a 3. Iet
sugp(do) be coq)ect' lben
a(do)cIlnfo.-2 ' J:o '
J
6Esi*z "*], >o rJ+l i:0 u
$here 0J = ar(ac) and YJ = YJ(dc)
.
hoof. Let A = lnf c., , J:O
or . Ihen by (1)
.k, - A+B #
B
=
""PO[: >o 'J+1
J:O
t]I
=
2'' A+R' rJ(ac,xo,t
'^' - fJ \- jlo to, +
(2)
6uP
J
+
n-I v. 2 tkn .E # !3-1(tu,xu.) rr(ao,x*) '-' J=r 'J
l'* - L#l
s
+.'
;f
.:"1 >O rj+l
J
.
.
Rrt here k= 1 andlet tr +@. Ey Lema2we obteln
a(dcl)c(--,B+2 3
If ve ?ut k = n ln (2)
and
let n + o ther
"tPiYil >O 'J+1 we get
Y]
A(dG)c[A-2
sup;4,-) j >o rJ+I
l+. Iet supp(ao) be coupact antl let x be fixed.. ff for every e ) 0, o takes lnflnltely nsny val-ues in (x - erx + e) then there exists a sequence of naturel lntegers {knl;l such that I I kn < n and Irqma
(3 )
|$\n,,,{*)
=
*' }}q,"t*l
=
o'
PAI'I,
Proof.
Sr4)pose,
G. NEVAI
wlthout loss of generallty, that for everlr e ) 0r o
taJces
in (x - erx) . Let for everV n the nr:mber Jo te deflned by Jr, = (k: tm(e) < x < x*-r,n(dc)) wlth xoo = +o . I€t k, = Jr, + I. lfs shnl'l show that tkolt* satlsfies the lequlrements of the lem8. Because of I€@a 2 k- < n for n large. If we can 6how that lnflnltely
esny values
(\)
ltu \ *r.r, = * n-€n-
then
I1n4- -=Ilex{ n* a* krr'*
Jn'D,
=x
and by the Me'rkov-StleftJes lnequaIltles
h-,"
"
S
\-r,n J* -.]r +r.n
do(t)
n*o(x - o) - 0(x - o) = o '
n'
that (l+) does not hold' Ttten there exlsts an e > o and a sequence [nr) sucb that pnr(&rt) has no more than two zeros in (x-e,x) for !, = L,zr.,, . Because" o takes lnfinltely nany values in (x-erx) we Sultpose nov
flntt three polnts xLrxzrx3 € (x - erx) n supp(do) and by I€ma 2, nnr(dort) nust bave zeros near each xO for every I large' Hence Pn,(a't) can
has
at least three zeros ln (x- erx)
This contradlction proves (lr).
B be ffu<ed' Sr4Dose that for every e ) 0, o ta.lces lnflnitely tDarJr values ln (x- erx+e) Ihen for everXr c € R I€ma 5. I€t Eufp(dcl) be conpact a.nd let x
€
2 ,T l* - "l< lln sr4r lor(ao) d - cl + 'J'*' J*"* +i#, J* exists, thet^ in partlcular, if a = ]-f ",t*l
x € [a - z rrn s,rPF, tJ
J--
koof. I€t us ta.ke (kr]
3"o6 1,sl'm
a+2
b. Iet
rin "* "*]tl' J-- 'J
M
€I\l
'
Then
by (r)
ORTHOCOI{AL POLYNOMIAI"S
l\,,,- "ls \,,,i1 t, - "ln3(*,\,,). iT" lor - "l * * ,\,o
i.et a€M(e,b)
Iema5.
Foof. rf
I.|.lth b>0.
[a-b,e+t] * sulP(ao)
llien
Gx
.*"10
Lnl-rtu"'1,)
T
,"$
'
[n\[a-b,a+t] n sr+p(dc)l
fhen by Theoren 3.2.3
,'i # S..^
-(t-a
d.t>0.
2 A contain8 no EDre than one \Ir1 for every = I . I\rrther A c (8 -b,a+b) and by ?heoren 3.1.9
tbe othe! hantt, by
Elnce
'
[a-b,e+t]csupp(da)
[a-b,e+b] 0
then
contalns an lnterwal Ar . I€t a - \o '
\))
-
then M*6,
n+@,
filrstlet
hr-1(do,*1 ,.,) rr(ao,1u,n,l
il?
A n supp(Ao)
Le@B
ttn
).rr(tb,x)
nto-r{*,*) = o
ll.b
unlfonfly for x € A'
Ttrus
the left side of ()) converges to 0 when n * o'
Thls contradlction shovs thet [a-bra+b] c
Ttreorem
let ffn oa(dc) = a exiet. J*' supp(do)=AUB, AfiB-9
?. Let sr.rpp(do) be ccrq)act
*here A ls closed (5)
sr+vp(ab)
[a
-
and belongs
2 ]in, swr
and
Ihen
to
q# ,
a
+ 2rT-:*
t*;i$',
.
B is at uost
gcfinf o.-zurrp $, J>l 'J J>o 'r
sup 0.+2sr4r Ff J:o ' i:r 'J
.
PATIL G. NSVAI
24
If o ( M(a,b) then A ts the interval (6). Proof. (he theorem follows lruedlately fmlr I€@as I - 6. Ihe only tblng rhtch we have to show ls that lf o € M(a,o) then a € supp(do) . If a f supp(ao) then n = supp(dd) and hence B ls closed.. B'.tt B can be closeti only If B is flnlte ald then s has on\r flnltely
Eeny
points of lncrease, that ls o ls
not a veight.
rf xf,supp(dd) thenthereexlst e>o arti theorenB. ret q€M(e,b) N>_ o such that for everlr n ) N, pn(dort) has no zeros ln [x-erx+eJ . hoof. I€t, for stEt)liclty, a = O' S Theoren f, x f supp(do) t4r11es x f, l-trtJ (or x I 0 if b = O). Sultpose, wlthout loss of genet.allty, ths.t b ( x ( o. If x I A(do) theB the theorem 88ys nothing since x*(ctn) €A(e) !y fheorenl for every n and 1 SkS n Now 1et x € A(ds) n (b,-) n supp(dd) is ftnlte
(lf,-l
a.nci
1t ls not eilptv slnce x € A(do) . I€t
tt
I€t e)O
besuchthat
x + € < t, -
e
( t, + e ( .'. ( tr-1 + e < tm -
e
( t, + e
1
and [x-c,x] n supp(oa) = l. By I€rEIa 2 we can fintt N = t'l(e,{tii) such that for ever1r n Z N(e), pn(ert) has zelos in each [tr-e,tr+eJ (l = L,zr...,u). thus for n I u(e) pn(dcx,t) has not less than n zeros in ltr-ero). On the other h8,nd d ta'kes exactly n + 1 valuee in
i, \;,
1
of o(tl)' Further, [tr,') doeg not contal-n zeros of pn(do,t) slnce A(dl) n (t^,t) = /' Ttrus bv lenrna 2 for every n: o, prr(aort) ha,s no noxe than n zeros ln (x - e't)' Hence for n > N both (x - e,-) and lt, - er') contain exactl-y m zeros of (x -err)
i: 11
:l
t
li 1i ];1:
f,i"
Y1
lf
we do not count the val"ues
pn(drart), that ls
[x-e,x+c]
contalns no zeros of Pn(da't) if
Iet us note that vtthout the assumptlon necessarlJ-y hold' (See Renark lr'1'6 ')
cr €
n:
N'
t"t(a,b)' Theoren B tloes not
4,
tlnlt Relatlons Polnfidee Lilllts
\.f.
l{e begln
vlth a s14)}e
resu.Lt nhlch we wlLL not a?p}y
ln tbe follorrlng but
nblcb ls $orth recoralllS.
lfheoleB
1. For every welgbt o ancl x
(1) 1n
.i
l(=U
€R
4.r,u",*) rfltao,*)
partlcular, for everY x
!
[1 + o(-) - o(--)]2
€R
(oa.x) = o ltur r -(&.x) p -n n* n+-L' '
:
,
.
hoof, Let x be fixeci encl let B = o + D* vhere b* is the unlt tnass concentratetl at x . I€t uB etrllaJxil pn(&rt) in a Fourier serleg ln pk(dgrt) We have Yn( dp )
p,(da,t) = il&,t I rn\s/
, Rrttittg t=x
weoltaln
pn(dp,t) + \*r(dF,trx) no(ao,x)
y.(dF)
pn(@,x) + ri]r(ae,x) pn(dcr,x) ' p.(do,x) = f,17} Ir rn\s/
. a
.. :::
ry an easy couputation rrr*r(
i::
o D D. --P {*r(cb,x) p;(tu,x) 5 n[(ao,x) [1 + o(-) - a(--)]-
t ll:
.:.
:::
i'*.
and
"it 4*r{*,*) t-a) o-fettfne n + 6
rfltao,*) 5 rlr{oo,*)tr + o(o) - o(--)12
we obtain
(I).
,:
::
a) 4..-a
ti-, :::
!;
25
I
[1 +
o(-) - o(--)]2.
PAUT, G. NEVAI
26
I€Ma
Ihenforevery x€[a-bre+b]
2. Let 0 € M(arb) with b>O n
tin E \"(d0) n* k=I '--
e\
I€t O
Proof. Letrforslqrllclty, a(xSa+b
lltren
: *1,
anal
z ri-t(!':1n] -z "-z " L
(.q;I-
r
*-..fu.*
by Theoten 3.2'3 (ta.ke t = I;x - €rxl
e3-r(o,x*')
r')sn o2 - rdo.* rn-r\*''-kn/
)
)
,3^f
1 lln 1nf I '}n (x - x*r)z - etnbt 4-e lel tr-* t-f if If x ( a + b'the rlglrt slde ls exactly of order lettlng 5 + o we obtaln (2)' exactly of orcler "-L/z
,
rin \(do,x) {(oo,*) -
and the @Dversence
.
Proof
To
get
(3
"
.
.
x = a + b lt is
o
for x € a c (a-b'a+b)
t-';;
) use I€ma 2
D
at
Thenfor x€[a-b'a+b]
1lheorem3.,I€t o€u(arb) wlth b>o
(3)
ff-[]
a'nd'
the fornula
y?(da) nj_[x .
1
lo.(do) r2"-rto",**l *.
ro(oc,x) p;(do,x) = T,^, k=l Y;-r(d0) k=1
,,2^'-_ ,., )2 (x t* - xkn/
l-
1
vergence
-1,,- --r = : &(*.) rn order to shov that the confron )*rr-(tu:,x) k=Ir-816)-' agplyTheoren in (3) 16 uniforrfor x € 6c (a-b'a+b) we xtill
3.1.9 .
BY
which fotlows
th&t theoren
13
i'rr(acr'x) l2n-r(ac"") =
unifo lly for x € Ac (a-bra+b) (3'
)
o
Therefore
rtun ;)tn(tb,x) l2rr-r{oo,*) + trn-r(dcrx) rzn-r(aa'")l =
o
'/.;.::
ORTHOGONAL ?OLYNOMIALS
also holds unlfonoty for x € A . uslng the reorrrence forrnula we obtaln
?. v;
{(oa,*{ 5 chl-r(ao,x1 * Pl-r1ao,*11 for x € a(e) . slnco i.n(dc,x) < trn-t(do,x) ve have i.n(ar,x) n'nt*,*) 1 cf\(oo,x)
for x € a(ao) .
Hence
nl-r{*,*)
+ tr-r(do,x) rfi-r{ao,x)l
by (3') the llnrt 1sfsf,i6n (3) holdE uniforoly for
x€Ac(a-b,a+b). There are two posslble ways
to define the chrlstoffeL functions for
com-
plex values of the argument. l{e can elther put
n-]
^ r-(do,2) = t'^i* pi(0",r):-1
r-(do,z) =
"k=0-
It ls
easy
to
see
n'
that the
second
n-l
[ r lp,.(d0,")l']-'
.
deflnltlon coinclcles r'rlth
= *" \- l(, * (z - t) fi
nn-2(t))12 do(t) '
n-Z^J--
To avold. confuslon we shall write
fi{aorz)
$hen we nean the flr6t
definitlon:
h-l
r](oo,z) =[ x p;(do,z)]-r
k=0 -
I€t for z,u ( A, Kr(do,z,u)
.
n-l r^ pk(do,2) 4fd0,"l'
,
=.
n-1
kr(dorz,u) =_f^ pk(dcrz) pk(do,u)
hoperties h. tr. is real- valued, :.
:t:
b,'
r
monotonlc
ln n
and
posltive, trn 1s nero-
norphic wlth 2n - 2 Poles' -l r-'(z\ 'YI '
(z,u) = T-ffi] = u,n'(z,z) n ' ' '. Kn' '
,
T,.
*_ .-t
;(t)-:
Ii: i:
T.:
F.,:.: s::
P,i;
E F
V
':::.
= kn.G'')
' kr(z'u)
= kn(u'z)
'
PAI'L
2B
Ko(dorzru)
G. NEVAI
- " '*(*#rffi
kn(cbrzru) =
.l-
t*t*f
ll=I
',ht*'"'
lg).'
fwther
ril{ao,") =
(l+ )
rr",t*l lpo(e,z)l'qj!? tr=r Yn(qc/ .}. "n-'(*'to') ,, - -*JZ-
and
(r)
r'l{ao,r)-t = rfi{uo,n)
nto-tt*'1*l
,r,"t*l $s.l tfit*l ;1 "]so'-' (" - **)t
}le obtain t-medlately fron (lr) a,I:d (5) the following
theorem
5. tet sqp(do) be coul,act antl let z f a(e) .
r
1tu
inf
1lm
lnf lrlto",") n",t*,r)1 t
n*
)rrr(rb,z) llfito",r)1
ftren
o
and
o
n€
Foof. If
.
strDD(do) 1s co'qlact then
y = Ilrn sup
(ao) v_ 'h- .| '
TJ&T
<
-
.
stnee xkn(atr) € A(do) for every I = 1r2p... a.ntl k = 1,2,...'n
arld.
z f. A(U) the inequality
lr-**l-'Sc
for
every xkn
.
llherefore by (4)
:.iro sr4r lrrr(dorz) lln(oa,r) l2:-1
n.s
The geconcl
part of the
theorem can be proved
ln the
t cf
.
sa.lre wey.
) holds for every z f supp(ao) ' lf o(x) + a(-x) - const , swp(do) ls coryact and o(t) = const for t € l-erel, (e > O) then Pr**r(dcr,o) = o although o f, supp(oo) '
Renark
6. It ls not true that
Theoren
For,
?= i= -
(i) for everv z f, srrPP(do) rin tr*(d0.2) lin r;(dCI,z) D2(d0. l2n{0o,")
;ll
n*
w
{tt) for every x
V,
'4. 7., %,
=^
= -,
lin
-.
€ suPP(do)\a
lis rn(do,x) nl(a,*) n€ -(tlf)
}.n(a!x,z) l2o(oo,r) =
ll€
=o
.
there exlst tlto weights d ana d in U(aro) such that
l-rn rn'(dd.a) = -n' ' o2(44.") n€
. ^ -lr(dd,a) ,-a ' Iln lnf ns . Iln surr - l.-(&i,a) n' ^
o
2,.: ni(dd,a) = o
,
2. pl(&i,a) ^ =-
,
!l-E
If z ls corplex use (4) anti ()). Iet nov z = x be real and. x I sr:gp(dcx) . Slnce sqp(do) is coupact and o is constant in a nelghborhoocl of x , ve have Iil lrr(darx) = o ' sq)pose that there exlsts a sequence n€ rt < % < ... such that rlu r. (trr,x) rl--h {oo,*) < - . k* "h hoof. (i)
We
have t6r the recr:rrence fornula
Y,n-L Y- , , ^ +2n-I r t^ x rl'(x) = r- %p;(x) nu(x) pu..,(x) *#1"-.,(x) rn(x) A r Yk+f Yn k=o k=o ' wlth lr,(x) = ln(do,x) , cb = %.(do) etc. Iet M be a natural integer. Inen
M-I
t l" - ul S i..(x) '
lon
k=O
+2
k>M
lo*
- al ,
M-r YL Yk tio-r .- , ?, ,.r/z , \ ^ + -i= tr"(x) nl(x)J 'r2 r.(x) x + |llr(x) lu-r(x)l " > M 'k+.I k "w;5 'n k=0 'k+L
pt)i. n-n " - "! t flrstlpf. x =a .
- al p'*1x) *.sul
.0'@
arldthen M+6.
Ef Theoren 3.37, x € supp(da)
Weget
l"-.1
This is a contradiction.
SO thatis
.
1,1
:ll 'ti; ,:;i
PAI,IL G. NEVAI
30
rf x € supp(ao)\a then by Theoreto 3'3.7 x ls a Ju4r of o 8r'1 consequently (il) is true. (1i)
-'t-'ht (11i) r-€t & be definetl tv ot(dd) = a a^ncl vr(ad)/vr*r(ad) = (i + 1) for i = O;11'.. . Ttren d(a + x) +d(a - x) = const and thus lr.*r(dd'a)=0
for k=0r1r....
Hence
(6)
Itun rrool(dd,u)
p3r*r(d,u) = o.
K*
l€t us conslder now prn(ddra) . * t:: recurrence formula
l?Hnru(') - #rr*-r(a) Y2k-t Y2k "'
'^
Hence
.
Dv
lD
.1
=o
p;k(') = (ft1"'r!n-r(a)
(7)
ry repeating appl-lcation of (?)
we obtain
.
D
'
that for every J = I'2'"''k t lo ' rio-tr(a)
2k-2(l'\ zK'z f2i-ffi ,ffi-nhl-/-
2 , ' ,2k p2k(a/ = Ihus for J = Orlr...rk-1
n!*r'l 1^,ffi that
1s
c
p;k(a) s
rlrt'l
,
t6 k-l p (a) l2 ,-r, ^, = i h;t", ni,
/;
,:^
^/
te defined by vhlch together wlth (6) proves the first part of (11i)' ret d ,-^, 1 ,-i., Repeator(d)=a and Yi(dci)/Y1+I(do) =erq)(-(1 +r")2) for i=orrr.... lng the above argunent we see that (6) holds if we replace d tv d '
slmllar)-y to
(7
),
r!*(ai,u) = e8k-2 n!*-r(a4,")
'
Thus
fntaA,'l
:+iifto6,").
Ncnrlet k'@. Iet us renark that both d ana d are continuous at a ' Definition B. For " 6 6\[-I,IJ
we
define p(z)
bv
trlxther'
w. ir:.
#. Y
omtocoml
+-, E.' ,_ t, 1 :::
=: u;, "...
PolYNol'{rALS
prt)=z-,F; where we take
12 that branch of Jz' - r for which ll(z)l > I
,66\[-r,1J.
wehave
tinlz/o?)l =*' z*
ttnP(z)=-, ,*
1;
I€@e
whenever
9. I€t lzl
r . rt
>
-k J-tn5z-=a f,s
tben
1n4 n+J-.t a_ = a. ^ K K=U i4 Z proof.
f,he
tions of Toepl-ltz-silverssn' s I€mB 10. I€t a € R, b pontllng
where pk, = (" - t;2k-n-f satisfies the condl-
Eatrlx u = tflr]
€
theorem.
R+
to r = fa-bra+bl
and
1el u,
Then
denote the Chebyshev weisht corres-
for every z € 0\[a-bra+b]
rlro r,r(vrrt) Infl{,r.,t11 = | or'
tl-s
i
'r l2 - r
and
rin r.l(v",2) rfi(v",2; = o{' ; ")2 - r
.
ll$
hoof.
Use l€Deoa 9 and the formu]-a
Pn(vr,z) =
for vv=I,21....
u. Iet eve r:y
.LPi--/ ,z - a,n + Pi--/ /z - &\-nrr
€M(a,b) wlth b>0
o
z
f Jfr
Then
f. su!p( d0 )
I lro trr(dorz) lltrtu",")l = lc{'
n
l_
rrn
{{0o,")
l2n(oo,r) = o('
i
i
t)
12
-r
t)2 - r',
PAI'L G. NEVAI
(ri)
for every x € su!p(do) t:.n ro(ctrrx) 11s
(lfl)
,l{*,*)
=o,
the convergence in (t1) ls unlforn inside (a-b,a+b) '
are contlnurf z I o(o) then both l, - al-2 and (,'t)-2 ous functl-ons of t € a(do) ' fhus by (1+), (t) and rtreoren 3'2'3
hoof,. (i)
/A\
|rn
q(ao,z) lnfltco,")l=
r*
fffff
_l
,u*t
*,-'
and
rrn r;(dd,z) rfi(0o,")
(e)
=
J"-b
_-F uffIlt -l;-:;z--
rf z € A(dd) but z € sulp(da) then ve take c frm
Theorem
orJ
3.3.8 and Put
I l, - tl-2 Io
for lz-tl>e for lz-tl<e
[ (" - t)-2 f^(t) = i ' Lo
for lz-tl>e for lz-tlce.
fl(t)
=
{
,
satisf! the conclitions of fheoren 3'2'3' ry Theoren 3'3'B ana (' - t)2 by neither (l+) nor (5) nilL change of we replence lt - "l'' ff(t) and fr(t) respectlvely for n > N . Thus (B) 8lrd (9) hold. for every z I supp(do) . To calculate the 1ntegrel6 on the rlgbt sldes of (B) ana (9) and f,
Both fl
let us remark that lt is the Chebyshev
serne
for every
cx €
weight corresponcling to [a -bra +b]
tu(arb) Now
,
1n
partlcular, for
the
lre use l€xnoa 10'
(lf 1 If x € [a-b,a+b] then use I'heorem 3. rf x € sr:pp(oo)\[a-b,a+b] then by Theoren 3.3'7 cx has a juqt at x whlch iqrlies (i1) agaln' (i11)
See Theorem 3'
12' I€t supp(dd) be cottpact and let a €R, b € E+ = - *u a sequence tz*)ir such tbat zn € a t il: "u
Theorem
If there exlsts
33
ORTHOGONAL POLYNOMIALS
e -n-I'-(b..2. 'K )
'1ln -n' ' E' n.s a-JE-:l-
for
z. -
a
rK \-f = p\_E-/
then 0€U(arb)
k=!r2r...
g!. Suppose rlthout loss of geDertllty that zk f A(do) for every k ' have yn-r(do) ,r^.\ - nl-r{ao,**), (ro) z?n-r(&,')= rTd"-f ,t.. * : .\oi*; "u,-;=q-:
J,
trd",Ef
nhlch
ca,n
easlly be checketl.
c = c(supp(do))
I€t
d.(z) = dlst(z,A(dd))
.
Ihen ve 8et lt'ith
pn-r(do,L)t tr-+(T) nar, \-IrI ' . fr r_ wu\&k/ tr ,_z;)-r >T.(E-)-
,fu rj-a
Iettlngflrst
!+@ andthen k+o
weobtal-n
y_ (dd)
.
. ! ri' i"r {t-or
s
On
the otber
hand we have
k# < rr"kt . .,rtffir . .,t$ffil
where Cl_ and We
by the recurrence forcr'rla snd I€ma 3'3'1
Ca
n+o
depentlon sulp(do). Firstlet
v- . (ao)
get
&tld'then k+-'
h
s; ' 'Tj*$e -ffar obtain
Using agaln the recu$ence forouLe o,,(crr) = z1
-
vrr(ocr) Prr*r(ctr:,zr) vrr-t(do) pn-r(tu1"r)
q;I6I p'lc6,T - TJet- -leT-d;T- '
fhus orr(do) ls convergent and lettlng n + @ we . - z--a = ", - E otf,-) - i
]t;""t*l
Ttreoren
13. r,et o € M(a,b)
and
nrr-t(tt:rz) ffu -'?;:-5 ns -n'
see that
z,-a
ot{-)-r
=a
.
let z € 0 \sulp(do) Then for b=O 0 f ) )
L o(';")-1
for b>o'
Foof. If b - O then the theorem follows imediately fron (IO) 3.3.8. If b > O then by (l-0) anal T,heorens 3.2.3 anat 3.3.8
and fheoren
4o
PAW
t2n CoDrlrallng
-
G.
MVAI
L,r(dory2nrt) = pn(&,t)[1" po(oorx) + ,rrr-r(x)J
the leaatlng coefflclente lre see
(1)
I
t,. -*1.1 -n_r'-,--lr, a. nt-rtue,*")
kFI
that f
.r L=
= yn(do)-2 . CoDsequently
t+5J9. f"r_l
Deand (f - *2) nn_r(ilg,x) tn a lburier sertes Ln n*(oo,x) .*) rt ls
to
(1,- ' "-) wlth
&n_1
pn_r(,ip,*) =
= vo-r(ao)/vrr_r(ao)
r' - {) We
easy
that
Bee
pn-l(do,\)
=
sd
*i#
"r,*l_
n+r
*l_, a l*(aa,x) = -vn_r(ae)/vn*r(tu)
pn-}(dd,\)
.
rhus
H*in,,*r(aa,*n)
obtain from the lecurolon formula that n
r*r(dcr,a)
rrr-r(do, xu)
Hence
ft - 4) PuttJ-ng
rn_r(0e,1)
=
this lnto (J-) we obtaln
Yo-r(f ) yn-r(tu),-e ; "\ nt"-rt*r,.*l -= ,-," *" {-r(*)'r'. ,.5-r!o) Lq-Ir@)'* l---T(*t-' 'J' Fron d € s forlows F € s
and
the rlght hanal Blde ls 2 .
ry Iheolen 3.2.3, 1f e € (Or1)
;
118
t r-k,r-sr-"
{e
cen use lpme p
to
show
.
that the rlnlt of then
v(t) .n \*t*,t'r*lH' = i t.rl-.
.
Slnce by the prevlous calcul-ation *l
' fhis ergunent 18 alue to Chrlgtoffel- and Is given in fol]"owlng, thle argunent lrlll be u6eil severaJ" tlm€s,
Szeg6, C'hapter
3. In
the
ONSIIOGO}IAL POLYT{OMIAIS
41
l.e ger
tln t I. (akr) n* l* l>t-. an' ' kn' for 0 < € < I [-r,1] n
let f be
Now
an
erbltrary
Riernann
lntegrable fr:nctdon
on
}Je have
p'-(&.:C
r Lllll' (e)31*.Kn'1$9= -
rf=r_
=
v(t) dt
.E
r6(ao)
)
'z
r - \rD P;-r(e,xlnl) I-:C
l1,l5r-'
for o(e(l
o' (do.*
r(6) --__
E Ur(d0)tt.u)r-El, -L Irc__l>t-r \n 'lgt'
KN
Slnce tv2t"--I+e, ][
€J
le
)
Rleuann lntegrable we obtaln from
Theoren 1.2.J that
rle
;;
n'o-'(*'*o')
r 16l=r-€
-\"lrn/ ')s!\*/ \-(do) ,1o-y
a-4
-= 3n at-' J-r*"
t+
r\u,/ t, vr-L -
fherefore
,*oj*
tol,
\*r*r,,*,
Eop lf(t) I rrn 5 -1
f,
lr.. l>t-" 'tgl'
+?
\-(dd)
?
S;,(,)
I#t
s
pi-'(o'l*) r sup lrttl l!\ .,1t;at. = 1-:f "{-c -r
Iettlng e + O the theoreu follows. Theoren 3
rll-t
Defl-nitlon 4. I€t
be useal cv
to lnvestlgate
sone llter?oJ-ation processes.
€ S . then
' "r roew(1J:tosw(*) rt")=-#Jr 1$r r-x T-{ vhere w(1) =c'(x)
rre,
I,e@at. If o€S
then
1T$
-r<x
r
X = coss ,
.
U"t*,cos0) ^^6lcoss)Ern? - rf;.oupne - r(0)ll2ae= o.
PAUL G. NEVAI
5+
zo - h- -(cct.z\ | '
l: exists for z f supp(do)
anai
--f......ro-r-'l-n'
equals to
directfy be calculated thet the l-atter expression equals "p(' ; u)-] but lt Ls eaoier if we renark that zpn-r(vrrz)/P.(v"rz) converges to tbe sa.ne
It
can
IlnJt when r" ls the
Chebyshev
I€t us renark that
weight corresponding to r = le-bra+b]
fheoreno 13 could have been fuduced
.
flon H. Poincare's
faoous result on linear difference eguatlons [17]. Using Theorens
J, 11 and l-3 we can prove llnlt relations for
n2rr(ao,") as {e11' lle wirl not go lnto ).n+l(do,z) lrfitao,r) I "rra {*r{ao,z) tletalls, we onLy fornulate one result which we vi1l use l-ater' Theoreml-4. Iet cr€M(a,b) with b>0
andlet z€C\supp(do)
. + ,z_a\_z -) rim tr-+](d0,2) li(oa,zt = .1 - p(E Lema]5. Iet o€M(a,b) end' z€A\supp(cle)'
Then
]
for b=0
pl(d0,2) t -
1i5 --[--n. n+@
t
I
L
tt {\z-a)
--'T "tr"ru 7"2
to for z>r
koof.
for z e o\suPP(do)
we have
-
o
'
pl(do,z) p-
for b>0.
D ,. -----:=,-". = f-dCI lT " -il=x'.a6T .r
3'
OH|HOGONAL POTYNOMIALS
Iet
e
>O
and
cn
f(r) ![r
Theoren 3.3.8
tf
e=
for lz - tl<
\" L;-
=(
c(z) is snall
e
ror lz-tl>e' and n > N(z)
enough
then
D'(&,2) , n -t,=' .=! r r(xKll'(do)) npn(@rz) o k=l_ Uslng Tbeorens
t.2 8rd t.3
we obtaln
for b=0
ni(do,z) [ r(a) i];rp'n= 1 re"*b f(r)
for b>O .
L " )._l
and f(a)=(z-a)-l
forb)o,
and f(t)=(z-t)-r
for b=o
t € [a-b,a+b] . Ihe calculation of the ebove integral ls silqrle: put o = Chebyshev veight correspond.lng to [a -bra+b] Theorem
16. r€t o € M(&,b) end z € c\surp(ori) , fhen
.Pn-1(tu,2)-
ni-r(do,z), I t =
for b=0
jf 'L:70"7 qftu=f' 1,,ry, Proof.
for b>0,
Ftom
Y.-.(do) pn(d0,2) p,.,_r(dc,z)
i
olr
,-.p;-l(da,xln) lffi t*(*)
follolrs
",S# If b = O then use 3.2.3, 3,3.8
,
H*#,
=
"
##,3
Theoree 3.3.8 and Le@a
and I€rmE
.
It.
15' For b > O ve geb
+
*!,
\m(dCI)
T#
If b > 0 then use Theorens
PAIJL G. NEVAI
anci
thle integral
ha6 been calculatecl
trYoE Theorens
in the course of proof of
r'-(&.2)-. f ur-";i= I PP(@rzJ= P*
The followlng resul-t
rl'
lf.
13 and 15 we obtaln
TheorenIT. Iet o€M(a,b) and z€C\supp(ao).
Theoren
Theorero
Then
O
for b=O
Lz - a.-f Lp(--J
lor b>U,
is rather surprlsing if
lB. Iet o € M(arb) wlth b > O
x€supp(do)\[a-b,a+bJ
p - (do.x) -n-J-'
we coryare
for
Then
,x -
it nlth
Theo"en J.3.
every
i'i;"ra*r = e(--)
ii,
a.
t:.::
hoof,
I
We have
nrr-r{oo,*) = \
-
nrr-r(oo,t) xrr(oo,x,t) do(t)
.
If x € sryp(do) \ [a-bra+b] then by Theoren 3.3.7 , x ls an isolateal point of s,&p(O) . Hence, we can flnd € > O such that t{ ag
pn-r(&,x) =
.l*-iS
l>'
Uslng r\r\sr^,
..
rn-r(tkr,t) xn(tu,x,t) do(t) + --i-#=r) vrr_r(0o) pn_t(do,t) ln(ao,x)
-t
rrr|d)--x
'we
E.
a(xto).-o(x-o) . Yn-r(e) ^ rvn_r\s,^/,. -(ao.x)r, --iJafri--=,.
obtain
{i*-f
vr,-r(ctcr)
xtl Y|j' Wi'
"'W
- pn(do,t) lrr_r(do,x)
-J -:Tr"-'
ft I
=
. pn(cle,xr
pn-r(do,x).
C
pn-r(&,t)pr'(tbr,t)
-
l*_)lr...r::#(t),
6.
nfi-rtoo, t
./Td'1- J ;-1." lx-t l>e
) oa\!J ,
,
5l
ORTHOGONAL POLYNOMIAIS
We beve
r:.in n€
(See
tteud,
rc -n'(da,x)
Srectlon
II.2,
=r
strpp(do) ls coqractl).
.
by [heorelo
Thus
l+.2.13
I 0 for n lerge ancl p- , (e,x) iun-= n€ -n'
ti
c(x+O)-o(x-o) -------1---79;1An\st^/
* SII ," - t)-1 rt2 - (t - a;2J-1l2 u' ,:\"a+b t1x - t;-t it2 (t - a121-L/2 * tr Ja-b
whlch equa.L6 Theoreu
,x - a' P(T-,
19. I-et o € M(0r1) and , be a flxed' nonnegatlve integer.
Then
n-I ltr! rn(.b,x) ,x^ pt(tb,*) p**r(tb,x) = rr(x) I{=u n€ for each x € [-Ir1] provttlecl that a I's contlnuous at x I ln partlcular, (ff) frotsa fe1 elno6l ever'lr x € sugp(do) . If o is contlnuous on (u)
1
c (-J-,1) tfren (If) ls satisfled unifor:nly for x € r
hoof.
.
RecaIL that
lirn )trr(dorx) = 0 n*
at every x where o ls contl-nuous antl the convergence is unifo:n on every lnterml of contlnuity of o slnce sryp(do) 1s conpact. (See tr?eutl, sectlon II.3. ) If !' = L t tben the theorem follows fron Theorem 11 8.nal froB the forsula
n-I n-L x - trn(&,x) _x^ nn(cb,x) nn*r(do,x) = trn(alr,x) .r^ a(oo) p;(da,x) kFo k=o "
n-l
y_(do)
l=0
'k+1\-'
+
+ tr,(do,x) - r- t2 V-::1EI' - Il p*(tlo,x) pk+I(tlr,x) -
"
v'l-i;r' - (tb)
whlch
ls a rlLrect
by Tbeoren 3.1.1
: $i
- Y-(@J p,(do,x) po(ih,x) -n-r'
1,.,(do,x)
of the recurrence fomula.
let
consequence
Now
X
> I . Ihen
PAITL G. NEVAI
3B
n-I
n-1
tr.(do,x) - E^ pk(do,x) pk+x(do,x) = ur-r(x) i.n(do,x) . x^ lo(oo,*) pk+l(t!lrx) " k=o " k=0
n-l - ur-r(x) + tro(itcrrx) Jo l*(tb,x) L*r,t*1(do,*) Since Ur_a(x)x - ur_r(x) = tr(x) vhere lt holds wlth I = I
:in ls
a18o
satisfled.
n-1 n_ro
no(oo,x) \+.c,k+1(&,x) =
ftnish the proof, ve 8.pp1y 3.1.(3)'
l1*n,**r(oo,*) |
for ,e flxeil
that (11) holds at those polnts x
e,rrd where
rn(do,x)
To
we obtaln
:' eo(x) tln*(o,x; | *
o
We have
llo*r(0o,")lJ
where
lin eo(x) =o h+- " unifornlyfor x€l-fr}] n-l
l:KU
p,.(ar,x)
rnus
. "-*, , 2," x el(x) ri(oo,x) &-.,,-,.(do,x)l< K+.f,r 1(+1 - k=o t
wlth
*_
J-un €klx
J
n* unlfornly for x € [-fr1]
Consequent\r,
n-l lln ),n(dorx) ,_r^ pk(dorx) \*1,,x*1(&,*) =o K=u n.s holds for everlr x q l-f,IJ for whlch lirn i.n(dorx) = 0 and the
congergence
n.E
is unlfom for x € A c (-IrI) for x€A
whenever
lln Ln(dorx) = O ls true unifornly
Il+6
39
ORIHOGONAL POLYNOMIALS
4,2.
Weak
Llnlts
Definitlon I.
We
wrlte 0 €
I€@42. If o€S
S
lf supp(e) = [-r,1] and vlogo'
.
then
llnv(oc)e-n= 'n' n* , tteud,
. .l r + \ r "'atNa
hoof,
See e.8.
Theorera
3. I€t o € S anii f be
"
€ Lt(-1r1)
v(t)
roe
o'(t) dt].
$V.5.
Rienann 1-ntegrable on
L-fraJ.
tnen
d.t
a1+\ r\e/ r(*r,') "rr_r(*;f*,7-t r. J-1 _r I - &r =3q, l3 J, ^/f (r - *2) ' proof. l€t F be deflned by clF(x) = 1f - x2) e(*) ' Then . tll_r{aer*) - yi-1(d6)*2"-21 is a po\monlar of clegree 2n-f B.Ird we have by
lr!n(da)
the Gauss-Jacobi mechanlcal quadrature formula
nr)
nlr(r-x[)tni-r(as,x*l
-
"l-rfuel.fl-t:\
(here xo = xkn(dd) antl \ = )fr]r(dCI)
tr
J,-,r
-
= r*v2n-rtoe)
S-,,*-r) t2'-2 oo(t)
Thus
{) p;-r(de,\) lo = 1* (-rtael tsl, t' c. )
do(t)
-
jr{' *, '
trUrther
I zn. Jr\ \ Hence we
=
:-
^
;r'''(aa,v2n,**) \
=
^t S-,
Ln(do'v2n't) dd(t) '
obtsln
\ c. , x (r - 4) ni-r(ae,1) k=tr'-r^!J-I
2 L = ] + vf-rfael \t- t.'" -
Slnce tzt - Lrr{dorv2nrt) ls a polynonial of tlegree 2n zeros
of prr(ao,x)
we have
Ln(dd,v2',t)l oo(t) whic}r ve,riishes
at
.
the
42
PAIIL
Proof.
See
G. NEVAI
ceronlmrs, Chapter IX.
fn the follor.ring, three applications of
Ttreoren5. If 0€S
we shau a!p1y l€ma
t
severaL tlmes, bere we glve
1t,.
then
^I lt' \u-l. p;(do,x) d[os(x) + or(x)J = o ns
Proof. ry the when n + o. Theorcla
Rl.emann-I€besgue
rema
and. I€r@a
7. Iet ct€s ana f€r[,
, !t
.
nlf *,x)oj"(x)a<
Then
, -l^ ^r r(x) p1(ao,x)ao(x)=r\ r(xi rtun\ n tlrG n*J-t ' hoof.
Use
Theoren B.
I€@a
t,
dx
lheorem 6 ancl the Rieroana-Iebesgue lema.
I€t 0€s, o(p(-, r.ie lnr n€
.r
^l_
\'--L
lp.(u",r)
s(:o)€tt.tt lP
e(t) do(t)
=o
then S(t) = 0 for eLnost every t € [-1rI] P"."f. I€t first
2Sp<-.
Putfor M>O sM(t) = 61n{g(t)'M)
Then B, €
,,;
i,
";
. n'ther
9'2 a,cl pnroo,E, ,.,P/n ..,-..,.2/n p. sM(E, '- oo(t) S t\^l In"{oo,t)lp s(t)do,(t)]elP . 5o1r) - o(-})l \r-l r-r ''
ff the hypothesis
and Itreoren 7
k
*
?.8
FEr'
Ell
Ii: &] g
F: p:
forevery M>0. r>O.
Then
Hence
!i q,t'r'ln
g=0,
[---7
I€t now f
Let e*(t)=g(cost),
l+3
ORTHOC'ONAL POLYNOT!,IIALS
-f ^1 e(t) d"(t) t w-I b.(ac,t;lp e(t) d'(t) dt = l! \r-l lp.(oc,tt \ ^ft
=
\o
Inn{uo, cos
t ) ^,io'(cos tfsin t
sr(t) = [o'(eost )stnt ]t-F **(r) . nl f\ b,.(ao,t) lP s(t)do(t)lrlp -J-1
lY
..- 2 - g+,(t) l0'(co6r)61ntJ
dt
Then
lprr(da,"o"t ) !ffTA;T)€rn-T;l u.1rh
> .J l€ \
=
g, >€
,_
"r/e
, \ tf,
cos(nt
-
r1t1y1r1r/r -
8tl'
- "'/p I f lnn{*,"o"r ) ^,F(6;'TJ6fiT - jl-? cos(DL - r(t))1n ur;rh slnce p < 2, llo"lp, l"o"l2 and the by I€@a t, {hus by the hypotheslE
t:. : .l: El
secontl
integral here
converges
to
.
o
I
*;
lln lnf
F. t::
\ "ou21r,t ns s1!'
thatiB neas(gr>e)=0,
,i Yr
O
=
( conslder -
.)"t tp
€ f:
a:,
and.consequent)-y
g=0.
If
< p < I we can repeat the previous arguBents, the only dl-fference is that ve
v
tr
Thus gl=O
- f(t))dt = o,
lnsteaa of {'J"-1 |n1r/r
.
rJ:
L-
F,.',
9. I€t o € M(a,O) Let tU anal tU be two sequences of natural. lntegero such that at lea.st one of them convelges to 6 when k * o , If f
rw
is
nr:
t::: :.:
t; :.. w
I€@a
contl-nuous
on A(dc)
then
W.a
6 f(a)
w',
w,.. via
-:
ur (- r(t) p- (&,r) p- (d,t) dCI(t) Ks"-@ x K
2
hoof,. In the flrst ffi w w
nk = \
:::::
b.. ffi
= k for every k .
I Lg
li-E (nK 5) =
kx
tf lirninf In_\l k* sr4tpose wlthout loss of generallty that
Because
of contlnulty
we can also srllpose
is a polynornlal. If f is constant, the I€@a ls certainly true.
v; 4::
case we can
=
lf
r;i
o
,0.
thet f
Otherulse
PAIJL G. NEVAI
eerr(l]t'l
r(x) = r(a). We BhaLL shov
(2)
thet for every J >
ac(x) = o
v-) do(x) =
*-
.
If J=2
i*%t*l=".
then
pk+l(&,x)
(oi(o) - a) p1(o,x) .
Hence, o- , - e1'{1oc,x) ,2 .2 5--,*
.
1
(x - a) r*(acr,x) tffi (* ')2 nflt*,*) = +
")J
'J
(* - u)J ,flt-,*) 1* J-kx \
If J=1,then(2)nea.nsthat
(* -
I
J=l
v.-
+
.(o) pk-r(e,x)l
jf*T
,* -')2^ *
z
Y-
f
'
t'
;
Slnce eu!p(dd) is coryact, (2) hottts also for J > 2 lf lt hoIaLB for J = 2. Ilhe seconal caae caD be obtalneal frcn the flrst one as foLlow6. If k ls ).argerthen %l\.
fhug
^- r,., p\(do,t) p\(e,t) e(t) = - r(a)l p%(e,t) t_- ttt.l S-
pnk(e,t)do(t),
that ts the absolute nalue of the lefb side is not greater than
- r(")lnl {uo,r) n1t1}/z - r(a)lpl.k (o,t) e(t) ,\t\J-o lrtr) J-- lr(tl \ E
Here
both factors are boundetl
a,ncl
at least
of then tenals to O vhen ks.
one
l-O. Iema 9 rerains trtre if f , lnsteatl of belng contlnuous on A(cb) , ts nere\y borural€d on su;lp(O) , contlnuous 8t a anat lt 18 dc
Iheorep
'/,
urabLe.
Proof. I€t c>0.
Then
t e$e do(t) < r &(t) S \Jg,-e ni(o,t) e(t) r[{o,t) \^'J-D
where g ls contLnuous functlon vanishlng outsLde fa- era+ eJ xrlth
g(a)=r
an.t o
for la-tlSe
' fhusbvLe@a9
meas-
rili
+,
OXTHOGONAL POLYNO}IIALS
li'\ l(€
^a+€
{{ao,t)da(t)=r.
v8-e
We harrc
l\ r(t) p;(e,t) o(t) - r(a)l t
srq' lr(tl - r(a) I
.
It-al<. €
.\ ^a+e p:(do,t)o(t)+2 Ja-e o I€tting flrst
k. c
a,ntl
^ lrttll ir - ^4.* \"9'- r[{a,t) t€srrpp(da) e
then
e
+O ne obtain
l* \ r(r) {(e,t) o(t) l(.s v- o ttle
case
o{t)J
s,rp
vhen I{l lnf ln* - \l k+o
= r(a)
.
t O fo}J.ovs fron tbe
ease
when lJ. (\-\)=0.
k*
for n = !r2r...
Deflnltlon 11. Iet us tleflne the nuobers o*(&)
arxd
k = n-lrnrn+l a6 fo]-].ows
o*(do)
for k= n-l
rYn-r(do)/Yn(d0) =J
L
a"(ao)
ror
k=n
vo(do)/rn*r1*1 for
k = n+1 .
IBma L2. Iet n be e nonnegatlve lnteger and. n ) ro .
xep.(ao,x) =
t
X
-1
''',tt..^
d . l'arv) *n\'n\+k2 a *t'o*\\*/
on+5+. .
.
(ary\
\*/
/ad\ do) pn+\*., -
+Kn-l, n+\+. . . +xr(
koof.
AppLy
Theolem
i3. Iet o € M(erb) wtth b > O . Iet (\l
quences
of natural Lntegers
.
( +k tu,x)
the recursl"on fonnuLa repeated\r.
such
that at least
wben k+6 ancl the finlte or lnflnlte .:=
.
Ttren
one
(\-t) ]tu tf,*
ard (\l
be two se-
of then converges to
@
exists. I€t f be
on gufp(tll) antl contlnuous on [a-bra+b] . Tlhen Tl^ -r,. l(+) . (3) 1r'\ f(t) p- (dd,t) p. (da,t) do(t) = Ir^: ^a+b f(t) \ -L*. -t '1r k* u-c k* " ra-b JO, _ (" _
dd
Eeasurable, bouncleil
"),
PAUI
t+6
G, NEVAI
Froof. Iet, wlthout l-oss of generallty, o € M(0r1) Flrst we shall prove (3) when f 16 continuous on a(dCI) ' rf lln (\-\) = o then the rlght slde 1! K.s
f ls a polynomlal then the lntegral on the left'sLcle of (3) equal-s O if k is blg. fhus for every contlnuous function f the left a -. Ihen we nay assume side in (3) also equels 0 ' Now 1et ltu (\-t\) K€ that for every k, \ = k, \ = k+J where '0 ls a fixecl nonnegative integer' (3) equals 0 . If
Because
of ltnearity
and continuity argulxents;
that i6,
because
of
B8!ach-
is of the for' f(t) = tltr where ls a fixeal nonnegative lnteger. (hus we have to show that for m = Orl,zt.,' r nI - T'(t) ndt . ,tB p*(tlr,t) pk+t(do,t) do(t) = i (\) ]i'\ \ , # k-EJ-o Jt-t I€t us lenark that (l+) ls true lf o ls the dhebyshev welght. For lf k > lI ,t nr(*,t) pk.r(do,t) &(t) = \n "o"te '|t"o" tB +cos(2k+.0)0ltt0 \lIiL.]^ e_6
Steinhaus' theorem, we ean s\*)pose that f
which equals to
1 a1
=\ tr J-1
\)) if 2k+ X>n'
(^6 a.m \v-o n*(*,t) =
tn
T"(r) a'
[--=
m
dt
If o€M(o,I) thenbyl€n&a12wehavefor k>n pn*r(
-r.i,.r - r-
=
tt,t*t1(tu) th*q,n*n
*5(dCI)
"' orr+r1+"'*tt^-r,k*/(&)
!=LrZr... ,v m
E kr=4
The
rlght side bere ls
fixed, lts llnlt
connergent when
ilepencls
k + o slnce
cr €
M(0r1) and l' ls
on\y on rfn or(da) and lirn v,-r(oo)/vr(do) , o j+6 u-
that
l*
thls lintt equals ()). Hence (\) holals if f is contlnuous on A(e) . If f ts contlnuous only on 6upp(do) which ls closed then f can be extendeal to a functlon which Is eontlnuous on A(do) If f ls a function satisfling the contlitlons of the theorem then we can write f = ft + f, where ls in our
fl
case
ls continuous on A(do), fZ j-s bounttreal and dcr measurable, further te
\7
ORTHOGOML POLYNOIVlIALS
vanishe8
on [a-bra+b] .
Hence,
lf ne can show that
(!- c. p;(do,t) do(t) = 0 Iir \ k* "t*O " then lre finlsh the proof of the theoren. I€t g be contlnuous functlon A(do) suchthet s(t)>o
for t€A(do),
, : "a+b \ s(t) "a-o
and
lsglven.
nhere e>0 na-b
* 5_-
n6
on
for t€a(do)\fa-b,a+bl
s(t)=1
dt<€
vo -(r-s,
Then
c
it
)
p;(dCI,t) e;(dCI,t) do(t) s 5r*o \_- u{t)
dCI(t)
fact that the theolen has al-readgr been proved. for contlnuous functlons ve obtaln (6) by first j-ettlng k + o ancl then e + O . and uslng the
Using one-sided appro:
14. Iet o € M(arb) vith b > O
on sr,rpp(do)
ri'\
11m J-o
and R:lensln lntegrable
r(t)
If f is ds neasurable,
on fa-b,a+b]
p2(d0,r) do(t) = l \"*o Ja-b "
,(.)
"
bounded
then
---A
IZ--(r-al ^ v0
1
,
Conq)are
t
this theoren wlth Iheoren J.
-
-;
coroflary lt.
I€t o € M(a,b), b )
in e neighborhoocl of x .
t.
:* n{ lt'* ns
,:';;
't:i,.
z
[ -
r)<-n
n2n{ocr,t)
.',
t::=
= li:
.= i:'.= Viii:::
'/i= Wu
W;:::
hoof. Ihe firnctlon
x € (a-bra+b)
anal
let c be
da(t) = *
1
'f_.AT ) 0
- _-.-, is tlcr reasurable for (x-e,x+8.,
1r__
Theoren16. I€t o€u(arb),
b>0.
e
sna11.
I€t f beboundedon A(da) Rienann J.ntegrable on [a-bra+b] . Then for every fixett integer !, n
rir t '-' r(x*) n* .k=l \*(a"l
contlnuous
Then
-J+h^
1,.:::
O,
nrr_a(ao,x*,r) nrr*x(o,1rr)
=
anal
\B
PAI'L
=
P"ggg,
We
G. NEVAI
^a+br(t) Tt. FG- (t (H) ^/b* ulrl-r r!-:lt -sis,, ^ \"_o
at.
#
obtaln fron lheorerns 3.1.3
('0
> O) anai 3.1.13 (, < O)
and fron
the recufience fotaul-a that lL
pn+r(do,xkn) =
-stsn, ulrl-, (+)
+
dt)
-a
+
tfurr_r(ao,5.,r)| + llrr-.(ao,x*)|J
1:,,
unlforrrlyfor 1
:.::i'
Itreoreu
tr, ,
pn-r(ds,\!r)
if
vbere fino(1)=O
I lsflxed.
fhusthetbeo-
t,,,,
a:1
:-
::1-:
"ii;,'. j : +r
17. r€t o € s, f be
lln . t-
n€
lr=L
\cn(tu)
f(x*)
I
#J-
;{l
,
W::,
w'
W wi
ffi tff
F
v
It w
K
E' trt Ft
V.:::::
ft'
EJ Yt. &&
ffi W w,E.
Wlt
M
tr{
&i
.
and '0 be a
P,,*r(tkr,xor)
-ff
Foof.
Re?est the proof
of
-sign, *
Theorern 16
=
r -
r nl
t
17,,,,
ln_r(oo,*1,,) =
tu
on f-Irlj
fixed lnteger. Ihen
,fl,::
ar;;).
RieEantl integlabfe
S_,
a.nd.
"lgl
r(t) ulnq-r(t)
.
ft: "-"
use Theorem 3 lnsteaal
of
Theoren 3.2.3.
5,
Slgenvalues
of ToE)llz l.tatrices
rtr crenander-Szegd the proof of rheorren.7.7
anal
the exaEple B.l(f) are Dot
correct. rn the ftrot the Gaus8-Jacobt nechanlcaL quatlrature forauLa ls for po\romlals of
tiegree
rcre than 2n-1, ln the
ten 16 constnrctecl, but Lt i6, ln fact, on\r sectlon,
of
we
second, an orthonornal sys_
normeti
but not orthogona.l_. In thl6
wlr! obtaln results nhlcb are a llttre blt
Grenantler-szegd, ve w111 uae so[!e nethods
used
roore
general tban tho6e
of the above book in a sirnplifled
forr, I€r@e
1. IEt supp(do) be
the Botlulus of contlnulty n
115*10"))l.ik=L
coEpact antt
ul ,
n-I.@ "r\u-o_ tttl k=O
Ihen
{{*,.1 e(t)lS n u,1n-1l3, tr * }latoll3J
for n > la(ao) l-3 . n-l ^ n.e2(t) hoof. since r e;(t) = j, ff n
r(.k")
n-l
-;t;
^6 S__
e
)O.
l.rl
ana )t* =
r(t) pi(t) tu(t) =
"=1
I€t
let f be contlnuous on A(do) with
3
!-
df
rl *trl -
r(t)]
^,[,,,,*)
we have
{ffL,,,
then
<no(e)+
I br
t
lr(+l -f(t)l
e2(t)
f'
oo(t).
1t_{12.
z
< n ur(e)
.
F q#t Yn
*1,
\
ni-rr\l \- nlr.l &(r) =
Yn-t u,( la(do) l)l - r.../-\ . =nr0\€/+T--J Yn n€ ,ii: qy easy cetcutatlon, vrr_r/vn < lalao)l/a .
4;
Ttrus
hl s n [o(e) + | lot*l12'(la(B)l) \g
t.
=o"
t'.?:,:':L
:"j'
.j '
i",iii.:: !at:.4:
,o
PAIJL G. NEVAI
tl==t,
Now
let , = n'r/3
and use the inequall ry
.;a:;aa.'
or(a)
1l4i:1:::
,
l:i::.:,
which holds greoren
for every
roodulug
^
(o
sG)
of contlnulty
ur
'
I-€t f be boutaled on A(dd) ard continuous at
2. I€t 0 € M(arO)
aa:
:::
:::
a.
Then
:: :::
!l+6
proof. Tf f ls continuous on a(do) then the tbeorem follows i@ediately froE Le@a'1 and I€@a ]+.2.9 . If f is continuous only at a and lt ls bourd.eal
on a(fu) then we flx
tions ft
e>
o
and we construct two continuous fi:nc-
snil f2 on A(dd) so thet
rr(x)St(x)!rr(x) ror x € A(do) and fr(a) - rr(a) < e
and' we
fZ the
theorem has alreaqr been proved''
Ttreorem
3. I€t
cr €
Rlenann integrable
M(a,b) wlth b > O
on fa-bra+bJ '
use the
fact that for ft
a$d
L€t f be bounded on A(do)
an'I
Then
; r(x'.(oo)) = l" f.o r(t) *ri'+ "a-b so- " k=t
"rt,
ln particd-ar, for every segnent o c
dt
,
_ (t - ,)=
a(do) i+
Foof. If f l.s continuous on A(do) then use
Le@a 1 8Itd. Theorem )+.2.I3,
otherwise app\y the one-slded spproximation roachinery'
Nowver'illtrarrslatethepreviousresultsl'ntoadifferentlanguate.I€t be real vatued and let us consider the Toeplitz supp(do) be co!4'ect, t a { natrlx A(f,@) deflned as A(f,dd) =
,tS: f(t) pr(do,t) pJ(do,t) o(t)lli,i=o
'
OMHOGONAT PO],YNOMIAIS
1et, further, An(frdo) be the truncated. tratTix consisting of n2 el-eeents. The characterlstic polynonlal hn(frdcrrx) ls d.et[An(frdo) -xE] , the zeros
of bn(fr&rx) , vhlch ve clenote tv x*(frdc) , (k -- L,zr...rn) , are caLl.ed. the elgnevalues of Ar.(frdo) slnce { = on aII x* are real. Tf f(t) =1 then A(f,dd) = E alld hrr(frdorx) = (1 - x)t, that ls 1 for k = L'2'...,n' tm(f,,dc) = a 4. I€t f(t) = t . Then for n = !r2r.,.
hrr(f,do,x) = (-r)n vir{oa) pn(tu,x)
Proof. (-1)n hn(fr&rx) satisly the sa&e recurrence foruula as y;-(dc) pn(dorx) , and for I = 1r2 the lema calt easllv be checked.. pefinitlon t.
and tbknltl , (n = L,2,...; Eh €n; b* €lR) ar€ t\"1t, equally d.istributed if there erists an interval A such that "k, € A and '\r, € a tot n = J-r2r... and k = 1,2r,,.rn, fi.rrther for every continuous function f on
d
un n€ We
].1$=].
ttt,u"l - r(bkn)i
=o
.
obtain frorn rheorens 2 and 3 the following
Thenforeverypairofweights Theorem6, I€t f(t) =t, a €1R, biO. ^ ohd ^ f-^h M/a b) the eigenvalues of A_(f,do,) and A-(f,do. ) nr" equelfy clistrlbuted.
,,n Definltion 7. I€t A = [[a.rJJ, a_. be real n xn metrjx. then n
TrA= I \u, l{=l-
,ta,
,
llAll
= l; r" Al ,
lr :'
((A))2 =
here s = (uir,.. run) , (u,v)-.f_\tt, k=I--
.w
9*#*
,*.J, u.,t:1,..
further
,
PAUT. G. NEVAI
,2
6 = riqrrl,n=, hopertles B. Tr
AB
= T" BA , Tr A = E (elgenvalues of A) ,
n ^ ...-P > !ra:( .t- ulr. , ((A))' - k=I,zt"'rn J=l
((AB)) S ((A))((B)) '
If A* = A then ((A)) = nax lelgenvalues of I-e@a
.
Al
9. For everv n € tr{+ . sup((r-(r,o111S -t€swp(dc) .lr(t)I
'
Foof. Let r be an eigenvalue of Arr(fr&) for wbich ((Anf,dcx))) = lll anil let u be the correspondlng elgenvector vith (uru) = t ' then ((An(f,do))) =
lll
(u,u) = [1lu,u)l = l{e,'{r,oo)u,u)l
n-I
^@
pL(tu,t)uu)z rt.l (t = l\'k=o-A 'J--'
ottlll
6up.
=
.lr(t)l
(u,u)
t€sugp(dd)
where u = (u'r\r...run_l)
Io, Let n € N be flxeti ond l-et ft (1 = 1r2r " ' rD) be glven' fhen for every J € (1r2r...rloJ m m [--'-" um Tr A.(f'dc)ll lt'/A"(4,dd)l' rtu. sup ll"i=lS .Trr! - tw,. ,lrr(t)l ' J tr ri=I t€supp(do) ns"'.rp n* Llt
Irr@a
Proof. For
m
rrc can sl4tltoBe
=I
the }ema ls certalnly true' Let n > 2
By
kopertles
that J = I . I€t B
Then
llnn{rr,ao) Bll =
=
rT A.(f.t,do) L=2 "
.r D n bJxl 1 t j=1 l*" k=l .r- %.r(ryoo) -"
s r*;rr,!, s
*,.,rfl.,"
{rrr,*) ,i, o3*r'/'1
,ri
r!*11/2r*
=
e,ri {tr'ao) )'/21 '
B
ORTHOGONAL
POLYNOMIALS '3
!y
Besselrs lnequalltY
n^ s !
n-1 oc /f eL{\_trg4/ !
.l^\
p.l(do,t) fr(t) l\J-o pk-r(&,t) - -
t
_ -
.t-o
Tbus
by ProPerties
l[n{r'o)
fri.l *1r; = 1*({,aol
n2*-rtoo,t)
s
do(t)12 S
.
B
t-'
((B)) S
n11< Ven{t'r,oc);1 1
m
lfqrFlll .rr ((An(rl,da)))
.
Nor use I€ma 9
I€@a
IL. let swp(do)
be colq)act, s
rtu
lh;('{,dCI)
-
€N and r
be a potynoniel.
An(,r8,do) ll =
o
Then
.
n+@
Itoof.
See crenanaler-Szegd,
0B.l
.
Iet us renark that in the prevlous ]eme it is sufflcient to suppose that for dcr the Donent lroblen ls werl d.ef1ned, that ls fo" *kf , xkg € L2do (g = 0r1,...),
A(fe,da) = A(f,dd) A(e,d0)
12. I€t o € U(arb) , f be tlo measurable If b=O end f iscontlnuouss,t a then t-rln lvAn(f ,d0)ll = lr(") | I€r@a
and boundecl
on
supp(do)
ll-€
If b > o anct f ie
RleDann
lntegrable -
rrn l[,6 rt'.o)ll = tl
ns Proof.
ia-b,a+bJ then
d\
{to
,lr, - (t"),
1t^
f!/z
-
See fheorens l+.2.10 and 4.2.1-4
Iet us recall that valued. functions f .
vre conslder
Theoren13. Let o€M(arb),
supp(do). Letfor b=0, Rleroarur
nc+u
\
on
.
Toeplltz matrices A(f,do) for real
5€I{,
f be do
xneasurebleend.bounaledon
f becontinuousat a endfor b)0,
integrable on la-bra+b] .
Then
f
be
,\
PAUL G. NEVAI
lin llA;(f'do) n*
An(fs'tu) ll =
Pfoof. Let I be a polynomial. Since Tr
-
An(fs,do)ll =
llt{(",tu)
-An(n",dc,, -
llnfi(r,o"l =
llaitr-r +r,do) -
o
AB
= Tr M , we have
An((f -,r +n)s,ao)lJ
=
.itll{,tr-,oolefl-J{n,ao)*on(-.i.,tl)tr-")J"s-J,oo)ll
=
J_:
= lhr * orr * Arr, 1\ . If b = 0, then ve shall
Let, for slnpl-lcity, b > O we can
put r(t) = f(a)
see from the proof that
By i€@a 11
rinllqlJ =0. n*
g
Theoren 4.2.14
(1) rln n*
We
lle,,-,ll
=
i t:l trrrl - n(t))i . r(t;s-j1 td----::-ll" \'.otj=1 "a-b -(r-a) d
t.
^lo
have, f\tther, by le@a lo
rim sr4,
n'*
lhrrll< rr" - i-t!l
tt'"* i6t1r-;1*)ll'
3=t-J n*
ln(tlls-j . -,op lr(t)-n(t)li-l-y tcsupp(do) t(supp(e)
Hence, by I€ma l2
rin
sup lh--ll rr
n.*
and
fron (l)
c : ,^8+bt< 1r\- ir(t)-t(t)l-u.1e - 'r .'a-t
,fl[],f
we get the sa.ne esti$ate
th
n-4
.ts sup ( lr(t) l+ln(t) l)"-I tGupp(dc)
for ffu llArttli
= n(frr )
,
:
llArirll < n(f,'r)
will be proved if llle show that for every e ) o one can fincl a polynonial n such that R(frr) < e . Ihls ls,tter can be shovn easi)-y' I€t f, be a function on A(do) such that f1(t) = f(t) for t € fa-b,a+bl , The theorem
lrftll s lrr(t)l for t € a(do) and fr e L-(a(oo)) . let us send' A(dCI) to t-f,11 by a l1near transforrnation and. thelr to [O,nJ by x = cos0, (-1(x(1, o:e
tu,
ORTHOGONAL POLYNOMIAIS
even extension
Wt
of g to [-nrn]
coslne potynoxdals, thev are bounded
they converge to g* ln e'g',
W
,,;
of g* ' {hey are ln nexlnum norn: llon(s*)ilsaaloi**llrr(t) l'
consider the Fejer sums
tlo[-n'nJ ' I€t us return
now
to n(fu)
artil
reEark that
bounded on supp(do) and 14. L€t o € lt(aro) ' f be do neasurable' at bounded on 'l c f(supp(do)) antl continuous contlnuous at a , ret 3 be
Iheoren El+
f(a)
Then
j, :tj i
Ti
f$ +.' T": 1"
proof.
observe
thet lf
e) ) = s(r(a))'
f(swrp(o)) c A then x*(rrdo) € a for
Tttls canbe shownWexactlythe
and k= Lr2r...ra. reroa 9. I€t flrst
s(xkn(r,
u{u) = us , (s = o'1'2'"' )'
true. To]| s=l
vv
= L'21"'
saDe argurnent as
For s = o the
in
theorem
ls
I xr..(f,cle) = tr An(frda) ' If s >- 2 n-
l+'2'l-O and lte apply fheoren
then
i {.tr,ool = rr A;(r,do) k=I --' is true 13 and\'2'IO ' Hence the theorem as is weII known and'r're uge rheorens lt ls t]ae lf U is continuous on A' lf J ls a polynomlal, and' consequently Otherdse
,we
use one-sliled alproxlnotlons'
I€t f be do Deasurable'boundfheoreDIS. let s€M(8rb) l'tth b>O' on [a-b'a+b] ' I€t 3 be contlnuous ed on sugp(da) and' Rle:lsnn integrable on A=f(suPP(do)) '
Ttren
" 'a*b 3(r(t)) lin=r It 5(x'.(f,do))=?\. "e-o tr-----v (t a)" k=]n* " Foof.
The sarne as
Jil -
that of
Theoren 14'
-
at
PAIJI, G. NEVAI
Theorens 14 a^nd
L) give us the follolu-lng
Iheoren16. Iet a€lRrb>O and f becontlnuouson R. Thenforeach palr of velghts q. *d q2 belonglng to M(arb) the eigenvalues of An(frda1) and An(frctx2) are equal1y distributed.,
Cbrlstoffel l\:nctions
6.
5.1 An lrter?olatlon
ProceEs
Tbe Hermlte-FeJer lnterpolation po\moela.I
HD(fu,frx) is the r:nlque
polyn@leL of tlegree at rcst 2a-l whlcb satlsfies tbe condLtlons
r1(cb,r,xor) =
for
k=
L,2,.,.,t.
Here fur =
n^
\(do,f;x)
r(r*r),
=
I€t us c€q)ute 4(x") .
,jr
\n(do) .
Hence
tt"",l tl - 2/kn(dd,xkn)
(x -
1*)J
lfu(aa,x)
.
We have
r
n
rlr(x)= x -
that ls
r{(e,f,x*rr) = o
t2_(* I
+, k=l
'"kn
n zt!&\ r^6) ^ E *'i'tn * r;(x)r;c(x)= k=I
.
Puttlne here x = 1rn we obtaln
(1)
-2t(xk) = r,;f"*)
Itlus
H.(.b,f,x) "k=l--'"s
n
= x f(\rr)
fi
.
D
trkn(dCI) + Li(ao,qor) (x - x*])J .
A-__(do,x)
ft6l-
ftrls ls lteualrs representetlon for Hn(dorfrx). (see [5]). In the brackets here we flnd an approrduate oqrresslon for lr(dorx) : r,r(x) = \,,, + rn(xkn)
1."( e)
(* - l*) - #
(x
-
1*,,)z
where 0 ls between x enci ar, . I€t us replace the oqrresslon Ln the brackets by trr(ibrx) . Denote the resulting e:qrresslon tryr Fn(tbrfrx) :
PAIIL G. IIEVAI
>o
Frr(tb,f,x) = trrr(ao,x)
For z€c
l
,**",
+H'
Put
jr r,-*, # -. , llfito","ll r("kr/ Frr{ao,r,") = [n(dorz) TGAr tr'n(d',r,z) = {tao,")
anai
L
.
k=1
(See
l+.I
.)
(li) rf f(x)>o for then Fn(frx)=1' hcDertiesr. (i) rf f(x)=I (i11) Fn(dorf,x*)=r("xn) for x€A(do) then t,r(f,x)>o for x€R (use(1))' (v) (1v) fi(oo,t,xOr)=o for k=I,2,"',n k=Lt2t...rD, Fnlsaratlonalfunctlonatdegree(zn-z,an.z),on.Iythenrrmeratordepentts on f
.
veights o, Fn(dl'f) coni's that the verges to f {henever f is contlnuous' The surprlsing resu'Lf consider convergence of a.bove class of velghts o 1s very large' We shau In(dorf) for o € M(erb) wlth b > o since for our purposes the case when Because
of (11) $e
cen expect
that for
Eany
we o € M(arO) ls less interestlng' In ord'er to avoid coDQlicated fornu]-as p(z) shal]. assune, without lose of generallty, that o € M(orf) ' concerning
see
Deflnltlon l+.t.8
Theorero
2. I€t o
tlnuous at
soEe
€
.
Let f be bounded on A(do) ' rf f is
M(ort)
con-
x € sugp(ac) then l-in Fn(tkxrf,x) = f(x) n.€
(2)
.
IffiscontlnuousonthesegnentAc(-1rl)tfren(Z)issatlsfledr:nlfornIy for x € A . rf f ls Rleroann lntegrs'ble on [-]-r1l and bounded on then
for every z € c \
supP(do)
Un In(tLr,frz) !1"s
=
2,,.^1 [--7 otz)4 \.,_1f1ttv._"rtit -nr (z _ t)_
a(do)
OBTHOGOI{AL POLYNOMIALS
F}- t,u - r [- rrtl ri.::! l:l f,'2 rinR(d.r,r,'; = le(')Jl 1i zft J-r lz t- - Ll' n.s
61
.
|
fron Theorems 3'2'3, 3'3'8, L.1.U aDd Properties I. Eoof. fhe theorem follows Then I€t us prove e.g', the flrst part of the theorem' I€t e)0. t'"-t -z , . * 2i# e-' trn(& ,*) p|(oo,*) ",tp lrftll. r(x) sl4r r(x) l lr(t) l S lr"(r,x) '' t€a(do) " Yn [x-t
lce
6grclthen e-O.
Itlrstlet
1+o
Definltlon
3. I€t
s(
:O)
€
ffTheoren4'1'LL (e) fottows.
Lt ' fhen ou ls
o-(t) t5
a"
cteftued bv
= \J_- e(") oa(u)
.
I€t us retrark thet og nay not be a welght, 1t
cen happen
that either
rrU
hesonlyafinltenrrmberofpointsoflacreaseornoteac}inomentofcruls finite. If S is a polynois,l then og certalnly ls a veight' If supp(do) -] udo then also o -'S is a weight' ls coryact a^no g .' -f l,eruna
c,
I
\.
o).
I€t g be a linear function, nomegative on supp(do) (e(t) = crt+crr Ihen
l-r.{dau,*) = j,1 .'n
(3 )
lfit-,*)
a."@l-6fO
Proof. (ry freud t?l). Let us denote the right We heve to show thet for every rrn-I
of (3) bv A '
nl-r{')uo*{t) nl-.,{*)SA\ tt-r D-f J_-
(4) and
hand slde
for every x €lR there exists a rf,-t which turns (l+) into equality'
have nn_I = l,rr(dortrn-r) '
,2
nf-rt*l
: jr
We
Hence
n \*(d0) .i-r(x*) e(xo,) '
nechanicaf quadrs'ture Sl-nce aeg r2rr-t g < 2n-1 we c&n use the Gauss-Jacobi
to obtain
PAIJL G. }IEVAI
60
"l-rt*) On
=^
the other hand $e
do(t) = n2o-rttl es(t) S*"tn-rt.) s(t) ^ S: can
*
..
.
:o_t(t) =
nn-l
I
W
I {*(dCI,t) rt
(dc,x)
j, aqrarr6car
5. I€t g'trpp(ec) be coqract a(do) . I'hen
I€r@a
.
and
let c be
d-r(*'**l s zftl', 4"t*l s l"-fu.| -v'o_rtool
k=1
one
.
of tbe en@olnts of
lc - o,,(ao)l
.
Foof. (ry freua [B]). since prr(dcrrc) f o i ,- . p'rr-r(e,\o)= to Po-t(ib,c) ' j, \"t*' " - -* :pfo-F)-
""-
Frrbher sIglpn*I(tlarc) = stgn fn-r(ao,,c) .
lc - o,,l Inn(ao,")t
-
#
Thus
by the recunence formul-a
In,,-r(ao,")l
.
The lema fo]-lows fron the above two fomuLas.
6. r.et o € M(0,1) Letg(t) = crt + "2 , ("1 / o), be nonnegative su!p(do) . Then for every x € srrlp(do) I n'(tb9,..x)
Theorem
lT ffi-T=*t'.r
and the convergence
Proof.
We have bY
ls unlfom for x € A c (-1rI) '
Iema
4
i.r(dr:rx) _I +F (do,g -rx) \n( &grx ) =
.
Tf A is Poaiti.ve on n(aa) we can directly a!E\y Theorem 2. Next let vanlsh at one of the endpoints of A(dd) whlch ve alenote by c ' trtlrst show
that
tr-(do-rc) =o {6"u-t n.s n'
un I€t e)O.
Ttren g+e
ispositlveon
(=
e("))
A(do)'Hence
.
g we
OFTHOGONAL
POLYNoI'{IALS
I-(do-,c) I-(do-, ^,c) osjffiSffifr,;
e(e) +e
61
=e
.
g-1 lsbou::cled. If x€supp(ckr)\c thenforsnallD>Q In [x-b,x+6J (.oa g-] ls unlfonory bounded ln a c (-1,r)). wrtting g(t) = A(t - c) ne have
Noslet e+0.
lrn(do,s-1,x)-g-11")l=l x + t ls l*-*ol1o l*-**lto
.-
*g(xl
Bax e-r(t) * It-x l:b ,
. x lx-x
.* + Yn
r,,{*) r'o{*) '
I
le-l(*) - s-lt*o)l trnl-rt**l l>o 2
.
rrDJ(
g(xl' - =s "'-
e-r(t) * u-t li
r'n(x) l2n(x)
e-r(*) *
lcs -? - *2 /-\ * A-1 b-2 $ nlt"l .; 4--lx*-cl k=r 7-t'^.; vi ^"t*l Yn
lt-"
ff l,emq t we obtaln
lrn(ao,s-r,*) - u-t(*)l
In;r
:
e o e-1(*) BE:{ lx-t $o
w-
* u-2
Lr(*)
Y'n
{(*)
Ttris estinate and rheorem 4'1.1-1
*-t(t)
e-r(*) * l-1 o-2 irr(x) 6hows
nZrr(x)
lc - onl
that
lin r' (itr,g-l,x; = g-r1x1 n*tr for x € supp(do)\c and the convergence ls uniforo for x € a c (-IrI) Lentna
?. I€t u € 0\(-r,I)
and z € 6\l-1, IJ .
o2(?)-tS:d*u.=;t
where n/z*-l->O
for z>I.
Then
-F;f,,
ro-I(,,)
- o-rr,)l
PAtJl G.NEVAI
62
hoof. Beca,use of continulty arguoents we ca.n srqrpose tlrt u € 0\[-],11 since t 17 r C' ^/r - t' *', "^= I
; J_I-T
z
I u
ancl
ffi
the proof of fheoren l+.1.1-3) we have
(See
cl
ffi =\ ;J-., Ifif@'r
r-
..
Dlfferentlatlng thia lclentlty wlth respect to z Theoren
.
"--1,(ur - P-r,,,, at = \z)r ' P-n; tP
B. I€t o € M(0,1) I€t e(t)
=
we obtaln the lema.
e(t - B)
be
posltlve
on
A(dc)
.
for every z € A\supp(do)
Then
ii{ao,z)
l.1na;-1 =s_r,, -r p(z/6S-I{r). tp-l(n) - o-t(r)1 , \z)+42rT---,,d n* trn'(doa'.z) ln partlcular
r_(do,
B)
Hd#=#_,) hoof. ry Iema
l+
ue have r
*"/an
on'l'-
'\
trn( Clog, z )
^l
^
= T^(&tr8-L,z) '
Therefore Theorem 2 ylelds
*
ur. ,-p= = f zlt -, Je(t)(z t ,4T,u n* q{ao*,2) - t)z r:..l
Thu6
the theoleE fo].lc'ns fron Irme 7.
I€@a
9. I€t g(t) = A(t - B) , (A I o) be norinegatlve on supp(do) . Y;_1(en)
T r*) Proof.
We have
rn(y). pn_1(d0,8) __1 = -I Y;tr&)'-eTdo,Bi r.(tnrx)
= Lln +(@ x€ An' A'.XJ whlch equals by l€ma
4
fhen
OFTHOGONAL POLYNOMIAIS
liB xs
:-4C;;t?
JlTe'r=
=
=
-Iro2,ri*(oo) ni-t(oo,\) Et
-ir
Y-(tu)
Ln(dc,,pn-r(do),8)
",,.-ft
r -(m.e) - v(da) -n' %fddTf
-II fnTdt)' -n-1'
I€ma 10. I€t 6
Iheoren
'
on A(tu)
be posltJ.ve
and
t 2 ,r/z ()i iT Yn_l(doq) v"Ttr = rA eGt'l S
' eTelE;=
Iet g(t) = e(t - n)
0 € M(o,l)
Irhen 0- € M(orl-)
Proof. If
=
J,
ts positlve on
= ur.p(
A(dCI)
, AI
-af \u-rto*e(t) '" L
.2
-gva-t
I
then B 1s outside A(do) .
.
Hence
by
l+. 1. 13
(6) Afrplfrine leEros 9 we see that the equauty on the consequent\r
v -(do ) n€ n' g'
'n-1"' -'a ' r]'&lGT=t.
left sile of ())
hofais and
I
Putting o = Chebyshev wetght we have o € S and aU ( S (tet us recafl- that [-1,1] c A(dF) for F € M(0,]) and hence g is positlve on [-].rrl )' Uslng t-e@lal+.2.2 ve obtatn the right side equal-ity 1" (t). Now we have to show that
. I€t us tlevelop
S
l1
0n(d0s) = o
.
pn(dcrg) into a !'ourler series in pn(do)
It ls
see that
g(x) p,,(ckr*,x, =
;+#
pn(do,x)
.
^
*ffi-p,,*1(&,x)
easy to
6\
PAT'I, G. NXVAI
Hence z, , z, . c \ e-(x) Pl(do-,x) ao(x) = #"lt*l r'
J-o
6
v-(a )
.z + a-
vflt*ol .
-:-ev- -(aa)
fhe left slde equals
(- n(* - n) p2(ao.x) o (x) = aa (da ) - AB. -n s' n' 8' J_s' fhus by le@a 9
on(&e) = u Bv
v-(ac)
p-,i(.b,8)
"ffd) freFr
(5) Ilnorr(oa*) exists n*
os € M(o,l-)
(
and
p-(dc,B)
+
F;fr4r,I r .
equols r - ]totrl * p-l(g)J = o.
.
g(t) = A(t - B) is
lL. I€nea 9 and the proof of I€ma 10 Bhcn that lf posltlve on A(dd) and 0 € M(0,1) then Renark
s(z)p-(d0-,2)
l}-dlt-"f-=F
^
.
L/2
lFi1"1-l
for z ( CI\g,rpe(0o)l\{nl = [c\supn(dou)1 \
tB]
4.1.U. give a
new
This reDark
and. Theoren
I€@a12. I€t o€M(o,l-) fhen
cYg
consequent\r
€ M(orf)
rP(")
-
proof of Theoren
I€t g(x)=(x-e)2+32
P(a)l
B.
wlth A€Rru2ro.
and
Yn(daa) L J/2 eryr -*r ^rl6;1ri;6;cl'l-/' (7) }} +fe =z = Proof. Iet us develop
C
c
S-,
roe
e(t)
1ln(dog) in a Fourier series ln pk(e).
Y-(do) v_(aa prr(do,x) * dn*r pr,*t(tu,*) * (S) g(x) pn(do*,x) = -+; TI;6$ n g' n+z' -
7br
We have )
pn+2(do,x)
Unfortulately, we cannot dl-rectly cal-culate do*l , Iet us note th&t g(n + fs) -
n
q6h^a
v-(oc)
qibJ u6
pn(do,A+iB)
Consequent\r
*
dn*r
!n*l(b,Ar
18)
Y-(do^)
. qffi-
P,r*2(&',n1in) = o '
=
ORTHOGONAL POI,YNOI4IALS
-cn+r =
(e)
T" vitoot Y;(es)
yn(dd) no(tb,l + tn)
TFJ
vo(oo)
pn+2(tlf,rA + 13)
t]-tu;T- lT')- - G;fd)- 4fdo',i.TB)
vn(tb) lr,*2(dorA+1B) !n+2(turA-lB)r - | pn(turA-lB) pn(tbrA+lB) ,-1 -4;F,j-:rBl,'L{I@E'I-{;Ida-ffiET, h+2\s/ Pn+lr
-=-,-L=--@f I€ttlng I + e
ahd uslng Theoren 4.1.13 ve obtaln
#tat 11' lnt*r - p(.q-rs)l . Ip-r(n-rn) - p-l(e*rr)]-1 *4I * n* vt(& ) = *tp(e*rn) 'n' s'
p(A+18) p(A-18)
left slcte equality 1n (7). fhe rlght side equallty 1n (7) foll-c*rs fron lrula I+.2,2. Now ve 6h8.11 shorr that for every z € a \swp(&) -
rrhlctr proves the
=
c\sugp(ur), (zl A11B)
(r0)
If (10) holds then by
Theorem
l+.1.12 og € M(0,1) .
.vn*r(e) =ifd,T,reJ Yn(dos)
vrr(tb)
tve
obtain fron (8) ena (9)
-
prr(da,n11n) !n*r(dc,n+1n), pn+t(dd,z) pn*r(do,z) .
,vr*r(tb)vrr(&) (--8,) eJ@TmT-
ry
(?) and Tlreorero
(A
I
Bl
!
t1,/z.,-,-, . - p1e*iu)J[p(z)- p(A-iB)]. 31-Tb=fi- = ihftsmffryp/' tp(,)
(1I) Nolv
il;@T;r6l'''5;roa]l-*5;@-,21
l+.t.t3 ror z € [c \swP(do)] \
g(z)Pr(dcrr,z) Ir
.,
(fo) follows fron fheoren 4.1'13. Hence og € M(orI) r.et us remark that by Theoren 4.1.13, (Lo) holds also for z = A + iB
r.enna
13. ret o € M(o,l)
Let e(x) = (x - A)(x - B), (n / a)
on sr:pp(ao) . fhen ag € M(0r1) v(oo)
iTiFfu
=
.
be positive
,
.r r/a '-: -n ( 'r/2 = e:
I
roe e(t) .at J-L ,C?
'
'
ob
PAW
further for every z €fc\supp(do)j \ s(z) pn(dog,z) rln ----;--d;;T"*
G. NEVAI
[A,B)
l_
Proof. Ihe proof of I€ma 12 car be repeatecl. Note that ArB F [-Ir1] but nay belong to A(do) . 1l+. I€t o € M(0,1)
let g(x) = (x -A)2 vhere A € a.\supp(&) , that is A nsy belong to A(dd) . Then og € M(0,1) arti Ierma
Y-(do-) (ra) iTffi
=
, rffi-l
Proof. Ihe proof of Lema e(A)=S'(A)-0. y-(do)
qqT ancl
and
=
12 has
to
be
modified.
We
.
Z#r
have (8)
a.na
Hence
pn(do,A)
*
y_(dCI_)
dn*r pr,*r(&,A) +
y_(do)
q*J
-*, ^1 S_r'.u sttl
"'at
ri(oa,a) n d,.,*r p,l*r(tu,4)
.
\:f& v_(oo_)
qffi
pn+2(do,A) = o
p;+2(do,A) = o '
fton here
v-(oo) p-(ao,n) y-(ds-)p-,"(co,l) ___ji_ ___ji_____5_ TdCI? tTd"T-* \*;fdl {fdo;TI
-*n+r- _= __-_
r1?\ and
y;(do)
rul *8, 'n. Now
=
vrr(do) . p;+2(&,A) pn+2(do,A) , .. pn(tu,A) nj(do,A) ,_1 'L Fffaa]-I 4ilfda;Tl'r
Y;@l ' ef;I@a-'- - %fdcfTl '
(12) follows from Theorems 4,I.13, l+.1,.16 and4.t.r7, f\rrther fron
\.2,2. Ustng (B), (12), (13) and Itreorem \.I.I3 we
we obtaln by
the
sa,ne
l.€nna
way as
ditl i"n the proof of l€ma 12 that
/rL'l
e(z) p_(d0_,2) , 1' I LP(z) "'2 il'--g'-' = l'2 | ;Al r,- "'-'PJdo-")-
f,]
for z € c\supp(do) \(al
P(A)r
which together with Iheorems
4.t.u
and
\'1.r3
ORTHOGONAL POLYNOIIIAIA
shows
ttrat 0g € M(orl-)
Theoren
I€t 0 € M(orI)
It.
N
e(x) = e be posltlve
on 6upp(e) . Tlren qg € M(0,1) , v(do)
,
$;ft tl.s n rin : and
a/
tT
Y
= "rat
\\g,
-+ \^I --r
rog g(t)
for every z € [a\swP(dCI)] \ tful
-l
:l
lg-*H,]=;h "*'-*
5-:
g(t)
r.os
' -1, "'] :
T?)
Proof. RePeated. appllcation Iemas lO, W, L3, I)+, of ras (Lr) ano (rb). Deflnltlon l-5. I€t F € S n(dp,z) =
for lrl < r rlopertles
Tben
the
Szegd
*"*t* S-
Rernark
11
e,rrd
functlon D(dFrz) ls
p'(cos
t I ' f-t'{
+*' of
folrnu-
defined' bv
atf
.
t?. I
e ut(
l"l
<
tin
r), for aLoost every t l(ag,relt; = n(d6,e
€ [-r,n]
tt)
r+I-v
exlsts end ln(ap,ult)
12
t ) for s'f'most every t ( l-n,tf t D(dF,o) > o. (see e'g' [!eud, Ctrepter
= P'(cos
I o for lrl. r, RecaIL that v d.enotes the Chebyshev welght'
D(dO,z)
I€@eLB. Iet ts€s,
z€o\[-]-,]l
rm Pn(aP,z) n-s ancl
P(z)-n
See
e.g. Freud,
I
=
-r-t-1,-1 spr P\a/ ^r---I,^ | !\v
-
the convergence ls unlfona for
Foof.
fhen
$V.5.
lp(")l :. R > r- .
'
V')
PAI'I,
6B
G.
I{8VAI
N
I€@a
z€
19. I€t s(x) = A fl (* - L) k=1
be
positlve on [-1r1]
f'lren
for
c\[-r,1]
(1t) n(e,p(")-') - 2N u,.nt* S-t, *-
ftj
'J,
#-t
It o = Chebyshev wetgbt a.ntt use I€EE8' IB' [ben for \ tn*] , (rt) rroras a,ncl consequentlv tt holtts for everv
hoof. Put ln
TheoreB
z € [a\[-r,r]l z € c\[-1,1]
Eft)
.
I€t us note th&t - beceuse of contlnulty argr.uoents - (I5 ) bolds lf g(:r) = o, further (1r) hords for z = o !f g ls only nonnegative on [-r'r]' f,lreoren
20. Iet c € M(orl-)
a.nci
supB(e) . lnren eg € M(0,1)
let I be e po\ynonlal- vhich ls posltlve
antl
Y- (d0)
Ih Vfut .n. n€
= n(s,o) ,
S
P.(cb,z) .-:l llm |;:: :r = P(e, p(z)--) Pn(a*rzJ 1* for z € C\swp(dd)
.
hoof. ry Theorem D antt I€@a 1!, the only tblng vhich tbat lf g(B) - o then p (ao.s) 16 "?)i'A = n(g, p(z)-l) ' (15) n*'n' A' L€t 6 > O be sDaIL enougb. llhen for lz - fl = O (do.z)
!r t-;H# -n' g' n*
we heve
to thofl ts
= n(e, p(z)-r) ,
that ls by theolen 4'1.13 (da.z)
D ,*{-tl*iii ns -n. -
= p(")-b(e,p(r)-r)
A-
f,or lz - al = s . since l,n-t(tu,2) = Lrr(tld*rro-r(do)rz) we have
rh#r \+# t ; =
u,*-) bn-,(ao,**,)l ' b,,-,(og'*u,)l s
on
onTHOcoNAL POLYNOI,IIAIS
y _(i!a)_ y1_1(eq)1" th ^o ^ f .- 'l-lll$' tus&))L/z r[ ot_r{ao,t) -J-o -n-L' S " ; - - T*,. , rs(t)]r/z. Yn(@g) € ltr. s' - +T6-f
s
r,
r8r4lp(qf)
for n ) N nbere e a.nd N are tleflnert W fheoren 3.3.8. Slnce both ro-r(tu,2)/ro (&g'z) and p(z)-l o(e, p(")-1) ere ana\rtlc 1n lz - al < o lf antl n > N lre can applv Csucby's lntegral fornu.]-a a.ntl lebesgue'e theoree D S. na ebout fln fn = * tn and ne obtaln J J
t*'*-+S'll lr.us by Theorem Notr we caD
lltreorem
easl\r generallze lbeoren
Then
€
B.
M(OrJ-) Let g be a polynonlal rrhlcb 1s posltlve
for every z € c \suprp(do)
n*
r-*
^llt'"i trn(cbrz)
rrn
I (.b .z) p(r)-1) P''=s: A tqg.zJ = lo(e,
and
n.s
hoof.
.
u.r.,rr,olrt;:."'
21. I€t a
srryp(d:)
= p(a)-r p(e,p(s)-I)
Atrrply Ttleorens 20 antl
= p(e, p(,)-1)2
n'
12
4.1.u.
(0 < r < 1) in (rl). rhen 22. Iet us put p1r)-1 = ""1€ 10 L -1 -10r --J + -e z = ;(re-- + r cos 0. ry hoperties { for alnost every r+l-O I € [-rrrr] r-(dd-rz) lin llF -ffi s, a = e(x) (x=cos0) Renark
'rn\
1+f-Q 1*
/
whlch suggesto
=g(x) hoperttrr23. I€t z=relg, In(as,")|2 = (See
o(r(1.
r-r "mt| -" \"
e.g. Ileutt, Chapter V.)
,o*
(-L<x<1)
Then
r - "2 u at) I - 2r cos (9-t) + r-
g'(cost,
PAI]L
7o
G. NEVAI
Iema 24. I€t f be Rlerlsnn l"ntegfable on
t [-1r]l
and.
1et f(x) > c > o
beflxed' {henforeve:y e)0
I€t o
for x€A.
A
ttrereexlsttvo
suctt tbat
polynonlals r(I and t(2
$S"r(*)
8nd
lir(r,z)
for l"l I
n
hoof. Iet
lt (, -
<
e)
lo(r,z)12
5 lo(nr,')12 {r * ')
.
e>
0.
We
construct a polYnonlal rr, such that
(x € A)
f(x) < rrr(x) srd
(n [n^("o"t)-f(cost)]at<e' J-,. '
(see szeg8, t.5.
)
Then bY
lo(,rr,z) anal
FoPertY
12
23
S lo{nrr-r,'112 lo1t,'112
by Jensen's inequafitY
n
lr(nrt-r,z)l' 1"=r"io;.
r n( no(cost) \_.
:*
f,"*il;;ff;t-
t - t'
ut
Hence
r- c" t!3t r-, - .r *2; -.rp l- S lD(rrf^-1-,2)
)
-l]@
.
t"
- "' _
ou, (l+consr.e t) + r-t fo" Since f(x)>c>o )_"
x €A.
(See Sze88,
I.5.
t
_2r cos(0
)
25. I€t o € M(orl-) I€t c(: o) be do neasurable' di o" Then for bountletl on supp(do) 8,nd' g be Riens'nn integrable on [-1't] Theorem
everlr z ( a\supp(do)
(au.z) \^it*';i r* n€ n'
= lr(e, p(")-1) 12 .
ORTI{OGOML POLYNOINA],S
hoof. ftr the assr:4rtions l,r(do,z) =
a
7L
ls a weight, sl-nce g-I
*" \ l{r * '--
(n-2
1"
- r)
rL
€
nn_r(t))
Reca'l1 tbEC
l' u(r)
.
Ihen fr.on do 5 dF lt follovs that ).n(do,z) S In(dBrz) for evenr z €
I€t us construct tvo functlons ft Rlena.nn
and f2
Euch
that both fi
and f,
lntegrable otr A(dd), fr(x) = fr(x) = e(x) for x € [-lrlJ 0
( c. ( f"(x) < e(x) < f^(x) ( c^ ( IJ a I
a
&re
end
o
-
for x
€
A(atc)
le(")-11 <
and n,
r
.
.
thls W Theoren 3.3.7. I€t z € C\supp(do) lhen I€t e ) 0 . Then by Iem" 2l+ we can flncl two poJ.ynorolals n, We can dlo
ttrat rr(x) 7 cr/z for x € a(do) , "r/2, "r(x) 7
euch
r.(oo* r_(dd ,, rJ-,z) -< r.(do_,2) u g' < - n' fiz',z) and
(1 1[hus
- e)lo(n'r(")-1)l':
lo(e,p(")-r)12 S
(r + e)lo(rr,p(r)-r)12
by fheorem 2l
tp(e, p(z)-1)
NovLet
t'
* s ,_*j", +H=-f s ,Tj* +H
s
1f
lote, p(")-r)t2
e+O.
Iheorem 26.
I€t
aJrd g be as ln Theoren 2).
o € l'{(0,1-)
.*
Y (tu)
= D(8ru/ n€ VTA-J n' g'
and so
''ns
v -(d: )
+-i;q
r
-
=+.
*e'ijtrtr
- ,..'i:",.i;
,!"'
'
ti' \t.'.'
:.,i,
'
hoof, Reca'rl thet fron tu ( dB follows vn(06) :: vrr(da) . NJ rr" ."""r, peat the proof of Theoren 2>, th.e on\r tlifference is that this time ve alply fheorem 2O instead Itieorem
of Iheoren
21.
27. Iet d € M(Orl) and g be as ln
rheorern
25. Ttren oe € M(0r1)
r
.
IAU! u. rw vAr
a,
hoof. If
we couLtl
clirect\y ca-lculete on(dag) the Proof
nlce. thfortunately
we caronot do
woultt lrobably be
thls. tnt x f, stryp(&) .
Then by fheoren 25
)\rr(doo,x) rn+r(ttlrx)
lTr"FF'i;te-*-r=1' r*t*t**")^n(*"")-r. n* 1+li{u,*)
that ls
\(ac,x)
fbu8 W llheoren
4.l.IL
13
n2"(aor,*) )'n('bs,x)
-
P(x)2
-r
-eg
o(*)-1
'
UB{ng lfheorern 25 we obtaln
l3J, \."(es)+#=#;= for x f
aqp'p(do)
.
get By lebesgue's clornlnated convergence theorem we
o
.,
l-ln f, \.( n-*lel --.*8'
nro_r(uo*,*or)
x-xn-
? , =3 [, flT = P(x) trJ-l x-!
u,
If f tgeontlnuouson A(dd) thenforevery e>0 $ecatr f,lnd e firnction F of the foll! Nl r(t) = .L 1x for xf,A(e).
!
d--
vhere a, € c anal xJ €n\A(dd)
t
sucb that
In&:( lr(t) - r(t)lS
(see e,g. Ahlezer, sectlon of Broblens'
then
e
.
€ A(dcr)
)
Hence
2 '-
lf, f ts contlnuous on r
A(do)
_
r(t)Jt-t'ar' '=3t n J_r
rtltr _ t_ \n(ddg) r(*rrr) pn-r(eg,*f"r)
n* k=l -'
consequently;W Tbeorem 3.2.L, de € M(orf) '
Renark2.B.IateTr'eshalLshow(wtththealttoftbepolleczekpolvnoDls^18), that tf w le ileflnetl
bY
w(x) = o.'p(-1:. - *21'L/21
f,or -l(x(1
and sr.ep(r)=[-1,1] then v€M(0'1)'
ConsequentlyWthe
OITHOGONAL POTYI{OMIAIS
?1:evlous theorern exr €
M(Orl) lf
73
g > O ls eontlnuous on [-1r1] .
Iet
us
renark thet the above v ls the "nlcest" welght whlch does not belong to S
.
lteoren 29. Let tl € M(OrL) and f,et g setlsff tbe coutlttloas of fheorem 2).
I€t r c [c U [-)] \ su!p(e) be an arbltrary
($)
fT}ffi=
closed.
set.
rben
o(e,p(")-r)-l
unlfomlyfor z€K hoof. If z e n\elltD(do) tnen (t8) follows {meillate\r frm ltreorem 2J, 4, 3.3.8 alrd 4.1.u. I€t K* be a reglon lt1 c U {-J such that Kc K*, tx nr,rle(o) = I ana x* nnl l. ryTheoren3.3.8 the functlone * --l Po(tb*rz) pn(abrz) - are analyblc ln K . If ve can show that lp- (ao-,,
(19)
)
|
(
Ino(o,z) I
- const
for zeE* ana n=NrN+Ir... vhere N=N(T*) thenthetheor€Br,rJ]l fo11ofl' f,ron IILtar!'s theore'" I€t \ be tleftnea W \ = arst(ft#, t1or(o)l-l*r*lr) . forsone N€N. l€t n>N and ze{. Ihen !yfheoremJ.J.B, \>o a r n@ ni(or,t; dd(t) : brr(ao*,r) l' < trn*r(dbrz)-' 5-_
< c ln+r(ctd,zr-t where c-l =
t
lnf € supp(do)
s(t) .
acr{t) S_ n2n{o*,t)
Hence
lro(o*,")12 S c lpo(ao,r)12 * c rn(&,2)-1
.
tr\uther we heve .
i"(ao,")-} = '^
consequently (19)
,2
t"-1td0)2 ;k=' ''TtY*i's yn(dd),u In.(ao,,)l' qf t:m\s/
.
rs setisfied wlth conet = tc(l + {2 lalao)12 .o.z51f/e
PAUT, G. NUVAI
74
5,2. A Sequence of Positive Operators Uslng the well knovn fornu.la
[or(oo,x) = \.',(d0) Kn(cbrxr1".) we obtaln
r1x.-) Fn(ds,f,x) = rr(e,x) n' fin an f(oo,x,x*r) -i A,-(do) k=t wblch
is the Rlenann-StleltJes
sun
for
Gn(dorf,x) = xn(dc,z)
\
tftl 4.(e,x,t)
dd(t)
For z€0 weput cn(dorfrz) (see l+.r.
=
)
propertlesL, (i) rf f(x)=r then Grr(f,x)=1. (il) rf f(x)>o for x € supp(dn) then Gn(frx) > 0 for x €R (lii) Gn is a retional fiinction of degree (Zn-Z,an-Z)
where the denominator does not depend
2. l€t o € M(OrI) I€t f be do measurable supp(do) . Then for each x € supp(do) \ l-r,11
Ttreoreln
rin G.(clrrfrx) = r(x) n*"
(r)
on f '
and boutided on
.
tf x € [-1,1] and f ls continuous et x then (J-) holtts' If f ls contlnrf f lscontlnuous uouson Ac(-1,r) then(1)holtlsunifornlyfor x€A. on supp((lr) and z € c \supp(oo)
f-
then
Iin cn(dc.t,z\ =U-: n n*
(2)
16-
Here Jz" -1>0
^r \
+/+\
------'w.lJ-t(z_t) rG
61
.
for z>L.
hoof. (1) L€t x € supp(do)\[-1,1] . Ihen by Theoren 3.3.7, x is an isolated point of supp(do) . Hence there exlsts e ) O such that
ORTHOGONAL POLYNOMIAI.S
co(r,x) =
tt*l 9Gtf*ll}k:.9)
(x) * r"n.-',
C rrtt r!n\-r,t) tu(t) . -\u/ €t"
, J
lx-t Here the
flrst tert
converges
supp(do) ls cor4ractl )
-
(See tryeuat, 0II.2,
that
va-r pn-r(t) po(1 'n
: :o(t)
po-r(*)
using lheorem 4.1.U ve see that
end
urn
(3 )
,.
to f(x) when n + a .
Reroenbering
K,r(x,t)
l>e
ns
(fl)
tt.l (t*,t) tu(t)
rn(x) \ lx-[
Ihen by Theorem l+.1-.U
Iet x € [-],11
=o
.
l>e
for every
€ > O,
(3) ls satls-
ld unlforn for x € a c (-1r1) . Ihus by hopertles Jthe usual Dacblnery of posltlve operators can be applled. We do not go lnto detalIs. (f11) Iet z € c\swp(do) . ry Tletzers theorem we cal sr4rpose that fleit
and the convergence
function (r-t\-z restrlctetl to su!p(do) 1s contlnuous and ne can extend lt to a function g whlch ls continuous on f ls contlnuous on A(do) . A(do)
.
}{e
have ..
cn(r,z) =
2
Y
.
The
aO
ilt") + J-o (- r(t) e(t) hr,-r(t) ln(z) - rn(t) rr,-r(z)12 w* 'n
=
'";t t^lt") Y'n
n'z"r"l
\"-o r(t) s(t) p2n-1(t) ao(t) +
2,, * r.lt,) rfir,l $ " ni(z)
- er,lt,l nfit,l f-n' Now
\-
J-o
,,r, s(t) pi(t)
\*-o
,t.,
dct(t) -
e(t) pn-r(t) pn(t)
e(t)l
ve app\y fheorens [.]-.U, 4.I.13 and l+.2.1-3. l,le obtein
--+]-= " |f c,,tr,") = fr to't,l - 1l[r + ,-'(d1 Ilu-'(r-t)2^fr-8 t r(t) - !cn |- o2(r) - rl o-r(z) J_l [! ,\z - r) .,2 vrF7- r
dt
.
oo1tl
=
76
PAI,L
G.
I{EvAr
But t&Ol - rl[r + p-2(z)J =,rrJrT-
"oa
Lp?(r)
ft ,-r^
r''c(r,z) t'' ' =Jnn
J-l (z _ t).
,*
- r] p-t(") = z{"
t(t) at. JL-
Iet us note tbat once (2) boltis f,or contbuous fiectlons thenitsLao bolds for
Rleroarn
lntegrable firnctlona lf
x €l\srpp(do) .
tel1s elnce In the followlng ve ghaIl concentrate on
We
sball aot go lnto iie-
co(rvergeEce
of Gn(ddrfrx)
for x € ErCp(O) , fhe folJ-ortng theoreo oqrlalns why we lntroduceal the opertors Gn(&rf) ancl rfty ve shoulil lnvestlgate then for B,s ttranJr welghte o aB possible. lheoren 3.
t"1 e(f o) € Lt . If os ls
a lrelgbt then
I_(cb_,r)
iffu
(t{ )
x€n
a^ndrr g-1 e r!
s
trr"tt
1
(r)
Gn(do's'x)
1
c;r(&,s-rrx) s
l_(da_,x)
+rfu
for x €R. Before the proof Let us re-q.rk
then 0_ ls a
that lf su!p(&) ls
conryact
"nu u-t a"t
welght.
t5
hoof.
trb@
).o(
= nrn nl2r(xl f-v-@
nfl-rtr)
\-
*t.t
follone thet
c- P. rn(&s,x) 5 f -2.(o,x,x) )__ {{*,*,.) 2-
= Li(o,x)
n-
\
--@
0t the otber
o
dCIs(t) =
{{ao,*,t) s(t)
dCI(t) = trn(ttr,x) c,r(do,B,x)
ha.nd
nrr-r(x) =
S-.
\r,*,*,t)
nrr-r(t) u(t)
OMHOGOMI, POLYNOMIALS
,.. xrEJ g-f (r,
dCI(t
\ that
(
)
"3
'2o-rtt)
,(t)
dos(t)
,
1g
llrtaao x) 1 rir(o,x) c,r(a,s-l,x) Frou Theoreos
when
e(t)
2
Enal
rmdiately 6$tsls ].rnll relatlons for I (rb .x) :r:+
3 we coultl
^_(o3rxi
both g ancl g-1 rt"
be veakened btr usrtg the
.
tourraua
on supB(e) . This colalltton
however uay
follorlng two results.
Ieme l+. I€t o € M(orl) .
vhichlsbounrleri: kn-(k
L€t [kn] be a sequence of uatural integers forevery n.
Then
I- (da,x) A . (@.X) n* n+KD'
,r*J_=L
for everXr x € sugp(O) Foof.
and the convergence ls rlnifonn for
Since
), (x) I (x) _A . tx,_A .{x, n+Kn' tr+tr''
have
to conslder on\y frr(x)/Io*r(x) whlcb
c (-Ir1)
n
J=n AJ+I(*, equats
r + )\rr(cb,x) nl(o,*) Now
A
n+k-f I.(x)
1/-_!-/--
we
x €
.
we app\y Ttreoreu \.1,1L.
Theoren
5. I€t
s(Z o) €
tL u
.
Iet og be a welght.
Then
for every po\mo@-
ia1 Pl of ale8ree r\
(5) Tf
(?)
4(*) r-(o-,*)
*
PZ Is a po\monlal. of ile8ree
ua
s
Gn-5(o,sfr,*)
arch
that 4. *-' , rl
dr-l o"|otu",e-L$,{=
ffi
(o
t ',-) . trt"n
?8
PATII G. MVAI
I€t us note that if supp(oo) is nonlal Pa then crg ls a welght.
cor4ract
ana { u-t a ,t
for
some
po\r_
Foof. hequauty (6) follovs fron rn(crr*,x) S vhenever n t \,
ft"l
{1*(ao,*,*)
S_l
{,t,
4-*{oo,*,t)
ao*{t)
!\rther for every,rn_l
nrr-r(x) pr(x) = \- n"_.,(t) pr(t) K-,- (do,x,t) do(r) 4 n+qD J_- .^-'
,
that le P
P.
n-
)
r@
\ "fi_rttl 'ri_.,(x) {(*). r'-r - j-' whLch
s(t) do(t) J-(dd,x,t) \ {ttl *-tf.l 4,_ -,-,
do(t)
tupLles (T)
5. I€t 0 € M(orI) I€t e(> o) € tt 8,"u sr4)pose that there exlst two po\rnonlals pI and p2 such that Ufi a,rru U-t4 are bound.eat on sqD(do) . ftren (1) fo! everv x € su!p(do) \t-I,tl Theorem
(B) (ff)
r* ^i(k'* 0a' x
n*
^rr(
= e(x)
.
')
if
x € [-1,1] and g is contlnuous at x then (B) frofae, (11i) lf e tscontlnuouson 4c(-1,1) and g(t)>O for t€A (B) ls satlsfied unlforrnJy for x ( A .
then
Proof. Iet flrst x € sulp(do) \[-]_,Il , Then by Theoree 3.3,7, x is an isolateal point of s\Ap(do) . Hence I nust be flnlte at x anat then we csn er4rpose that p:- iloes not va.ni6h at x . We obtaln fron Theorens 2r) and Le@E
4 that
(9)
rin
sup
n*
whlch 14r11es (B) does not vanish
if
at
g(x) = Jc
O
HAn(oQ'xJ -S uf*l
. If g(x) > O then we can assume that
. Ihen by the sa.me argu,nent
pZ
ORTHOGOML POIYI'IOMIALS
l.-(dc_rx)
'Ti* i;&I:
s(*)
If g lscontlnuousat x then g(x)(andtbuawe pr(x) > o . Hence (9) hotats a€aln, rf g(x) = O then (g) can sulrlrose that fol-Icrys frcm (9). If g(x) > O then we can sqgpose fr(x) > O vhlch lrrplles (f0). If e ls contlnuous oa A c (-Lrl-) then the above arg@ent cen be used lf ou\y g ls posltlve on A .
ISoYL€t x€[-1rI].
In ortler to l]-]-ustrate the strength of thls
theorem we glve a fev exortrrles.
Def,lnltlon 7. u d€notes the Jacobl welght, that is swp(u) = l-l_,11
&nal
u(x) =o(e,b)1x; = 1r - x)a(r + x)b
fo! -1 (x(I
where a,b)-r.
Ifence ncr/z'-l/z)-v,
In the following lt wl1l- alrays be clea! if u\-'-l
or
n
8. I€t o be the
arb are related. rrith
M(arb) Chebyshev
velght (aa(x) = v(x)ax)
l-
w(x) nA - x' = g(x) be posttlve
and.
contlnuous
on [-IrI]
and
tet
Ttren v
=
v
r\)rther, by easy calcu]-atlon,
Iti'(v,x) whele Un is the
Chebyshev
=
i i' - j
* prr,-rt"lJ
polynouial of second klnal.
Hence
for
every
x ( l-1,f1 tn _lrn ns
-
] *]
urr,-rt")J rrr(v,x) =
(J-ater we shall show that the convergence
n'(*),.f17
is unlforn for x
€
= lrs(x) [_1,1]
), ln
partlcul.ar
I1mn).,(wrg)=ie(fr). n€ D(ar1t)le
9. Iet
w
=
cpu where g >
O is contlnuous on f-Irll
rrn n trn(v,x) = nfiT
"(*)
Then
s
PAITI
BO
urlfornly for x € A c (-1rI)
G. NEVAI
Ihis ls, of couse, not new.
(See
e.g'
contlnuous D{a[ple IO. I€t b > O, supp(a') = [-brb] eJrd !t > O be
[-brb]
I?eud' )
on
Then
,rb1*1 = "1tr.; 18 a selght
FTm the
on [-Irf]
tleftnltion of Chrlstoffel functlon
we obtah
h
-l ).o(nrx) = b).o(worxb-') Hence
urn n).o(w,x) = n,F*'t(*) rrnlfonnly
for x
€
A
c
(-brb)
l-I. IFt v be continuous on [-1r1] and w(x) ) o for x € (-I,I) Iet e)0,6>o anat A=[-1 +6rl-bJ' lIhen
Er(a,4)Ie
r.o(x) w(x) S v(*) S v(x) +
for -1<x<1.
e
Hence
n i.n(lowrx) < n trrr(w,x) S n rn(w + erx), where
euprp(w+e)
prevlous exs4tles = sultp(w) ls assuEe'l' I'hus W the
we he've
ltn lnf w(x)
r
fw(x) + e] s rrlT*nr,r(w,x) n* Slnce e)0 and 6>o arearbltranlweobtaln
=",rc
urrlforo\rfor x€\.00.
llrn n trn(wrx) = n'E? untfom\y for x € AI c (-1rI) '
RecaIL
n,:x)
that w(+1)
rnay
vanlsh'
velgbt $(a'b) ls Definttion 12. I,.- arb €R wlth e t ltl . The Fo'.leczek ateftnetl
by Bu!p(w(a'b)) = [-1rr]
r(a,b)1coe s
0€[OrrJ,
an'l
+ b)]l-', ) = 2 ""p t;i*T'('cos0 + b)][r + e)ret#(acos s
x=coa0.
ORTHOGONAL POI,YNOMIAIS
hopertleB
Bl-
13.
(f ) lle have
o-(v(a'b)) n'
for n=0rlr2r"..
hll
=
-2n-ffi
I,2,...
r \
^/ 't?;
L a +u(-:) ^,1. =z-F
)
.hjF[ n=
-zn- +
ancl
Yn-I(*("'b)
for
=
(See Szeg8, AyrpendJ-x)*,
(t'b) s . ; (u) *(a,b) € M(o,L) o,rt n o - r\ (111) {ni(w',"',x)} ts unlforn\r boundeci for x € A c (-1,1) . thls use ltreoren 3.L.U and Dra.u4)J-e It. (
To prove
lv) Iet q(x) = w(a'b)(x) e*p1 (a + b)tr * (a - b)r ,E ^/ffi.^E
.,
/r-
Then g ls contlnuous antl posltJ.ve on [-I,l]
(v) Iet w(e) - *(a,o). tt"n or(t) is
even and
p:("( "),1)
I1n F
n-s Vn erqpt (See Szegd, Al4lendlx,
^/a
Vn j
=re +f,
4A
)
(v1)
y-(w(a,b))
;;
r\af
nre =r\--l
* 1r-L
where f tlenotes the f functlon of Eul-er. (See Szeg6, Apend.ix. )
(vri) untf,omly for x €
(a), , (a) = n r/1 r--z llm tr.(w'*'rx)n - x- wt*'(x) ns" A
c (-fr1)
a.nal
rtu lrr{*(e),gt)n
ns
.:rp(r+^6^61 = 2 r ea
.
rhe fLrst uh{t relation follolrs f}on D
a
smewhat tetlious calculetlon.
(vr11) r-et z € c\[-t,r]
{hen
_-I€t uE note that the fo:mula (1.1
) Ln the Alpendlx of szeg8's book ls not qulte correct,
lt shor]-clberrltten nPn(x;arb) = f(2n-1 +a)x+blnr-r(x;erb) - (n-r)err_r(x;a,b), ree,J,\,,..
.
B2
PAUI
Iln n*
lrh\o)
p.(w(a,
_az+b -rTz'-r p-o(r)
,r)
(See Szegd, Alpendi*.
G. NEVAI
T41 b ;rs4z= r(* * - 9\z) ' ,2^/z- -f
I ez+b -t+:
-
az +
oZJ
'1"2-t
)
(rx) rf z€a\[-r,r]
then
_
t- n' ,r1-r n$ ^*1rn(erb)
az+b
rT-
^/z--L
p-2n1.1 n
az+b
=
r(*
'
*
az +
b ,-2
AF;
antl
,zFrrJrTP(z)
\k'- 1) ' -l--
az+b
lrn
n*"
rn(w(a,b)
,4-L lp(r)l-2'
62,F2;r-z _
= i
fr(++-az lp(")12-r '
These
follow fron (1i), (vilt)
Dra.np].e
14. I€t w
az+b
+btf' n#7' rffirr. " o\z)'
^m''
and Iheorem 4.I.U_.
be atefined.
(u)
l+
by
sulp(w) =
[_1r1]
antt
v(x) = e:
for -1<x<1
ffi
!y Property Il(iv) g(x) = w(x)
"(*),*,-t on [-1r1] We have S(gf) = * "r.pf _*l by Properties (il), (vli) anil Iheorem 5
ls positlve Hence
aatt contLnuous
]f unlfondy for x €
6
rrrt*,*)
c (-lrl)
g
rheorem
"
''"[l?
) n o
6.L.A7, w € u(o,l) ulrr"u
Theoren 5.L.25
=
t1*;
and
Iln I" (w, rI
!l€
"
(\
I =r
,1,
"t7'
€ M(OrI)
. ry property 13(vl)
a.nd
Y lW)
..-
n* -T ^n z;
!\--T,w, l -l;_r =i't=*l)n-L'l-r =
\;,
lrlv
and
by hoperty f3{jr) and Theor€m ti.I.2i, for everlr z € 0\[-1rl] , r , ,'2n ' z 12 J-rE A_iw,z) lp\z)l n-@ n',11 ^za\lz -L
=(lp(")12-llrtl* : I\rrther by lheoren 5.L.29
'
tV-
z*lz_
sJ.d
lr:
)12
-|
Property
^ lr4z
- Il Ir--?.;rr
lT.-Ll r-, w f4z I
,.-1,r2 ) lu\---i-:,0(z) (=) |
l3(viil) for every z € 0 \ [-I,1]
-zL
,\-n-tlin po(v,z) a(z\-" itn4t-z -
n*
nrL = r \F +
-
z
znJr7l ^, , t v\ - ,-1,-r | 1 /.\ \-/
r-1
^ /2 - LZr4Z
wn
Uslng D(8ltrI)le 1l+ and rheorems 6,L?7-27 ue i-unediately obtain
1>, I€t w be d.efj.ned by lI). I€t g > O) € Lf w' be equivalenL to a strictly positive aJrd Rlenann lntegrable fr-rnction. fhen every result in Dte'rrtrlJ.e ll+ rernains trre lf we replace w by wg = gw, in particuJ-ar, w* € M(Orl) . Now we shall i,nvestigate G-(urf) vhere u = .rr(a,b) is a Jacobi weight. fheoren
L€rma
16. Itrere exists a constant
C
= C/u) such that
nfl(u,xi 1 cr^,,{-:l * !1-2a-1
for l*f af, Proof.
n=Lt]t.,..
See Szeg8,
fr.32,
I€mE.17. For m=n-l-,n
and
nax
l" Kr
i.[T;
* !]-:b-r
?AI'L
8l+
hoof.
AlrpLy l€@as
$
I€t us note thet
G. NEVAI
and 6.3.5.
fheorem 1.1.11 and tenoa 4 glve
r9*_ _, tro(u,x) nflt",*) = x€ Ac(-1,1)
tB. I€t g e Ll n r] ,
r€@a
ot|)
(n =
n-I,n)
.
fhen
rt'* ' \^1 le(t) | pi(u,t) u(t) dt = o . --I
n€
1
hoof. I€tuBconslate" \-. Jg e>0. Ihen
Jf a(o
thenu6el€@sl-5. I€t a>o
") ^ ^r ^c < * ltEK p3(rr,t) [lo JO f l*f.)ln?tu,t)u(t)at n Oct<._" -n' "-rt J_I l*f*ll,,1t;at*c\ Jl_e agaln by I€@a
16. Elrat let n.E
IheoremIg. i€t geLln{,
Ec f-L,Lj
and
then
e+o
Flx
le(t)lat
,
If I iscontlnuoueonecLosed6et
then
Itn cn(u,grx) = s(r) uniforntyfor x€[}. koof,
We
have
to
show
thet
tln Da:C [l-(u,x) [t (* - t)2 ((u,x,t) le(t) I u(t)att = o n J-l n* lxl<- n' and then the naehlnery of posltlne operators can be app1letl. But tbls ls -
so try
I€@as 17 and IB,
Theolem2o. I€t g€*nrl
Iet x€(-L,r)
beal€besguepointof g.
Then
lis cn(urg,x) = e(x) n* hoof. I€t x € (-1rI) anal e ) O be *Lxed. If lco(u,B,x)
- s(x)l S
.
e 1s snaLJ- enough then
w; ORTHOGONAL POLYNOI,IIAIS
W Ft' 1,.,,
-
-
\
l"-i14 | ,n
!;*1t1<. ts'
le(t) - e(x) | S(u,x,t) u(t)d.t
s lx-t l>e
ry remas r5-t8 (r2
( .* r(u.x) n' ' J, r
)
1r:
tr(u.x)
)
.'J
*
[
lx- tlG
4S
l< l*-tl<" and the
thlrd tern
converge8
l*t.t - g(x) lat ,
\
l*-tlS
\^ l< n-' l"-rl<.
to O lrhen n-s .
].1n
v/
We have
(right sitle of (12)) = o
ll.E
because x ls e Iebesgue polnt of g.
To esttnate the
right sltle of (13)
we
lntegrate by parts and renember that e > 0 iB arbitrary. L€ma
21. If arb > -l then
l1 lT
nt*'
rrr(ur1) = (a+1) ea+bl r(a
r'o-t \(u,-r)
* r)2
= (b + l-) r*rb+l r(b + 1)2
Foof. Szeg8, $ll.) ancl s@e calcuLatlon. Ttreoren
P
such
( r.l+
)
22. I€t
= [-Ir1] that w-l f e r,11-r,r; , supp(w)
and sr4rpose Then
for
that there exists a poJ"ynorniel
al-most
evety x € [-]-,l]
ff w ls positlve ancl continuous on a closed set I$ c (-l-,1) then (tl+) hol-ds unifornly for x (8. If w ls continuous ab x € (-I,1) then (t4) trotas. If there exists a Jacobi weight u on lc[-1,I]
then
such
that w/u is positlve and continuous
tr-(w,x)
.,/-_\ lin='^. A tu.x).="tr1 u{x}
n+on'
.
1
85
PAI'L
G. NEVAI
n
1
unlfom\y for x € A
rin
If w/,:. ls posltive end continuous at I
n2a+2
n.5
If w/u ls posltine
t (v,r) = #{ t. + r;
anal contlnuous
rln n2b# r-(v,-l) n'' n*
=
at -t
e4b*r r(a + r)2
i
then .
1
then
#++ (b + r) e*b*l f1u * r)2 u(-r)
hoof. Pttt g=w and. a=I€gendreveight. Then w(x)dx=ao*(x). fheyefore by lheoren
3
\-(w,x)
i
qtbFT
c,,(ttr'e'x)
Appfylng fheorem 2O we obtaln )r-(wrx)
t*n*l* rJ-ro;)
< g(x) = w(x)
foraLnost evely x € (-1r1), thatlsbylxautrle 9 (q= f, u= I€gendrewelebt)
tTj*
n trr(w,x) S r w(x)
r{:7
for l]mst every x € (-l-,1) ry the contlitions of the theorem *-1f e w-lth sme flxetl polynonial P . Iet n = aeg(f ) . Then by 1heorem 5
r,r
f 1*l c"lrtao ,e-Lf ,*) s iffi where g = w end o = I€gendre weight. Using agein fheorem 20 we get
w(x)=g(x) for e'lmst every x € (-1,I) .
Ttrus
w(x) <
for al-nost everry x € (-1rI) x € (-l,t) the lnequality
SrTj*#; by Ierma
rm tnf
l+
\-(wrx)
fr.","I
AlplylnS D(BmpIe 9 we Bee that for al.rnost every tr- (w, x
) r w(x) lr-_,- x- l rin lnf -L An(@'x' n.*
holds, Therefore (Il+) rs satlsfied. vhenever
"-lf
e * , Tbe statenent ebout
W: F=' w'\,
ORTHOGOML POLYNOMIALS
B?
Y..
polnts of contlnutty can be proved. ln the
of
theoren 19 shoulti be usecl instead
sa.ne
Theoren
way.
20.
The
Norv
only cllfference 1s that
se wlLL shov that
lf
*-1f e L1 vlth e suitable polynonial p and there exlst e 6 c [-Ir1] snd a Jacobl velght u Buch that ,,t/\ ls posltlve and contlnuous on A tben (w.x) l,'.n\.,-i
\cT
i':
=
v(x)
"Ct
, If 6 c (-1rl) then thls follo{s lmed-lately fron D(a:rple 9 and the first part of the part of the theorem. Othel.lrlse we can assume vlthout loss of generallty that a = [Or1], (0 < b < l-). We can also holcls
unlforuly for x €
A
tbet u ls of the forn o(a'o), that is u(x) = (r - x)a . I€t g=wfu. lhenbytheconttltlons ge rln* *a {s-1 e {nl1 withEoEe polynonial P1 . Actually, ve can put ?r(x) = (f - x2)2p(x) By Theoren 19 lin Grr(u,grx) = s(x) assrue
tl+@
and
Iln ns unlforr\r for x € [O,f]
. ?-1-,x)
Gr(urPle
2,.-r,. = frr(x)e -(x)
Co[sequent\r by Theoren 3 and >
tr (w.x) 1r{xJ I1&- Ag(urx.)= -J-? u(x, n* holds unlfort].y
for x € [o,r]
Here ve also had
flxe
tr
to use that fact that for
(u.x)
rrx0
n.s TT=r n'-
= 1
unifom\r for -l < x < l" . Thls follows fron l€ro&a 17. FhaILV, uslng Iema 21, the last part of the tbeoren can be proved ln e sieiler rray, Corol-Lary
23. I€t
c [-1r1J ,
for
a].Dost
lio. $rlt n tr.(v,x) . n,uf,l?
t1*;
suprp(w)
nt
hoof,
I€t
coror-Lary
€
> O.
.l 24. rf ;
Then
rtren ).rr(vrx) i )rrr(w+.1[_f, 11,*) e r,11a; then
every x € (-]-rf) . 11
antl (w*.I;_trrJ)-'€t-.
B8
PAIJI, G. ITEVAI
]ln
sqr
n€
'I
<;Ttr?T n'
foralmostevery x€A. hoof. In(cirrx) > trrr(o'rx) > trrr(o'lorx) In the follo$lng,
we
anal
transform A to [-trl]
shall i4rrove both corollarles. Corollary
very strong resul-t. To see this, cotrpere Corollary 2\ with
.
2l+
is
a
F?eualrs r€su-Lt
(See !Yeud., QIV.5.)
Tleorem. I€t sup;l(v) c [-lrJ-] cr lv(cos(e )o
for h s-oll wlth
and
+h)) sln(s +h) cos 0
1{(
e
) l- .
w(cos e ) sin
I
Then
for
al-Eost
suD - -..-l'.- a n€n' n A tw.x,
Iet us nentton tlro appllcetions of 2t. I€t
supp(do)
If
j
fr)
.
CorolJ-ary 24:
c [-1,1J € Ll(A) where ".rrd + 2t lim sup T n-* J=n
(See Definitlon 1.1.1+.
ae = o(roe-6
every x € [-1,1]
Ilm
Theoren
I
) sln I
c!,b1oo;2.
A
c [a-b,a+b]
-
) tnen the sequence tffitO"r")] is
bounded
for
almost
eveqf x€A,
Proof. lheoreu 3.1.U and Corol-J-ary 2b. Iater ve wl-lL
see
that ln both fheorens 2J
and.25 the condltion
;l e *141 eay be neakeneat to [o']-€ € Lr(A) for some e > o . Now we wlll consialer cn(dorf) for weights o which are less nlce
tha.n
the Jacobi weights. In the following,
va1s,
Recatl-
that
"O
etc. lril]- denote closed inter", "ldenotes the lnteriox of r .
Theoren2T. I€t o€M(0,1), rc(-1-,1-)
I€t or(t)>c>0
forahost
89
OFSHOGONA], POLINOMIATS
everT
t
€
r . tet
the sequence
trfito",t)t
be unlformly bounded on
evel'
. ^l . Irt f € r,r* and. n be a polynonlal ve,ntshing at tbe enQrclnte of ", "u Ihen fot do al-eoctevety t€sr+rp(do)\t' r.I€t lf(t),,(t)lcM
,**\**o lr(t),r(t) - r(x) x(x)lao(t) h*O"*
It ie weIL ]moltn that e]-Eost every x 6 rO is a d} (see Freud, $Iv.2. ) Elrst ve viIL 6how that
=o'
I€besgue
point of fi '
gc'(x)(o - tTj*n\('b'x)<-'
(16) I€t vA be the
Cheby8hev
welght corresponcllng to A(d(r) '
rrn((b,x) S rd,2(rro,*,*) S _
slnce x € "0. t c
A(dct)O we have by D(adp1e
n \(on,$<
cn[o(x*]) -o{*
Then
4,"0,",t) do(t) .
B
-llr .i, 5,, #k lx-E l=;
antl lntegreting the lntegral by parts we obtain
(16)' Slnce d' € L*(-1'1)'
polnt o'(x) < o for aLoost .rrcry * € "0. L€t norr x € t0 b" tlo Irbesgue of fir end let o'(x) < -. We w'iLL prove (11) for such points x ' I€t If n islargethelr e)O [email protected] x+.€"0.
' c * (
cn('b,rn,x) - r(x) r(x) = L(do'x)
t._)ts**
l.
q*)r1=.
**
( *t -.1" ,*;iJi.
tr(t) *(t) - f(x) r(x)J ({ao,*,t) e(t) ry (r5)
Itul-(dd,x) \ =o' n*'^ rlx_tYr-I _ l1i we have
further bv
(L5)
'
90
PAUL G. IVEVAI
i lr(t) ,,(t) - r(x) n(x) I --gsGf \ 1.3n ' ' |*s l"j*1.. (x - t.) ].l*!.1..
)r.(do,x)
.
Integratlng by parts lve obtain
tr:.:*
rn(da,x)
n*
Notr
l,
\
f s l*-t
conslder
. t lx-t l>s ,
J,
lS c "* *-- \*.n lr(t) n(t) - r(x) r(x)l oo(t) . "x
lg.
lu
l<.
we have
t€r
in(aa,x)l '-
t
I
l*-[ 1>t t €r
tnfi{*,rl*nfl_riao,t)ttlr(t)n(t)l S* n"-t t €
t
.,. !, + S$lrt*)"t*,1 n€k=n-I ne-
!
t tr
nflr*,.llr(t)n(t)loo(t)
Here we cs,lrnot use slrEtle eotlnates since the sequence bouncied only for x € r, c
+ lrlx;nqx)llao(t) <
.
tffltO"rt)l is unlform\r
bot not for x ( r . Br Theoren 4.1-.lI
"0
:-im nrex
k*t€r
t (dc,t) p3(oo,t) = o . ^
Hencefor k=n-lrn
* t€r$ nflr*,r) I€t t,
lr(t) n(t)l
do(t) =
o{r)
S t€r
lr(t) n(t)lao(tl . ffif*L5K'
tlenote the Chebrshev weight correspond.lng
to r .
Then fron
o'(t) ) c > O for elmost every t € r follows that l,n(dd,t) for
(SeeFeudr 0III.3.)
t€r I
i
a
t
\t
).
p'nt0",t)
lr(t)
) fi ""(t)-r
Hence, for
n(t) I do(t) =
k=n-lrn
o(t) \a v"(t) lr(t) t €r
e
Iet us recaLl that I vanishes at the endpolnts of r . Ihus 1t!otrn(do,x) \
n-ErYr
lx-t +
=0. l>€
a-
tr(t) I oo(t)
.
ORTHOGONAL POLYNOMIALS
fllalry, by the conclitlons and
(16)
I n'(da.x
[ <s.
)
I-
+
x€r.
Consequently (L5; hold6 for alrost every
M(0,1), r c (-I,1), o'(t) - c > o for alsost every t € r, tffitao,t)t be unifornlv bountlett on each t, c to . 1g1 f € ** , ,r M < be a po\ynorniel vanlshlng et the enc$olnts of r 8nd. l-et lf(t) "(t) | S for do aLjdrost every t € sr+ry(o) \t . If f ls continuous at x € ro and Theoren
28. Iet d
€
for l*-tl snal1 then (15) holds. rf f is contlnuous no. on rtc r" a:ral o € Llp I on ra(rrc rj) then (1t) ls satlsfled unifo:mly lo(x) o(t)13 xlx-tl
for x€rr. Proof.
Repeat the proof
I€me29. I€t d€S. r
-JlL
bounded.
^t/+\ u\u/-f
= 1
on each
f^, rvr
of lheoren 2l ltlth
sone obvious modifications.
L€t r c (-1rl) and o be absolutely contlnuous on b ( r . Then the sequence tfl{oo,*)t ls unifolrm1y
r.I c r 0 a!ti l-Lo
ns
n
Ln(dcYrx)
= "R
untfornly for x € r', c r0 . !{oreover, o € M(OrI)
hoof.
See ceronimus, b 'l+.
I€mt30. I€t B€S. I-€t rc(-L, 1-) &nd B beebsoJ-ute\rcontlnuouson r wlth 1/B' € Ll(r) . Then for al-Eost every x € r
lt-:'
^,,,*,*)
Proof. I€t us define 0 and g
and
r
e'',JJt-
.
by
do(x)
with supp(do) = [-1,1]
=
=
[aet"l Ldx
ror
x € f-1,1J \t
for
x(t
PAI'L
G. NEVAI
for x€[-Irr]\t
tl
s(x) = I Then o€S,
e(*)$=1(o
for x € t
I o'(x) for x€[-].rfl\r
ano. B=og.
F\:rther o
sati,efles the condltlons of l€r@a 29 ana coneequent!.y cr eatlsfles algo tbe
21. I€t us put f, = vl2 where vr ls the conesponding to r . fhen by I'heorem t allt1on6
of
fheorem
Chebyebev wetgbt
vi"(x) xo(d8,x) _, _\ S Gn-2(do'sv? 'x) TTd"-F)-Slnce *r;u vanlshes at the endpolnte of r fheorem Zf
fron Iemee l+, 29 ard
we obtaln
Iln er4r n lrrr(tlgrx) < nJ]- - x' B'(x) n*
forahostevery x€r. on the other hand, puttlng P, = uf,2 alcl uslng ftreoren t l'-(dg,x)
-l+, cifr(aa,s-'"1*) -r -1 -h. i(x)
S
'
rtrue by the
sa.me
argunente 1*-r e
*G
lll
r-?dtrrT 'Yr+2
.
.
e,(x) S rin i'r H* n.s n'
I€me 3L. I€t q be an erbitrary welght' Ihen for alnost every x € [-lrlJ
u-n trn(v,x)
hoof.
We
have
a[o"(t) !-t, (,",*,r,
+
oJ(t)] = o
nx+h
. ,(+h
h'o
h-o and.
l|e can use standaril argument. In thls
foflowing.
.
for almost ever:f x € [-f, f ]
u.+\-" +o.(t)Jl= ri'*\'" rx-h la6o.1t) - -*-h
)rrr(v,x) tfn('r,*,t) by
const
ut"^(.) +or(t)J =o
case the stantlaral argunent
We replace
ninfnr--:---5) n(x - f,,
con-
ls
the
ORTHOGONAT POLYNOMIATS
fl i\=\+\+\ {
-'
{
. =:=
Here the
ir'
$hen n*
xl
Tr,
flrst
,rnd
l*-tl
o where q -a -
33. If
"-L
sugp(o)
+
c [-l,IJ sup
n.s
hoof.
seconal
n
tben
)rrr(ciorx)
<
r o'(x) J;=
^
r
every x € [-Ir1]
Use I€@a 32 and the lnequaHtY
r (dd.x)
ffi which fol-lows
fron
Now we can
suftp(Oo)
< rtn(v'x)
c f-1,1J
\-, 4("'*'t)
oo{t)
.
prove the foalovlng
Theoren3\. I€t o€s,
tc[-3.,1J. ltn n
tt #ei,](")
Ln(aLrrx) =
then
r-'-"xn o'(x),/r -
n+@
foraLnostevery x€r. hoof. ry (t7)
lntegral
we
Theorero
Jl
x € r-r,rJ
or(t)J = s(t)v(t)d't+d[o.(t)+or(t)l
the }ema follows from Theeem 20 and I.e@&
Ilm aLnoet
O
({",*,t) do(t) = JL - x' o'(x):""*
hoof. slnce oo(t) = o;(t)dt -l v--
to
eonverge
* -':T*;"o'"lunr. *.":rr*..
ns
'f.
for
lx-tl>e
thlrd lntegrals on the right slcte obvlously
trn q(v,x) \
Theoren
c J.
parts' Elna{y ve let €io '
r€t o
::
=
J 1 j< n lx-tlce
for alrrost everlr x € [-1r1] . To extltrte the
use lntegretlon by
?. rc*v. t::
c
ct cJ. r J r
a;a:.:::,:
93
we have
to
show
that t----'-a
lln inf n )tn(do,x) > r o'(x) Jr - *' n€ for alrnost every x € r . We can asstae r c (-Irt) . Since
31.
PAIIL
9B
G. NEVAI
1 u:(t),* - tf r-(d{r,x) \ "-t t2 [-rB;;)-=B'(x)+o(1)') ( ^, ur(t) InJ} C*t at wtifornJ-y
for x € t,
(B' 'r1 € A0
(B' €
)
t)
.
Proof. If r 1s Bmall then d(8")* = ,lB, -tt is bounded on [-lrl], +1 ,ul,-iD, or +1 respective\r by Renark 4I. Frrrther Fr Eats:- €- {(B;) si'€ A:-({) t- 'r isfles the contlitions of I,emr 2p and. consequently g" satisfies the conditlons of Theorens lB and l+0. Final-Iy, app\y Theoren 3. Iema h4. If
1
c [-1, 1-] then . rn(v,x) ^\ ((v,x,t)
t-----V I de"(t) < - xt + o(f)
J-1
uniforuly for x €
"1
.
"/r
n ln(dn",x) unlforrnly for x €
_P"""1.
trleuct,
swp(dF) c [-lrl]
then
.n.,67 * oq}l
.
$V.6.
c f-1,11, r c (-1, I) . Iet exist a poJ_ynonlal that n"/O', < r,11-r,r; . Then
I€@a such
See
", - "0
if
a.nil consequent\r
"O
4t. I€t
Eupp(d8)
' Il:dB;F uniforraly for x €
"1. "o
e = A'r/v
Theoretn
).
We
put there a =
that dou(x) = ei(x)ax . ret pz = u'Zn.
-4, , n2,(xl,
-
\+n(v,x) ---IfF;'.f--
:_,*, where
* n\;/
.
hoof. I€t us consitler (7) in so
: r----. nJr-x'
n=degr+z .na rr-3nz/B:ei,1. "T V
can slppose ths.L
n
<
Chebyshev
weight,
l,Ie obt&tn
-?2
G..-(v,frrx) .T
Since Br(t)=lT'
n has no zeros in "0 . Hencelor
xer-o
for t€r
tre
ORTHOCONAL POLYNO}4IAI,S
99
u'3*2 -l-2,. l+, , .-1 , ).n-(dF",x) 1n -(x) v-(x) rn'r(v,x) cn*r(vr!f,x) Nolt
ve should. apply Iheoren
.
l+O
lrith o = Chebysbev weLght, f=.r-3n278' eff"
anct rl(t) = t, but we cannot do thls d.irectly slnce in ou! case Ls not bounded on sugp(v) fhis suall problern can be avolile
lf
",
"O
that the
polyaooiale are rmlfornly
Chebyshev
\ p. Ittrl
lx-[ whlch
ln our
case
IttrlI € \"-r
Iemas hli anti
"0
l+5
glve
and thus
v(t)at
Hence
tn*.{'"ff'*l "l.
on [-1,I]
I rF,,(.,,*,t) v(t)dt < *
is flnite.
unlfornly for x €
bounaled
= "-35i),,12'tr). o1l1
whlch proves the lema. ua
c [-1,1J, r c (-L,1-) and. let exl"sts a polynonial n such tuat ,tz/g'- € L1(-1,1) . Ttren i'-V r n rn(d8r,x) = n/1 - x'+ o(f,) Theoren
45. Iet
supE(dp)
unlforaly for x € We
"r.
to.
obtaln lniedl.ately flom Theorens 43 and.46 the foJJowing
4?, I€t
= [-Ir1] anal slqrpose that there exlsts a polynonlal rr guch that nz/a' < f.f1-frf1 . If x € (-1rl), c is absolute\r continuous Theoren
sul4)(dcx)
nean x, d' € 4(B:)
ana oi(x) > o then
-1(l I*2 (-nJln ).rr(ib,x) = ro'(x)
J;=
-
o(1)
l : ,r(t) L1f ". ; \1 ---I- uu .. J= n
r c (-Ir1), c is absolute\r o' €A;(B?) and0'(t)>o
contl"nuous
for t€r
u,
then
(d' € A0) ' x'
(CI' € B:)
in e neighborhood. of i,
l"oo
PAI'T G. NEVAI
(o' € A0) ' T' n ).n(da,x) = 'ro'(x)uil?
*
o1r1 ldt
r:nlfors\y for x €
. ", - "O
The reaale! shoutd coq)are Theoren r+7
hae
c Eo\
to be assteed. In Iheore' 40,
(See we
Feua,
ha'e
$v.5.
ehown
sith l?eualrs r€Bu1ts
rrbere
n2/u, eL-
)
that cn(dcxrf) wlll
con.verge
to f rlth
rate = if f ls good. Gr the otlrer hand for f € L1p 1, we heve onry obtalnett logn/n aa coDvergence rate for Gn(erf) . (see theore'o lB.) $e nay ask two questlons, na,ne\r, rhether rogn/n occure becs,use of our weak technlques antl hor to l4rrove convergence. Iheoreu
48. I€t f(x) = lxl ,
Ihen
un\vrrrvr: t ^/..+^\-^logn --
for n)1. hoof.
Since
crr(v,f,o) = re
hane only
to
show
2
tr,(v,o) \t , ({.,,0,a) v(t)at -ro
that for k Lo
odai
(k>3).
S:#dt>crosk Ihe left sltle here equals sin2kt *--Jo u. = € €* glnt d. :€ --1'\O JO hr
Theorem l+8
lf
we
rant to lnprcve the
then we have to nodlfy these operators.
Iet
"t"" &r > c rogk
convergence
us
put
for
proPerties n<
.
of
m
Grr,^(&,f,*) = rn(dg,x) tttl Kn(tu,x,t) \o(do,x,t) do(t) J__
For z € A , cn,n(dorfrz)
can be alefined by
.
Grr(dorf)
IOt
ORTHOGONAL POI,YNOMIAIS
ctr, B'(do.f.z) (See l+.1'
f(t) kn(do,z,t) k (darz,t) do(t)
=
)
Properttes l+9. (1) cnrr(&rnr-o) = nr_r, . (ii) onr, tu a ratlonal functloo lhe Iebesgue functtr on cl,r{Orx) of the of, clegree (n+n-2,2n-2) . (tll)
operator co,r(&) is not gleater than [rn(da,x) ii]{u,*)lrl2 consequent\y
lf
f ls
good global\y anal
c ls
ndce loeal
.
ly
(near
x € supp(dc)) then for e.g., D= 2n, cn(dorfrx) Eav converge to f(x) very rapld\y. Gr the other hanal, lf o ls nLce near x , then the kernel fi:nctlon of Gorro(db) hes the sare naJora.nt on\r
aB
that of
Go(da) = crrro(do), naoellr
CD
l-Tn' 1+n(x-tJ
l11\
- es iE ltell" knorn- Ls too neak to assl&e good convergence plopelt1es fo" Go,2o(ttrrfrx) lf f ls nlce ody a,t x . For this reason, lte lntroaluce snother c4rerator c"(ctr)r (N = (nlrn2r...rnk)).
whleb
L-I
x
{-1
IEt k>2
befixetl 8ndlet \1%,
("J-f)Srf-1
fov t=2,...,k.
c*(cb,f,x) = a.nd
for z € 0
we
k-I
n-l+nr-l
lDgeneral
We1nrt
a@
,Ir. ^r,r(*,")
a.nal,
rtt) \__
k
Jr
K. (
define cN(abrfrz) ln .exactly the
eaJne va,Jr
as we ditl
when
k=2 I€t
uB
note that lf e.g., o = ChebyEhevweight then the kerTtel functlon
of %(dCI) oalr be naJorized by Ctr
:--ffiI r + n lx -
rl
(xrt € [-1r1J antt aIL n, are of orcler n), whlc]r clifferes very uuch fr@ (21)l (ze) iryues that for k > J, c"(vrfrx) converges to f(x) r*-tttr rate I tr all n, are of orcler n antl f e d wrtU o(t) = t . It shou.lil be possible to {EFrove most of, the resuLts of this sectlon using cN(da) ineteatl ot crr(ilz)
by
At the present t{me we camot tlo thl8.
1(D
PAI'L
G. NEVAI
I€t us nention I sl-!q'le result whlch we shal.l L€r@a
tO. Let x
.
€ lR
for every po\Ynonlal r'
Then
l*n(x)lSI
rn{oo,*)
koof.
Use Property
49(t)'
Theoren
5I. I€t o € M(0,1)
later.
neeal
the lnequality
l"ntrllt({oo,*,t) * dn(aa,x,t)l o(t)
\
holtis.
Then
(23)
Iin sr4r n L.(clo,x) ( r o'(x) ,t;f n#
for aLeost everXr x €
supp(do)
Proof.
3.3.? it is
Ey Theorem
enough
to
shorl
that (23) holde for al-rcst
every
= [-1r1] then (23) follows fron (heo:ren 33. I€t now A = sra4r(da) \ [-frl] be not eDPty. Ihen there exlsts .l ) O such that for every e € (o,er), a. = sr:pp(tlo) \[-r- e,1+ e] is not exdpty. ry ftteo"em 3.3.7 t A. contal-ns flnitely nar\y polnts. i€t a. = [eO)[t where n = n(e)
1 6 [-rr]J
If
sr+xr(do)
Iet r be tleflnecl bv ,(*) = .3- (* - \) Ihenfor n)m ^
.
- If -1rr-,*,t) - n2(t) 6"111 u--nt(*)
'n' ' 1\ 4(ao,x) - J-- ff-S (-r('*'.,*,*)
for x € [-1r].] where ve alenotes the Chebyshev weight correq)onding to Slnce I vanlshes on Ae we obtain [-]-erl+eJ t*t l,r(ttr,x) - )'n-*(v.,x) ,*)-2 n2(.) ao(.) , nt^' \__(r",*, " ) -5;; \-!n\ '€'4l \J-r-. (_r(v.,*,t) \T?.fF) Transforrl.ng [-I -
rn'(&.x) _.
q(v,i|) -
er
I
+
e] lnto [-I, t] ]te get
rr (".I)
_n-re
rn(v,i})
-/14\-<
.
' rn-n(v,I*) l-, 4-rt";,t)
n2((1+e
)t)
do((r'+e)t)
POLTNOMIALS
OFf,HOGONAI
I€t p betleflne.tby dF(t)=n2((r*€)t)dd((l+e)t) Hence by Iemas lr and l2 on f-Irll
],OJ
Then F lsawelgbt
I (tu.x)
IF--2- *.-P ,x\ f--Z---I lI - ----5 n(x) ' ts'(I;) = ,r(l+e)- - x- o'(x) (t+e tl-s In(v,T;) { for alsost every x € [-I,1] , that ls by DrartrI)le 9 Ilrn sw):-.n.
J
r{h Eup n trrr(do,x) S,r c'(x) J(t*")2 ' x2 n$ every x € for abost [-1,1] Nos l-et e-O . Norr we can prove the follovlng theoreE whlch ls one of our naln resu]-ts. 52. Iet o € u(0,1), r c (-J-,1), L/a' e LLG) . Aseule that there exlBtE a Bequence (e"(2 o) with lin er. = 0 such that tos o'1"11^@=
Theoren
'K*
for every fixed k .
€ *(-r+e*,r-e") (24 )
13
o
Then
^rr,*,*)
= ,r
c'(*)^Fl?
foralnostevery x€r' Proof.
Because
of
Theorem
tl lre only
have
to
show
that
lixo inf n trrr(do,x) 2 r o'(x) Jn.b
for al0ost every x € r .
we
have lrr(tlcrx) > trn(o',x)
I€t k be fixed
and
c'((r- e*)t) for -1 S t S I ltlth supp(w) = [-I,1] gr the conclitlons, w € s and r-1 € L](rr), rilhere rf= [(r-en)-rcr,(t-e*)-rcrJ
v be tleflned by w(t) if
r = lcrrcr],
ff
=
fheorero
Jl+
un n n€ for
aLmost everJr
x € r, .
We
)trr(wrx) =
here by E(sq)l-e fo
- %)-1 rn(rdo',(r A= [-1 +1r1-eoJ. Hence
trrr(w,x) = (1
where
lirn inf n rr.(o',(r
-
t(*)
o,Gz
e*)x)
: -m
en)x)
<
(1
.n)-r
trn(o',
(r -
e*)x)
-
e*)x)
o'((L
lol+
PAITL G. NEVAI
for alrost every x € rl, thet ls lln inf n lrr(o'rx) I r ns for
a^Lmost
every x € r .
Irter
we 8ha11 thow
Non
let k*.
that lf e.9.,
for each 1 c (-1rI) a].l the contlltions of Theorem t2 ale BatlEflect anil let us note thet T'heorem ll+ folthus (21+) hol(lB for aloo8t every x € [-1r]-l
tben
lors froB ltreoren
52.
corollary 53. I€t Cr aDtt t satiaf! the contlitlon8 of llleoTeE 52 and let be a ftxed nonnegative
lnteger. Ihen for al'ost every x € r
r n-I ut * _f,^ pk(do,x) Pk+[(.b,x) n*"k=o o nhere T, ilenotes the ,-th
hoof.
Use I'heorems 52
Theoren
tl+. IFt o
€
eld
Chebyshev
Tx(x)
n
o,(i^G
potynmlal'
l+.1-.19.
l4(orl)
llhen
for alrnost every x € suprp(a)
l--v limer4rnlrr(do,x)=ro'(x)^/t-f
(2t)
I
'
Proof. Iet Is be ctreflnedby I$= (x: or(x) > 0J ryTheoren33 $e have to show tl8t (25 ) hol-ds for aLrcst eve4r x € E . Since c ls aL@st ever;nrhere conttnuous, for evel-y x € (-frl) rre ca,n tr = o, [x-er,x+er] c (-1,r) flnd a sequence [e^] such that .^ ) o, l$ and o is continuous at x-e, and x+eB. ryTheor€nlr'2'1lr 1 \ e(t) = :nICX+et li' .X+€\. ;T-jIE;El -{-., -.. ;; {-",
dt
uf}
for
ro
= Lt2t... . lhus W Fatou's lema
r0t
ORTHOGONAL POI,YNOMIAIS
,(+€ - J+ € -'Ii' inr o,(t)at S# [Jx-e^ ' + + rrAr.lcrr'l ztr t' - 'no[ 4,-", n* =r+=
;/, -]
Lettlng ns
a,nd u61ng
l€besguers theoten we obtaln
ltn lnf n*
for elmet e\ret'y x € [-frl] noet enety x €E tTj*
o'(x)
-T#;t n.
< + _ nJL
3Y fheoren
*.
3.3.?, Ec [-1r1] .
n )to(tLrrx) > * o'(x)y'l - x-
Hence
for eI-
.
-
lltle conrerse lnequallty ba8 been proveat 1n Theor€D 33.
5). Irt 6WP(do) = [-1r1] antl o'(x) > o f6v qlmsl svery * E [-trIJ . Then (2t) bolds for a]-roost every x e [-I,1] ltreorero
Proof. If, f 1s continuoue on [-IrlJ
tbe!
*
Il
,t**t*rl = rri u-l t(t) u", +.1s* u g=1 Jt _ t2 (See
tleuil, 0fff.9.;
(26)
llence by l€t@s
.
t'L
r r(t) ;-rh:rr u'\ ^I n \\u'E' n* J-l
r nI \'J-1r(t)
dCI(t) = + t(
*
dr
J-
Uglng one-6laletl a14rro:tinatlon ve obtain that f is contlnuous on [-1,I] (25) reDains vaud lf f ls the characterlstic functLon of, a tlr @8.6urabLe lnterval. A c (-1,1) . Now ile ca,n repeat the proof of Iheorem )4.
lf
IAIII
r06
6,3,
Chrlstoffel
Generalized.
Deftnltion 1. I-€t
NEVAI
tr\ulctlong
( p( o'
O
G.
Then
the generellzetl Chrlgtoffel fluctl-on
Itn(do,P,x) is al€flned bY rn(n,,p,x) = see why we do tl^
Iater lre w"iIL
\(&,1,x)
= inrl
S* 1""-r,.,lp uo(t) .
"::l "i,;F
not lntrotluce a nomallzatlon:
l't'
dp the! trr(dorPrx) < ln(aF,?rx) ' tro(do,2,x) = trn(do,x) . (11i) rf supp(e) ls coqract then
hope"tles 2, (i)
rf
ds S
trn(do,p,x)
dn-:,r" =nrr_,
Inrr_r(*)
(ri)
ln,,-rtt)lp o(t) l" \J-- '-r
.
koof. I€t us flx ornrp anal A : A(dd) . I€t us show that trn(dorPrY) > ).>o for y € a. let n be an lnteger such that m>p. Iet y€ArA(do). Ttren
n|-rtv) for y €4.
=S A
Hence
ln--, tvl I' 'ax €n
Y
where c = c(nrnrtlorA)
does
= l,rrr-a(t) lp Inrr-r(t) l'-p
rarr
not (ro-p
lrr(dornrv)>l\>o
-
S c \Ja In.-r (t) ttepend
: o)
1,r,,-r(v) In S
J\
lnparticular
d'(t) "l-r,t, K*'(e,v,t)
I'
uo(t)
on {n-1 . Wrltlng Inrr-r(t)ln
=
we obtaln
t
oo(t) So lr,,-r(t) lp
for y€A
,
Ttrus
-1 -
r
c ' '\''l'-'D'( n - tr-- lr_,(t)lPoo(t)lp. )--lnn-tt"'ld0(Y): ^'-' n-I then we obtaln fron the prevlous lnequalrf we wrlte trrr-r(x) = nfo lPo{*,") ltythat I ^@ l",-r(t) lp ao(t)lp (o) \ s cr r\__
laf
ORTHOCONAI POLYNOI4IALS =l
i :::
:]j,,' '-.
does not where c, -f = C.(n,p,do) !' "'
depend
on n-n-r1 .
Now
(1i1) fo]-lorre
Bolozano-Weierstrasst theoren by the fo]-lovlng argunent. we fl:(
:.]:,
lrben
ne can fincl a sequence of polynornielu nn-f,, €Fn-',
such
that n-It_f,, -(x) = f for every u u
(o') for
r,(oo,n,x) ^^
ro
=
c-
!\
J_o
from
n antl x
.
(n = Ir2r...)
and'
l
,
,'D In,,-1.r(t) lp a"(.) < \(dc,n,x) * fr
L'2"" ' r.t nn-l,n =
=
-t *pk(e)
rtren by (o)
lUl s c, [\(dc,l,x)
+ IJ
for k = OrLr,..rn-I and 6 = 1r2r... . ry Bolzano-Welerstrassr can choose e subseguence m, 6uch that 11u
r*
theoremne
a.*J = a-ll
for k = O,1r,..rn-I, Consequently, on every courpact set the sequence Since nrr-tr*.(x) = I unlfomly cotwergeg to sone nn-l *hen J'f,n_l.m, n-rru-l - -r-J
for evlry J, ,rrr-r(x) = t
aLso hoLds. Flnall-y,
r-(do,n,x) = \J_6 I
lx._r
lt
foLl-ows
(t) lP u"(.)
fron (o') tlat
.
Flrst of aIL ve vllJ- investlSate the siqrlest case, that ls when o 16 a Jacobl weight. Let us recall- that the Jacobiweightls
,p(x)
We
- ri(x), n - &. etc. have sirnllar
rneenlngs'
Definltion h. I€t a,b €)R. Then un ls
alefined bv
. /a.b). 1r2b+1 1"2a+1. f,-l ur(x) =u)-'"/(x) . = F t^/1 - x +f,J-- - [y'l + x * ;J-- for x € [-1,]l
-
r0B
PAI]L
I€t us renalk that lf I€@e
t.
n
G.
IEVAI
- n then un(x) - ur(x)
.
I'le have
I (a.b), . , (a.b) - 'rx,, ^';u;' '(xJ, ^n(u for -1<x<1, koof.
See
[IL].
rcma 5. I€t cos $* - xOr(u) for k = OrLr...rn+I rr"lth xon = I
x-.,-=-1.
and
Ihen
t',-L-r,n-* for k = !r2r... ,n+L .
hoof.
See
corol].ar.lr
e.g.,
[12J.
7. Iet x
€ [1rrr(u),x*_r,o(u)J, (k = I,2,...,n+J.) (ar, b,
(a.,,b., )
_ un ^ - (x)-\'
_ ^ (xo)-urr''
(a., ,b., )
)
,
lhen
(\_r.)
for arrbr€R. I€t v
be
-
as before
-
the
Chebyshev
welght.
l[o(v,x,t) | < c nrn(n,;f1
l* -
for xrt € [-1r1] , this extlnate ls t close to the endpoints of [-1,1] fonoulate
tl
lnslde (-Ir1) but not for x anal We wlll neetl e better estlhate which we
good
as
rcma 8. I€t xrt € [-1r1]
Then
lq(v,x,t)15 Proof.
Then
The irlea comes
rn(x)rrr_r(t)
c rornln, lx
rl
fron FoUartl probably:
- rrr_r(x)rn(t) = [Tn(x) -rn_r(x)]rn_r(t)
+ trr_r(x)[rn_t(t) -Tn(t)].
r0g
ORTHOGONAL IOLYNOMIATS
=
fi
anal a,brc€n'
Le@a9. I€t O
k=1
Ttren we have
unlfornly ln n
aad
lk-Bl
x/n 6J-
-\
- ro*" II
rog (n+2)
ii:'.il
/LM a+l
.1
\
* rt*o J los (n+2) L ."*1
i:::.il.
a+b+c +1
[' ,-
ii :r:i:,i
+ ( Io&t-+2,
L
hoof.
We
$rite
,ra+b+c+r
.m,^-I _.,
tct uln
n m_1 l-t(b +rJJ X+t+X t+ k=n+l trm/. -I +rrj+r . i \ 'l . .k=L-J+r .Itrr . k=1 x=Lztb 1
and eacb sun can easily be estiratett' Now
we can coqrute the fo]-lolllng l4)ortant
l-o. I€t u = u(a'b) be glven' u, = Nr(arb) such that for evelr fi:cecl fheorem
(r) for -1 Sx<1
n
I
K-(v,xrt) *
Then N
there exists a naturel integer
> Nt
s\v/se I " n -n lqG;Ftr \.-t l#-lNu(t)at<e1"("'b)'*r
\"/
anal n=Lr21....
P"""f. Iet us note that if (I) hords for NI = N then lt holcls a]-so for every fixetl N>Nr. I€t N beanevenlnteger. fhen n-nN=M a'nilbyDeflnltJ.on l+, 5(x) - 5(x) for l*l S r ' ret us coryute the lntegral on the left side of (f) ty tfre Gauss-Jecobl mecha.nlcal quatLrature forur:la. We can alo thls
IIO
PAIJL G.NEVAI
I'I ls even. {tre above lntegral. equa-l8 to
6ince
Y .,..,
(2) antl lre O
.Ko(t,*,ag(o))rp
fr.kt""-EF,rr-' wiIL estl&ate thls suto.
<xSI .
Ta.ke
we uay suppose
nlthout loss of genelality that
a sultable vahre for N such that (L) holds. If
then by elmilar ergr@nts, we obtaln another value
for N
6uch
-1 < x <
O
that (I) ls 8at-
lsfle(t. Taklng the Eaxieus of these trro values of N rre get a nev N lthich i6 good.fo!evety x€f-I,Il llehave q(t,x,x)-N-l.1-N for -rSxSI W n
5.e.8. I€t B be cleflneci
by
l*-*r"l 2lx-x*l (2)-M-Nt r
+
11 *ks-i
for k = Lrzr.., tM .
+ r)\(u)i{{r',*,**)
x
-i
ry I€ma B the flrst sum here 1a O(1) sum is of ota"r lfl-l r:"(x) . Hence (2) S c tliN *
ryr€@a6ror
-ts1a
kf n, l*-*rl trfi^[lz
., ,
-*< e-
t
snd CorollarV 7 the ].ast
1*(u) i{(v,x,x*r)
.
'nu l*-*r."1 :$lx-^llt*'|.
B
q*,(u) r{(v,x,fu,n)
S
n-<1 a
kln
u*zI
S cu-N-r
(1 -\M)
L '|
-
i
S**
Cl,t-N-] (]
f'---?i'
f'-7-'
JL-*'n f-i*,* 'TM - --1;-/
->
( ----;-
#r
N
a
ry I€ma
* 5(x)l * u-N ., ,
conoequent\y by I€@as )16 and
M-N
.
- x')'
crM 2a+I .(t-m)(t<+m).-N - ^ + -D o \fi/ l-:t =fl k=l " M.
tln
where O(c,(1 . To estlmate AandB
rre u6e Ieme 9.
We
obtaln that 1f
OBTHOGONAL POLYNOMIALS
::::
' l
::::
then
2a+I-2N<-l
and
N>1
N
I
I
a,i l-z-2a 2a+I-N-(r-x,r t1 --2\Z r t-1 , A-(1-x-)-}f,'---n frtvr-^+frr
t12a+1-N
r .>"fr\ \^/' ^ ,,(arb)/-\
if
anil
N>1 entl 2a+1-N(-1 ;-,
then
iaa
" Thue
for
N
"-2-2a '2a+I
- fi 1i",o)l*l
tM-N
* t,{"'o)t*lt s . *
#'o)(*)' f, ''l"b)t*)'
flnlsh the proof of the theore!0 l.te choose
o(P(o'
r€ e l. I€t "Z-r1
-,
^r
l,rn-r(x)lP < c n2("*1'So
ro" ] 5 x :: r
and there exlats 8, numbex
Foof.
N
>
l'"-r,t,lp
end by LeE@a
N
1r
-
> tnax(1r24+2,2t'+2\'
t;aat
c, = cr(e,P) > 0
't ^L (1 - t)aat S, ^1-""-2 fJg \JO |".-.',(t)lp u-r I€t
o that
fhen
(3) (r*)
.
> EBJ<(lr2a+2J
(2) s c To
-
k be a natural intege! such that
such that
|,r,,-r(t)lp (r - t)eat
2k: p '
Then at8 nf-t -
n
5
2k r*) <.,r2(a+r) (t nt*.(t) (1 - t2)adt ,,n_f {n_}\^/>uu J_1
for -1 :r x S 1, that ls
E+{ lr--,
l"l.r''-'
(x)
l'n s ,'T. lnn-r(*) l2k-p c nz(a+I) !-t, l-"-rtt' lp (r - t2;"at -l"lsr
Therefore
-,
In,r-r(x) |n
^t . . n2(a+r) t-'. l""-rrtllP 11 - t21uat
for -l < x < I . r€t us put here
*2M
nrr-ti"2) insteetl of 'trr-t(x) '
for M fixetl l*2M nn-r(*2)lP
< c n2(u*r)
.h-* f l*n-rttlln
1r
-
t1a at
Then
'
PAITL G. IIEVAI
Ietnow M besoLargethat Un-]20.
for o::xS1. We
obtain fron (3) that for
"1
t
Hence(3)holds.
O
., ^l -J (- cl (1_*).ar, ( fJ- cl r"n-l\-, l, _(x)lp(r_*)ad*. | \- --l -- ^: *z(a+l) -. ''-n-1' ', J, ^ .- --' -- JO ln_,(t)lD1r-t,;.at I
L-n
L'z
hn
and
c, ) O 1s snall then
lf
(r)
.
,,2(e+r)
\r
o_
(1
- *;"
a* :,
|
,
-^2 wbich
lryliea
(l+).
Ihen I€ s12. Iet roin(a,b)a-r, o(p(o. (6) q(u,x)
for n=Lr2r,..
ard -l-S*<1.
Put ar=(a.il?-* Foof. I€t k bea.nintegersuchtha,t 2k:p. 1 I 1'2k -; . Then ul,br *d trv terea i ard u there exlsts " + i) and b, = (t l-i ? €>O Buchthat (e 'h (a-.b.) ]-9'f * ,r'-1'-r'1t) c"S nrr-r{t)2k Jr-"'$ nrr-r(*)2k u\'1'"1/(x) lr:? -"8 s ')
^/r - t-
-I*?
for l*lS1-"?.Hence (4.,b. ) --..- + f lo'-1'-1'1*; Jr - xzJz)^ nn_r{*) lP S 1-3^ o1'11; ,R . . ,' S_r: ; ;o('r'
*
r
F
r,,-r(t
-t*rt?
that
) le
J#
1s
r
,.rD -
l,ro-r(x) l" S
. By le*a lL the latter tneqgality * l*l S r . Nov (6) follo$s fron I€ndma 5'
for l- | : 1 t - *S
(a,b), r-1 fI r I u(t)at " tl"'o'(*)-' \_, Inn-rtt) lp a18o holdls
for
at
,
u3
OETHOGONAL POLYNOD{IAI.S
fibeoreo13. I€t u beeJacobil'elghtand o
Tben
Itr(urPrx) - ).rr(urx)
for -l(x(I. P*"L. Iet N be 6uch that (1) holcls. I€t D €N be so blg tbat ry > N ' If we put "1 = f*f then . K* (vrxrt) u(t)at S c{",,r(*) -}uo(x) - tr,(u,x) (l*lSrl ).o(u,p,x) S , *f6ll4 = "rtry l€ma 5 antt fheoren
10. Nor we sh&I]. shon tbat
for
\(u,x) S c i.n(u,p,x) -|l't"il' lxl 5 r . rct u = u(a'b), o* = rr(*(t'
U=
,,-,; +-],
(?)
U
Inrr-r(x)
-]r,,
,.r -
to;*'-1,
-1,,. nne
-|ll
heve by
t"u
rema, ard 12
( \(i,x,x)le 5coiol(*)r* S-; l,rn-r(t) Kn(0,t,t)lp o*1t;at l*l:r),
lq(a,t,t)lp o*(t) < c nP u(t)
(
l*l:r)
antl
{{*) lr,r{t,*,*)lP - nP u,r(x) Hence
for l*lSr.
.^1
Ino-r(*) lP < n u,r(x)-'
u(t)at 5-, In,r-r{t)lp
.
lbon thls lnequallty antt I€@a ? we obtaln (7)' There
ia a very lqrortaJrt
colsequence
of
Theorem 13
vhicb ne foroELate
14' r€t u be a Jacobl {elght sndl 0 < p < 6' Ttren there exlgts aunber c, = cr(urP) > o such that ., _ ", t ( 1*-1.) lp u(t)at I , [ ':- l,ro(t) lp u(t)at ,_r.1 rheoren
J-r-'-'n"'
for every rh Proof.
.
We heve
-
es
a
PAIJI, C. NEVAI
Ihus by Theorem
tr
Iema
Inn(*) lP
5 {l(u,n,x) \J-It h",r, lp u(t)at
(-I<x
Inn(*) lP
' ^r 1""(t) lp u(t)at < c r.i'(u,x) \_ ' J_l
(-rcxcr).
13
5
if r_x_ rcrr2(a+r) t' l]1(''r,*) < I l,*cn2(b+r) if t+x--l
1
n
I
Consequently,
if
e
) O ls fixeti
n7.'
then
-r --9 n-'^2 J {\J-l '*\ ,r ,) Inn(x)lpu(x)axS -+
-r.3 -' "t (:-+x/'b.cx + n2(a+r) cl \ u,
c I n2(b+1) \ S J-1 Now
lf,
we choose e
-f €
>O
-r.3 -'o2 . cr *\ u.
e n'
' ^1 ) In.(x)lpu(")0"
the thecren follows.
Coro].Ia4tIt. I€t u beaJscobir,telght, O(p(o, "n
(t l,-(rl lp u(t)at . ' J_I't'
snalJ- enough ve obtain
,c (\
Hence
(r -x)aax]
^r
\J-l
^1 lp u(t)at S c n2' \J-l 1^.ttl "
l,-t.l
lp
e)0.
Ihenforevery
u(t)(r - t2;'at
wlth C=C(prure). Corollary16. Let o
Iet1P
.1^,r1\,
beaJacobiwelgbt. Ihen 1
(( In,tt) u(t) lpat)F S c n'iA-F/ r(' In*(.) u(t) lqat)q J-I ''-1 -' for every rrn vhere
C
= C(prqru)
.
OFfHOGONAL POLYNOMIAIS
W Iema )
Froof.
anci Theorens
lJ,
Il+
1-1 01
u(t)1nu.5, { { In.(t) u(t)lp-s+qdt lr.(t) ., -I -1
*?
S
n
c1
^!-q
Sc
IFt
uB
D-o 1--;
-
L*^a
n P (\,_l In.(t)
u(t) lpat)
n 1 n"
-, nF
In"(t) u(t)lqat
.
note that Corollarles It and15 for 1Sp< - and t< q.
are Dot new. (See Kha[Iova [9]. Before we begln
)
to lnvestlgate the generallzed Chrlstoffel- firnctlons for
$eight6 tllfferent froro the Jacobl ones we will
need.
a few lennas.
I€ma 17. I€t o(x) + o(-x) = const. I€t 01 and. 02 be tlefined. by
Io o.(x) =l| , f '
-
-, L0tvx t
0(o)
for
x
for
x>O
for
x
for
x>O
anal
or(x)
Io
=| ^*
/
\ t
LUO
da(,/t)
fhen
tr^ r
\ D.
{oo.xl
end
n^ L
P*(@1xJ
Kar(nodz /
=
,^
n&x n =r{
for
2. .-J" "n+1 \qt2rx )
for
*+f
l
I x 2
k+r(nod2 )
(001rx2,)
trrod ?
_^_
where r n = n(nod2) mesrs that nn ls even lf
n ls
oaLt.
P:noof.
E8,sy
calculation.
A "1(t)
) "1f"1 "- o
n is
n
od.d
oo(t),
even
and r'
is
odd
if
116
pAUL c. NEVAI
Harrlng
ln nlnti later alp11catlon6
we shal_L lrove tbe
follovlng
I€me 18. I€t o € M(Orl) antt tet o(x) + o(-x) = con8t. Lt be tliefineai as ln I€@a 17. Then o1,% € M(21,*) .
%. and
02
Proof. Ihe leme follows from the relatlons
^ ,r t.th,
\
frr*l
Y:,,-r(&)
* T;-, Y;n+r( do)
Un(dCI)
Yn_l(el)
hrry
=
Y2n_2(dd)
Trl@T
and
, *.1i*er " <
Y:n+1(d0) fn(da) . -Z--:
Y2'n+2(&) Y;n+I(dd) Yn_r(doz) Yzn-t(d0)
rrqr
=
%"-'r-&tI
vhich calr ea8lly be ched<ecl. RecaIL that y_I = 0
r€ma19. ret w(t)= ltlflr-tz;t sr4rp(n) = [-1,]-J . lhen trn(v,x)
- | t l.l t.| lr
.
for -t
wrth f,a>-r
t.vr- * *)t"*t (,FT*
antt
*)2*1
for -1(x(1. Foof. Apply Iemag t
anai 17.
20. Iet Bupp(do) be compact, a c sr4ry(tu), t* € AO, f > -1 . I€t o be absolute\r continuous ln A and let Itreorem
o'(t) - l. - ."lr rn(dct,x)
- *t l" - t*1 * ])r
for x€Al.AO. Foof.
Recal1 that
rn(dg,x) S \(aa,x)
1t e a)
.
w| ORTHOGONAL POLYNOMIALS
W. w=
Tr $
whenever de S
dCI
. Therefore by Ie@e
(s)
\(oo,x)
for x € a]. c Ao .
Iet i
19
bave
to
shorl that
Sc*(h - t*l **)r
denote the chebyshev velght correspondlng
antl m be a natural- lnteger.
to a(&)
Ttren
K;3,
7:L
re
r,r(ao,x) 5\^ ifl.^ --@
(+'"'
t
)
-1lt'*(r)
[g](v'x'x)
traaaaa=
i= :,:il' l.l ''it. 1
l"j,. t
"i.i
=
i
,
anA hence
aolty *c \ tr-t*1r urr u=at 1+n**(x_t)-* tfo t(a unlfomly for x € a, c a0 if n is flxed. The second lntegral on the rlght sltle of the ratter lnequal-lty cen be estLuatecl by standarit methoris. Ftnal1y,
\(ao,x)Scr,-2' \
we obtain
that (8) le satisfled lf
we choose
m large
enor:gh,
I€t us note that the calcuLation in fheoren 20 vas siryle etl estlaates on\r for x € q c Ao and not for x € o . I€@a2l-. t€t O
o€Al,
1)0,
Inn_r(x) lP
.
c ,,r*t
unifo:rn\rfor x€\. koof. I€t e)0 I€@a
t
ancl rheoren
be such
O
.
Thenfor.r"S
\.a0.
J^ l.n_rt.l lp lt lr
[-ee,ze]
nrr-l
at
c a! . rf x € Ar\[-e,eJ
then by
ll
ln tJ 'n-I' -(x)lPccn " slnce f:
that
because l{e lcear-
l,rn-r(t) lp at S
r\ r-.9 3t "" 2t2' I€t now x € [-ere]
I€t
.,
,,t*t
V"
at So l"o-r,., 1p 1t 1r
alenote
the
Ctiebyshev
weight cor-
to l-2er2el, M be an even naturaL lnteger and n be sn tnteger that 2n ) p . Then W lem" 19
respontllng such
...
.-M
--2n -t ^2e f* /+\-M -trn_l(r)J /+\.12m t. tf dt rrn_r(x)J'* v.(x)-' Jv.(x)-"' Lv€(E/ S a tr'*t \,_ag lt l^
for x€[-Zer2el.
Hence
r1B
PAI}L G. ISSVAI
lv.(x)
r-r , ,.rrD ,., -Mp-**t 4\ c t'*'I -2c Inn-t(t)ls v.(t)
-M-+ tt nn-r(*)l'.
J
r*rf aE tEI ,*
-ze
for x€ l-2e,ZeJ. Norlet M besoJ.argetbat -l,p-t+1(0. J:'om
l€me 2l-, ve obtaln the folLor.ing
Il,d!. 22. I€t o(p(o, such that
o€Ao.
llo,
Thenthereexlsts cr=ct(PrlrA)
c rIntt-tt"t 't) 1z \ at :5. z lp lt lr of, )o '" '
t
In,,_r(t) 1p
1t
1r
at
.
i\"0
Itt* r€t o(p(o,
r.€@a23.
t€A
with suPrP(w)=4.
(e)
o€ai,
f)0,
41
.40.
I,et w(t)=ltlr
rot
Then
*,t s . \(v,l,x)
*( Fl .
(n = 1,2,...
)
unlforu\rfor x€A, If x € Ar\[-e,ei then (9) that t-2e,2eJ c {. follows from I-ernoa t,and Theor€n 13. If x € [-e,e] thenwe finii an tnteger *. , ,2nr/n ,, 2 .2,(n/p\-(I/2\ ^ rc such that 2n > p We put w (t) = ltl-*'/r (l+e* - t*)\-/r/ \-/-/ for ftf < a" lrlth s\rpp(w*) = |2e,2eJ. Then by r€mas 19 antl 22 Proof. l€t
5
) o be
6uch
,2m
*,
[-T(x) lrn_r(x)l'*S c n ^Ae- - x* w
C )
rInn-t(t)l-*w r,r r2rtr __*r, (t) at
3: n- l.lg.
C361 J< lxl < ze . Hence ffi r J4"' -*' w(x) Inn-r(x) l" S cn
c )
c,
lP w(t)at s l"n_r(t), _ cn\J1lil-n_r,(t) lpw(t)at
fsltl="
4c'c'
ror ]5 n -' lxl'-5 ze .
Hence
(9) hords also
ror t'1 l*lS . . rr l*l St'
ve aXply Le@a 2I. iema 2\.
I€rora 23 remalns trrre
hoof. I€t e>0besuchthat
if
-i- < l. < 0 lnstead of f :
--t Ac(-e,e).Ietv"(t)=ltl
1 "
for
v4
+
OFTI{OGONAL POLYNOMIAIS
tLg
tt!=
Itl < e vftrr s,44,1v*; = [-ere] .
Then
ty rema
19
f ra('n*,*)
for x€4.
ryI€!o423
"''-r'* 'tD''cE An (lt ,Erl\! ,^/ | - v u JO l"Ino_r(x) /h 411**,*1lp
? "t
t: ,:=
- |t l.l . *,-F
for x€\.
Hence
1,r,,-r(x)lp
tl*l * ]trt n_JA c " (
Inn_.(t)lp
ltlr
at
fo! x€A1. 2r. I€t er4rp(u) be coryact, A c supp(dd), t* € A0, O(p(o. I€t q beabsoLutelycontlnuousln A anaJ.et Theor€B
o'(t)
for x€At.A-. hoof,
trn'(da.p.x)
o
- lr - .*lr
(t
€ a)
F
> -1 ,
.
!)f - rn'(dc.x) - fr n.'l* - t*l' * n'
The lnequallty
(x e a, c
\(do,r,x) S;( l" - t*1 * ]lr c8,rl be proved.
exactly by the
sa,ne way
as
in
Theorem 20
for p = 2
ao)
For the
estloate fron below we can sulrpose tha,t t* e Al anit then we apply I€mas
23
and 24.
Corotlary 25. I€@a 22 renains valiti for -1 < f < 0 and consequently if
e>OrO€A0
l)-1,
\^
and O
l"orty JA-JA where
C
=
C(pr
f,
thenforeverXr r'
lp l.l" dr ( c ne \. 1""(t)
I€t
= lcrrcrJ,
1r*' at
e, A)
Itreorem2T. I€t sufp(do) becdrpact, O(p(-, A(dCI)
1P 1t
D
>O
enal
a>-l-.
I€t
let o be absolute].y continuous in [cr-DrcrJ
PAIJT G. NEVAI
(t
0'(t) - ("e - t)" ).n(do,P,x)
for
X C lC^ -1 -x,c^J 1'1
Proof.
We
- it^r;= * *)'"*t
.
heve by Theoreu
lo
and stendard 8'rgurentg
'I . ).r(dc,P,x) SfrtVct
1.2a+1 - x + ;)
2,
let v be defined
bY
,,(t)= 'k=l4 wlth sr:pp(s) = [-1,1]
lO>-1
2f,+I N-l
llbenforeverY
\(wrr,x)
for l*l S r
il l.-r.-lrn
for k=Ir2r""N
(-r
I€t
io{t) = (^6-.*)'" for -1
and Theoren
8!ld.27 we obtaln
Theoren28. I€t l=tl>t2t...>tN=-I, antt
fron leloa 5
follows The converse lnequalJ'ty -
for x € l.r-|r.r). tr?oD Theorens
€ [c2-6,c2])
r1*llt
t,lk" ,(^/r+ o----= t +;)f ,2f**t - t*l +f)
0
-
trr(v,x)
-|qt"l
.
Renark29'l.lecanestablishinequalltlesslnilartothoseinTheoren14and to the corollarles I5 and 15. T,tte exact foroul-ation of those results ls Iefb reader.
Recall that v clenotes the Chebyshev weight' I€@a
30. I€t p>1 .
^r for -I<xSL. hoof.
The estlEate
fhen
lxrr(v,x,t) lp v(t)at
-
n?-r
-
trntv,*)l-p
t21
ORTHOGOI{AL POLYNO${IAIS
^1 nP-t< c \ lxn(",*,t)lP v(t)at r-I follows lmectlately tYon Theoren 13. The eonverse estlEate 16 obvlous antl can be obtalnecl by a strryle cal.culatlon fron Ietma 8
g 22 I€@B
wben
rhen L
31. I€t a be 8n arbltrery welght, p > I . Ihen for qlbst ertty
x € [-1,1]
^L
1{n ll.s
Proof. 6.2,32.
\"_L ltt"(",",t) lP ao(t)
t
IJ-l lx-1.,,*,t)
U6lnA IF@8 30 the We
lema
,CF
o,&).
ln eLuost the ss.E ltay aa r€!@E
can be provetl
wt1l not go into tletalls. and o
Iheorcro32. I€t supp(dd)c[-L,Ij
(10)
=
lP v(t)at
lhen
J-Lmsr4rntr(da,p,x):ca'({,G n*
for alrnost evety x € [-frl]
wtrere c = c(p)
.
Proof. I€t n be a natural lnteger sudt thet ry > I K - (vrxrt) r-(c!rrfrx) < ^1 \ Now
t{ftt*tt-da('i) '
we &Itply I€mas 30 and 3f.
33. I€t o € M(orI) anal O < p < 6. x € srpp(clo), (Io) hol-ds with c = c(p) . Ttreoren
Proof.
Conbine the argr.roents used.
Then
for aL@st
everT
ln the proof of fheorens 32 end 6.2.t1.
34. I€t o be an arbitrary welght. I€t A anti e ) 0 be given and let vo alenote the chebyshevwetght corresponding to A. iet ;s'1-€ € r,I(o). Then for each p € (0,-) Iheoren
ltn 1nf n trr(dorprx) > c o'(x) .ro(*)-1 ns
)22
for
PAUI al-most
every x € A vher€
C
C. NEVAI
= C(erAr!)
hoof. Iet q = sp(I + e)-1, n and M be natural- Lntegers such that 4)€>1 end ZE)M ) I + e . Iet N = t*] . We car sr4rpose without ].oss of generality that
A
= l-1,11 .
Then
by Theoree l-3
r'-Er---r --'--'-2M v(x) *r.-l\ (vrxl
,r(t)-2M nrr-r(t)lq v(t)at . r \: lr((.,,*,t) .-r(*)lq S .,_f rl -
Hence
.
by H8lder's inequellty
/, \ lP ., r, r,, ,-- --rPo --r--r2lutP | l1*\ |lP < .9 ,A^rtvrx.)v(x, - C- -n ,^ tr.h i\^/ \t_.1 lrrr_1(r/l- G'(t)dE rr
.
r
.
.
(('
.J --I r
qpq
lK,(.,,*,t) ll
Using I6mo 30 we obtain
Ino-r(*) lp
.
c ,, v(x)2Me
.( S]
\.f- o
lffi
ll
v(t)
_
2I&q _ -g_ p-q P-q o'(t) P-4611 a
l,rrr-r(t) lp oo(t)
1q,,",,.,r)l^"p
.
.'r(t)r*'-2MsP
*'(t)-€
at
L/" ,
S-, 15,",*,t) l"P ',(t)
at
Consequently
^1 cn
n In'(clf.D.x, ---]-a
v(x)zW
I
\_, l5(v,x,t) I \
leuP ,r1t;1+e-2M"P
t- r-- -- . r tD€P --r' r)dt l\(v,x,t) I ^ v(t
1+e-2Mep -(o), . ,-e is a weight, ht the condittons v
Thus
o'{t)-tat ,r" l
the theoren fol-lovs from
I€ma 3L.
In lneorem
J+
the nost lrDortartt case is rlthen D = 2 . I-et us forloulate
lt separately as Theoren
35. tet
€>
(a')-" e lr(a) then , '-1 0 (x/ v^lx/ linsr4)- n I [email protected], .r.r {L-fA) =,r=l ns n'
o . rf
::=:
OFSHOGONAL POLYNOMIALS
:-::t'.
!:k=
,e, :Ui;.
'.=
':=
I€t us note that Corollary 6.Z.Zb ts Tleorens,6,2.2, and 6,2.26 renain valid. if l ldt € itt A\
contal.ned. 1n
ltleolen 3r.
Hence
[o,]-. € L](a) instead of
7.
Ihe Coefflcients in the
TheoreE
Recurrence Fo:no:l-e
1. I€t susp(do) c l-1r1J anal - v.,(aa)
r:, lirter il 1
'-'
Then o€S,
hoof. I€t x+o,
O
L€t us divldre both stdes in 3.1(2) by x
anc
J_e!
Weobtaln
^-n yn\Q.)=2^-k Yk(ofr+ '2
t 2-r y"(do) [1 d J=k+r
2
Iet us flx k so that t J=k+I
- r
t^ .l-Il lf-z-*--lat. J
Thenforevery n>k+l 2-n v'n < 2-k y- + -
^
+ 'ur( e-J v. ' k+I<;
'J
andthusforevery m)k+l t-. 2-l Y. , !)a:( ,-n y- ( 2-k YL + + * k+{J$ k+l(n(n inparticularfor
m)k+1
2-nv'm-<21-kr.$ I€t us fix e > 0 srat aleflne P bY dB(x)
=
f
oo(x)
*.
ax
-I(x(1
l*lt r
1
Lo
.
Slnce 6u!p(do) c [-lr]J, the Beasure dF ls greater than all . fherefore Yn(dF) < Yn(do)
for n = Lr2r,,. .
Hence 'I
-k 2-- rr(ae) < a*-" v*(do)
for
m
) k+I .
Consequent\Y
rz4
ORIHOGONAL
POI.YNOMIALS
- vr(ae) < 2-I-k2-m
tTj*
T4
Yk(do)
Slnce I € S ve obtain fron I€nne 4.2.2 that that
i3'-*
v'(ao)
;3'-'
r'(ae) =
=
v(t)1ogs'(t) * "*t-* 5:
dtl
'
l.s
Hence
f,or everY e )
v(t) rog [cr'(t) + ] dtl * "*t + S-l e
O
* a1 \l J-r 'J; ",.pt-+
v(t) roels'(t) + e]dt] S 21-k ,r
.
i,ettlng 6+O weseethat \J_1 "(t) logo'(t)tit > -o ^l_
.
o €S
= iu-
Hence
=:
, r I l€ 2, I€t lt be of the forn v(x) = p(x) lxl-, (t > -], -1 < x < l) where g(> O) is an even functlon of bounded. varlation wlth e(f) > O . Then
:., i:r. :,.
L€@a
for 1=l-r?r... . Yn_l(lr) f
e.- ,.,n+1,) n+5Lr+t-r/
fi"1-=z
rrhere
rn = ^l_ 5_r.
lrrr(w,x) x l*
prr-r(w,x) r. = -L 5.,
:. I
pn(w,x)
ag(x)
l*lt
a+(*)
.
koof' sinee vrr-t(w)/vn(v) = Yn-r(*)/vrr(c*) for every c > o pose that q(lr) = r . t.Ie have cr * .2, ,n;t',*)l'
5_, ii
.l.li,
lt
On
we can sup-
w(x)dx = t '1 Pn(w,x) [n1n(w)xn + "' ] w(x)dx = 2n ' 5_r
the other hand since w is
even
.l'.lD_;r^lo
\ xtpzr(w,x1l' "n' ' J-I
w(x)dx = epl(v,r) -n -
- J-I \ - li(",*)dx "
w(x)
=
126
PAUI, G. NEVAI
o = 2pi(w,l-)
^ - r - ^I \ p'(w,x)x tu(x)
.
Ihus
2n + Because w(x) =
c
^I
\_, ni(",")*
q(x) lxl€
r
= zp2(w,r) "J_lr
- \t nltn,*)x dr,r(x) .
we obtatn
aw(x) = ^I
^ \_, ni{',")" lxle aelx) *
(t nl(",")*,p(*) * . J-L-t" (t ll(*,x)x lxlc ae(x) + e. -\ ' l*le-l | | slsnx* = r-1 Consequentl-y
(1)
rl{on,r) = n *
1i ..
i
!_tr
rl(w,x)x lxl"
aelx)
for n = 0rlr... . Now we shall consider another lntegral3 .1 ^I 5_, f%t",*) l,r-r(w,x)J' w(x)dx = 5_, n;t",") prr-1(w,x) w(x)ox = S_t,
nr-r,",x) fnyn(v)xn-l * ...1
=
w(x)ax =
"
o}s
EUt
af
^1 )_, lnn(",*)rn_r(w,x)J' w(x)ox = 2pn(v,1)pn_r(w,t) - \_,
nn(*,*)ln_r(w,x)aw(x)
and
^I p-(*,*) p- (w,x) dw(x) .
\ J-l
=
If n
=
"
cr rr,(",") \_,
,
I . , ,t lxl'aelx; * . S_r. pn(w,x)
Frr-r(w,x)
p,r-r(w,x)
$
*.
prr-r(wrx)x-f ls a po\mornlal of degree n-2 and. consequentfy the latter integral equals O . If n is od.d. then pn(wrx)x-l ls a Ls even then
of degree n-l- . Thus I 'v) ^ nr .. --n-l, \, \J-1 p-n-r, (w,x) -n" Il f o" = \' ln-r(t,*) [Yn"t-l ' p-(w,x) "'] r w(x)ox polynonlel
Hence
for
a=
!t
Ir2r,., nn{*,*) p,,-r(w,x)
{d * = }#
1r
* (-r)n*ll
vn(w) =
"=il-.
=
ORTHOGONAL POLYNOMIALS
i/j;.
::,/i=
=
:t:,::-
..
T'hus we
obtaln
r-(w)
=,
., D' ', €.. , r . rn*Ir, (2) (n+iiI + (-t)-'-Jlf-rfi tn-l\"/
= ?..'
Puttlng
.',7:
(r) rnto
3. I€t w(t)>0. ".rtb (3) theoren
=
rf
':
bouricled
for
hoof.
We
w-€ €
('^1" rn(w,x)rr_r(w,x)l*l"a,p(*). -\J-1' -
= 2p.(w,r)pn_r(w,t)
(2) we flnlsh the proof. srrpp(v)
r,1(r),
c f-lrlJ,
I€t
nr
w(>
"
O)
'T\_p;(w,x) J-I n>l (e >
o,
almost everxr
c (-r,r))
1
t
be even and
of
bounded.
varlation
la"1x)l<-.
then the sequence I lpo(v,t)
l]
rs
r
€
.: :.
-
(3
apply lemo 2 wtth c = O .
)' fhus
v
Both [arr] and {bnl are bouncletl by
- (w)
]. orlr +# n' =
slnce v ls
even'we
obtaln al'r{*) = o(+) ,
.
Now we
use Theorem 6.2.26,
, r t re 4. Iet w(x) = 9(x) lxl-, (e > -1, -1 < x < 1). I€t cp be even, continuoue and positive and let g' be also continuous. Then v'n-+';' -(w) (-r)n*l 3 * orf) 4n 'n' Yn(trj = *z *
Theoren
:
tl:
fo" tr = Lr2r... , If g ts constant then hoof. I€t us
"(*)
2. S the conditions, i\uther
u6e Le!@a
Thus w€M(orl) €.t . tn+ri[J.+(-J-] e_
h,t
can be repla,ced
an =
W O(+) n
0(t) and bn = O(1) .
Y-
_- yn_l
l
't =2n[I+2i(an* = 2n[r +
'
#un*e+1)
L _ ,.I/z -t lc l)]-rfr+fi(an_1 + e - I)i-l- -b,.= +
o(*l: fr, *("',_r+e-r) * o{-}tl - t,, =
. -2bn) +o(l) =2n+ e*l(r e'n *an-Jn
: I
e+
Thus
.
.
L2B
?AUL
.\yn_I=, .-.#:*fr n+![r*(_r)..-^]
G. NEVAI
-
o,#)
+
o1{v
.
Ftnall-y ve obtaln = + * (-r)n*I ;. orJ&--'?{l +\z+lr'^n 'n
If g ls constant then have to shov that
an =
(\)
an-l = bn = O ' If
t""
fj
ry the recurrence
+ an-t
-
* o(*)
.
ls not constaJrt then $e
2b,r) = o '
forstrLa
vY
'n+l ' 'n un={Jon+r*f,"a
Ilence an
Slnce v
* en-r -
€ M(Orl-),
lf
r, Ey the conditlons,
yn 2bn =
yn+2b
ffn*r
lln bn exlsts
^,yr* - r)b + yn_r * at-\_, r-t n and
it ls flnite then (4) frofas'
= S_t, p,r(w,x) pn-r(w,x)
ffi
tq'(t)
rinb =I(t ;;; " n.,-r41yuilT
.
Uslng ftreorem l+'2'13
g'/g is contlnuous on l-1r1]
obtain
Consequently
*(*)o*
(\) ls satisfied'
t. I€t o be such that elther strpp(oo) c [-Ir]J or I€t 1 c [-L,1J a'nd P be clefined bY 2. (x€t) ,p(x) = sr4r P'(clrrx) n>O -for
we
dt(o.
Theorem
Then
But
aLjoost
cr € M(0r1)
every x € T
o'(*)
^,fT:;*il
,
>o ln partlcular, if 9(x) ls finlte for aLrsost evenr x € r then s'(x) then foral"mosteve4r x€Iscr endif q(x)SK(aLmostevery x€r
t29
ORIHOGONAT POLYNOMIALS
:1'
;' =l
o'1*1
(r)
f*
aLrcst
**g. Eheorene
-,,
= i. ' =,
rY x€IB'
eve:
for x € r ry the deflnltLon of 9t nlrrr((b,x) ) tp(x)--
tt
i
and fle agltlv
5.e.33 ana 5.2.11'
I€t uB note tbat puttlDg o = Chebyshev ln (5) ts not exact' Defltrltlon 6. I€t su!p(a) c [-1']'l '
weight we see that tbe constent
t.et
clrctuference associate'l wlth cr In the usual
,^,
t,
,'{-? 2 fi
unit F = Po be the welght on the way:
fotrl-o(coso)
for o
the correspondlng systeu of
Iet Q-(furz) = z\ + "'
(n = o'1"" )
o.trrogor,ar polynoela"ls'
rt Is boiln that the coefflcients of €n(fu'z)
:i
:l
"u*r.
(see
e.g', rYeutt, iv't')
l{e put an = -Bn*r(d!,ro)
i
iff
ielooa?. o€s hoof.
clenote
4'-' n=0 -
See G€Tonfuus, Q8'2'
rema B.
We have
,,L/2 Yn(do) l .,. _ \/r al')(r ,2 + a'n*t)J' = i rtr - er',-,)(l -
#T antl
on(do) =
! t"rrr-r(r * %r,-r) - ar,.(} - "rn-r)J
for n=Lr2r.... Proof. calculatl'on. For the first rel-ation ,
i: =
=i, :=
f"on
Le@as
?
and 8 we obtaln
fheoremg. I€t 0€S'
Then
K
see
e'g' Geronlnus' 9'I'
are
u
1
PATJL G. NEVAI
130
Reuark
IO.
The converse
of
Theoren
9 ls not true' Exa,glple: the
Poll-aczek
weight.
Iet us note that
Geronimus has proved
that lf
i^
1t" I
< o then p is
n=O
aIr't lvt (a*'z) l < c < absolutely continuous, u' ls continuous end ?osltlve d'enotes the corlresfor n = Lr2,.,., -r ( 01r, z = e10 ' Here gn(fu'z) that nelther a'(x) > O for ponding orthonorsrsl Po\'nordal' It ls obvlous followsfrolo -r(x(l -1 <x<1nor lur,(dorx)lSc
weight' later we wiLL t C:'-(do) < 6 . To see this, consid'er e'g' e Jacobi J=o nl that supp(cl'e) = [-1'r] nor that sho{ thet t c;'-(do) < o neither iraprtes 6 i=Od cl''{ao)<-' frorn follovs .ro o 1s absorutery continuous but supp(o,) = r-trrl Ll. tet
Theorem
converges
cr
€s'
Then
the serles
(oo,x) p2oroc,,*; i ,r- - *2) tr',. 2 2"+L k=I € x [-1,I] unifonrly for
Eoof. ff
Theorem
J'I'B
(ao,x) ?2urd'",*1 S ct2-k ' (r ' - x2)r,2^.1 2-'
i-r"
-L k-l
\g/ ^urf,/a^\tI U.
J
-rl
and we aPPIY Theorelo 9' Theorem
12. tet either o € M(o,l) or
be such that
--l
i {t < - '
Then
supp(do)
c [-]']l
I€t nr (
( "'
the series
kta.)
(do'x) E l- (do,x) P" -l( k=r ',]r converges for alrost every x € supp(dr') supp(d'r) n [-1']]' Proof. By TheoreIil 3.3.?, we can assme that x € Beppo
n2
Using
Levirs theorem ve obtaln that the serles
i *"'(*) p? (a'.*)
k=l "k converges ano I .4.:r.
for
armost
"tuo "^€-""pp(dg)
'1(
n t-r'r]'
Now
appry theorens 6'2'33
13r
ORTHOGOML POLYNOMIAI"S
13. iet P-(dorx)
Ie@8
,. , n-l
,- . n
Y'n'(OAIX - + Un'(@JX +.,. p-(do) n-l
=
I Ar(dO)=-:;-ri:;T r-\*/ J
J=w
for
n=
Foof.
L,2,...,
tn
n"r.r":::
pn(do) pn+r(&) on(e) =rJ-T-r'l;rdo_I
We have
n i o 1 atr-'rd'r)\^- xf an'(do'x.)do(x)= ). %(dCI)=\ x r;a(do,x)e(x)=-rn -n-t k=o k=I Rn' J--
=-#
3 *"-t*r =bt".'n\+/ Definitlon ll+. I-et t (
lR
. lfhen Dt
ctenotes
the rmlt
mass cotrcentrateal
at
t, tbat ts
for x(t for x:t
Io
a*(x) = {
"
I€maIt.
(6)
Then t€n, F=o+e8t y-(dd) €".(dc,t) K..r (dort,x)-
I-et e)0, pn(dp,x) =
where
r,rf6t
- -T.-,L-+fdoTE]"-J
[n,r(da,x)
Y:(dF) I - rr *= Yn(0)
(z) further
(B)
il
o,,(de) = oo(crr)
+
"-$?,
rr o(x)+o(-x)=const.
"p2iao,t;
eK'-trtur-ff)', n+r
+|ffi
t)0,
e)o
-,+# \*"*#,,*9i
anal B=o+eDt+€b,t
then
K-,r(tb,t,x) + (-r)ntq*r(tlr,-t,x)
v-(aa)
p-(dF,x) 'n' -n ' = *ti;.t rt ' ' - €Ph\du,E, Yn\upiiP.\tio:,xt L+ e[Kn*r(th,trt) * (-r)n\o*r(tlr,-t,t)]
and
,',' t ue I
't^
( "n2n = - - - ----;
Yn(cl
i*.-r^ L
ao,
t
)
^ p[1ao,t;
t-n
k=r(nod.2 )
PAI'L
hoof. shown
We
wl11 prove only the
exactly ln the
sane
G. NEVAI
first part of tbe lema, tbe
secoad one car be
way. Iet us note that both (7) and (8) foUow tron
(6). If ve nuLtllly both sitles of (6) by pn(do,x)do(x) ancl we lntegrate over lR then we get (7). U8tng le:@e 13 alld cc'q)allng coefflcients ln (5) we obtein (B).
Non
l.et us prove (6). Develop 1ln(dgrx) lnto
pk(dorx) .
& Fourler 6ede6 Lr:
Then
y-(do) apn(dF,x) = _ \J-o ln(dg,u)xo*r(tu,x,u)clrr(u) =*@len(da,x)-e?n(d8,t)Ko*r(&,x,t). tn.-
futttng here x = t
we obtaln
pn(d8,t) =
(e)
(5) follolrs f,rm the I€!@a
"torr.
Y- ( d0)
fu
pn(e,t) [r + eKo*r(do,t,t)]-r
.
t*o form:las.
15. I€t a € M(o,l-),
€
) or t €3, Pt=0+eDt (
l3+H )i,,.,,-, L
the convergence is wrlform for t €
A
Ihen
::: :;ffil:]
c (-1rl), further
0 Ilno n'(de.) t =
n+6
B anal the convergence is unlforu for t z € o\sugp(do) \[t] then for t f supp(do) for every t
€
€6c
(-l-rl). If
) Fu^#=lp(t)l i'-#e*itlr n* n. (d8*,2 Pr,.t
analfor t€supp(do) 1r-:..:-1
n-(dB*,2)
1* Pn(@'zJ unlforn\y for t €
A
c (-1rI)
lurtherllore
Io
for t
€ supp(do)
[ i^i ltl'- r
for t
F
linpn(dpt,t) pn(dd,t) = I ,^ ^ n+6
If xrt
€
sqp(dn), x I t
anri the sequence t [p*(oorxl
11
ls
sr-rpp(do)
bounded then
1??
ORTHOGONAL POLYNOMIAIS
ll :.
Iin [p,r(op'x) - no(aar*)J = o . ns (1o) holtis unlfornly for x € IB = S c supp(oa) tf t € suFp(o) \A a'nd. { ll*(aa,x) ll is unifomry bountled for x € {s . rinauy, 9t € M(0,1) for each t e R. hoof. ltre Iema follovs llrmdletely fr@
Le!@E
lt,
Ttreoiem6
4.1-.I1, 4.1.13,
4.1.14 a.ud (9).
Iet us note that alt the llnlts ln I€@a 15 - except for pentlent I€@a
of e
one
- are lnde-
.
1?. I€t o € M(orr)
atral
-
n t
.E^cj'-(e)<-. J=v Iet for o S k S n, Rr,*(do,x) be as ln 3.1(2)-(3).
Ttren
Iin R- ,.(do,x) = o tD-k* "runlfornly for x €
A
c (-l-rl)
Foof. ry Theorem 3.I.D, the sequence { ll*(Aarx) l) ls unlfortly Novr we app\v CorollarXr J'1-'!' x € A c (-1,1) Iema 18. I€t B be a functlon on [0rrl . I€t +rr(x) = coslns +
for n=0r1r...
B(0)]
' Thenfor lSkSn prr(x) = ,q(x) urr-o(x) - q-t(x) ur,-*-t(x)
(-rSx<1). hoof.
(x =
AIE,IY Theoren
3.1.1.
I€@a 19. IFt 0 € M(ort) and @
nl
r c;'*(do)<-.
{-n
cos 0
)
boundetl
for
PAIJL G. NEVAI
Sulvpose
autl C on [O'rJ an'l a ths,t there ex18t three functions A'B'
n.(n^<'..<\ c L
.-k*
sequence
* o sudrthat
un
(II)
tc(e) nn*(tu,x)
-
o
K€^
anal
(12
)
r,rhere
x= cose,
- +n-1(x)J = o rrn tc(e) ?.'.1t-r(drr,x) K k-rc -1Sx::r' vrr(x) = a.(e)coslne + n(0)l' rin
/r?l
[c(o) pn(do,x)
-
-'
c(g) <
fhen
,lrr(x)J = o '
uliforn for x €Ec A c (-l'I) ancl If the convergence 1n (I1) and (12) fs then (l-3) hold's uniforo\r for x €Ig ' C(g) ls unlfolsly boundeA for x €IB P"*g,. ry
Theoren
lc pr,
3'1'1 antl I€@a 18'
-
qnl
v(lc pr -
:
| ql + lc pk-l - ek-rl) * clRn,t'l
|
n let k=k(n) vhele v lstheCbebyshev{eight' I'orafixeci Nolr we use rema v' = -' n, s n) ' rhen be clefined. tv k = na:<(nr: l3* l+'2'4' has been deflnetl in Definition RecaIL that the functlon f for l(k
Iheorem 20.
I€t supp(m) c [-Irl]
!
.^
J=v
and
c!'r1oo; < J
-.
fhen for ahost everY 1 6 [-I']l
r----:
t n ---'"r-in t^/c'(x)Jr - x' pn(dc'x) --.8 "os(ne
(Il+)
- r(o))l
o
n€
{here x Proof.
= cos 0. By Theorem
19 &nd Thus the Iheorem fo].lovs from l€r@as
I, o € S '
4.t.). I€r@a
=
21.
We
have
uo'',*t(x)
r' ') D . =Tn+r' -E;---Tp;(v,x)
n
t
k=O
k =n(nod2
antlfor t>l
)
ORTHOGOML POLYNOMIALS
2,
Pn(vr!)
.n +* t1
x
p;(v,t)
o(r) n p(t)-2'
#+
K=U
=n(nod 2 )
( where lo(1) I S C rurlforrnly fo! I + e ( t o' Proof. Calculation. Recall that v ls the lheoren
22. I€t o be the
for
tt = LrZr,,.
Proof.
that l-c,cl c
$
C
>
Le@a
and
0
A(do)
Foof. For the welght J = Orl_r...
t^n-E [p(t)-' * o(r)
8=cr+bt+D-t
fhen
n e(t)-2nl
2l'
ig interestlng
Colollary 23. I€t such
t>1,
velgltt'
.
AltPlY lema8
Ibeorem 22
welghtt
Chebyshev
(de) = vn'G
Chebyshev
F
because
of the follovlng
) O . fhen ther€ exlsts a welght o €I1(orI) - o. l nrr( , i. oo) Bnd c;, e) = o( €J ), ln particular '
a,nai e
,=roci'-(
constructed
ln
Theorern
22, o*(d9) = 0 for
.
24. I€t d€M(or1), e)0,
-r
9=0+€bt
Tf
2\
ura6q, r 3c!'r1u;2
oo(de)=on(oa)+o(|) and
Yn_r(dF)
Yr,-r(tu)
^,r, -f6.r=f,16r""';' for n = Lr?r... Foof.
.
The I€ma folLows from Theoren 3.1",8 and I€!@a
esttng to renark that the estlnates do not
depentl
15. It Eight be inter-
on e '
.
'I
PAUL G. NEVAI
?6
M(orI) follovs that, o + ebt € M(0'1) for every N E eob, aLsobelong6to M(o'I) antiW!eHence fl=a+ e>0, t€n. ' k=l ^"k eetlsates peating applicatlan of the prevlous resul-ts we obtaln asylg:totlcs and for prr(op,z), c!'I{ae), rn(d8,2) etc' Let us x0entlon two resul-ts' By I€tma 16
fron o
€
for k=1'2'"''N'
tkto
Theoren2t. I€t o€M(0r1), tk€R,
I€t
N
B=0+
antl
E et6* . Then B€M(orI), k=r " -k Y(d8) gn ,,ni::rr = TI lo(tu) l-t n* Yo(ct1l to f ewn(oo)
for every z € C\61rPP(dB) rl (de,z) (le(tk)l ru 6-r-6.2I = toFuLraol n.s -n' 6
* - ff
""1 ;
;:*"''
nt
< -, fl be aefl-ne'I as in lrheoren 26. r€t Bu!p(dd) c [-1,r], .t^ cl"{ao) J=u for al-most every {heoren2t wlth tk € (-1r}) for k= Ir2r"',N ' Then pn(clr'x) W x € [-I,I] the esyq)totlc formula (11+) hol"tls lf ve replace there pn( dF, x ) We
suggest the reaalelc conbine these results wlth those
of Sectlons 4'I'
l+.2 and 5.1-. fheoren
Zl. Iet
o
/n
1\
r 3c\",'/1ocr)<-.
J=o Then
bhere
exlsts a Positlve nurber f = r(ao)
Buch
thet
t"'--7
J]--*'llr(ao,*)l1r for -1 <x51
ancl n=O;1-1 .."
for abost every x € l-IrI], proof.
n8ther
. r o'(x)ltJr'" trfi in partlcular, ot € S
'
of S. Bernsteln t-nar( ln-t")l S n ,tqx l'/r - x' nr(x)
By an lnequellty
l*
l5r
lx lSr
I
137
ORTHOGONAL POLYNOMIAIS
n>I
for eve4f nB if
lfii
#-
Iet 2 < k < n .
pn(do,x) |
t
tfi& +
or ln
|
Then we have
by coro]-lerle8 3.5.1
llx(o,x) | + lr*-r(ac,x) lJ +
z I i ,!O")tdo) ,'q* hG J=k-] " lx l9
r.,(ao,x) | ,
Bhort
An
r
JCJAJ
J=k+I
o'1' ara c(t) ls a constant l.r rr = ,H Wl1l.,(&,x)l c;'-(e), u _ l*La l ihen depenctlngon k and ds. Ietnow k besolergethat l=i.rJtiSi' \+1S2 c(k) ' Sqryose tbo,t \< 2 c(k) for k( m( n' If \>2c(k) then An:2c(k)
where cJ =
6 = k+Ir..,rn-I,
8o
n
J cr Aj
A-
Re'erk
28. rf
+2 I
we
*,
put ao(*) = *F?
fttr every J = o,rr...,
c'(x) = rf=
*H ffi;z tf n > nr(r),
(nr(t-lrrJ) = o)'
= [-1,1] then to'r16s) =o and for each r c [-1, 1] supp(do)
pn(.kr,x)
| = lz;
Hence rtreorem 27
- except for the constsnt -
carnot be lrproved.
t 8nd 3.I.L2 we obtaln -c r c]'r1ao1 < -' Then for each 1 c (-I'1) there exists Theorem 29. I€t J=o d rn suchthat o'(x)ZK foralmostevery x€r' number r=r(r,ao)>o partlcularrlf o' is continuous at t € (-1rl) and o'(t) = 0 then r?om f'heoren6
r c:'*(&t)=-.
a
?AUL G' IIEVAI
138
t c?'l(dCI) < o then swlp(or) = [-1,1]
corollary 30. rf
J=o
hoof.
Theoreloa
3.3.7
end 29.
€ M(o'r) 3L. Iet o be such that eltber sw!(ds) c l-1'1J or o r,et p I 2 ancl w(> o) € f(-1,1) ' fhen froro
Theorem
cI , .rD *(x)ax < ffi \-r lr"(oa'*) lY follovs
t^1 to'(")^/r-x-l \,
(rt)
-* - w(x)dx(-'
Roof. Slnce P/2 > I lte be've ' tUit*lll*,5o1',q - qg lU2,ta"tlt " n,;n
- n k=o '-n ;,8-s*"i
lu,-rr,-1(ao)ll rr' 'h '-'' Because
\ ut rnr lrnl I rrt "'o t lrrl
the theoren follovs fron
Theorems
5.2.33 and 6.2.rL. Theorem32.
Iet s€s, 0(p(o, Itn inf \^1-
w(>o)€r'11-r'r;' lnr,{oo,*) ls v(x)ax
fhen
<-
lq)ues (lt).
OcpS2'
Proof, filrstlet
q"
be deflned
fhenwewlllusel€@a)+'2't' let N>O
bV cl*(t) = rolntN,w(cos t) sin tf sin tc'(cos
for O(t(rr,
Then
1f 1'Vftt "."t't -r(t)l 'JO
1_I rif( lP,(do,"ou ,-J6"r-l
-1. t)l 2l
I
I 12
v*t')a'lF S rr -
and
f{J0 ff
cos[nt -
t)'6'-(-osTFfr'T-UE"ostnt
r(t)] lp qn(t)atle
-r(t)l
lp 'n*(t)atlp *
,F f(Jo"lp-(ao, cos t) "Erc;;EfffrTle ,r*(t)atlp
<
5
'I
ORTHOGONAI POLYNOMIALS
l].
"O
2-n
--E
f\JO
lnrt*,cos t)Gr(co-sOffiT
' u/l "ort""-r(t)ll2atl2
*
lrl
+ ee
W le*a \.2.t, 1t1s
l:ne
t\
lr"(o,*)lP
w(*)oxlp
flrst tern ln the rlgbt side
llnlt, lnferlor of the second term is finlte.
the ]-eft siale conve"ges to Jo
bnr
By
the
to 0 when !l + o '
Riernann-L€besgue }e@a
I
(r .)1 \ when n+6 . fhus
converges
o--(t)at)P
the above 1nequaIlt1es
' iI It
aI , ,,D :11 JO \ q,(t)dt < 2 1ln lnf J-1 \ lpn(do,x) l- w(xJdx ' N* By BeIDo Levirs theorem, Itu q, € Ll(orn), that is (1t) ho1ds. If p > 2 N+- " thenfor N>O P..r pi(do,x) e ia'(x),/:- - x'+ N-'l ' v*(x) o'(x)'/r - x-tlx ' \
2-P -P 2wn{x)o*J n t\-L S ' ,S], t"'1*lufl? * w-11 lr - Jr
r
-.-FaD
where n*(x) =
nrn[rv,v(x)]
Rlenrnn-I€besguer s
Le@E
4.2.1
and from
lema that
"*,5: ro'("1"6lT Hence agaln
Lettlng n-* I{e obtain fircm
lnrr(oo,*)lp v(x)ox1p '
* u-11 E,n*{*)a*)2 1 ,rl;1,
Ii]
b,,*,-)
lp w(x)ax '
by Belpo Ievi's theoren (r5) nofas.
33. Iet lt € M(Orl), supp(w) = [-1r1], w be.RlelDann lntegrable on I€t s(> o) be aloost evenrwher€ continuous on [-]-r1l and p > 2 [-1rI]
Theoren
Then
lqrlles
^r t*t") Jt - fl \-, --P
'g(x)dx <
-
hoof. In the contlltlons, the functlon q! ateftned by
'
PAIJL G. NEVAI
140
r-
'-B = iw(x)r/r - x'+ N-'l 'ntn(tl,g(x)]^/1 - x(N > o) ls Riena.nn lntegrable for each N > o ' Now're ca'n repeat the secontl part of the proof of theoren 32. Appry'ing Theorem 4'2'!+, we obtaln the tbeoren' o--(x) 'N'
fheoren
3l+. I€t o
€
M(orl)
and -
nt
C",r(&)<@.
,
J=o
')
rin t4(do,x) - pn-r(clo,x) pk+t(e,x)l K+@
f61. almst'
every x € sulP(e)
Foof. I€t O(k(n
=
t-";i;fil
.
anti Ac(-1r1).
fhenbyLe@el7
pn(do,x) = urr-*(x) !k(do'x) - urr-*-r(x) ry-t(d+x) + o(r)
vhere ffn o(I) = O unlforn\y for x € A' W fheorem 3'I'P' n)-k* {lr*(ao,x)l) ls untfornry bounded for x € A ' Hence
the
sequence
{(oo,*) = t{.-*t*l lfl(oo,*) * (-*-rt") rfl-r(arr,x)
-
2pk_1(do,x) pk(do,x) un-o-r(x) urr-*(x) + o(1).
Thusfor E>k .kcB-koou-k-I * -f-.({*) nftuo,*) * .r^ I4(x) p;-r(do,x) E pl(da,x) r-'-(tu,x) -K ''e+J-'-'' ' = .-n .t=t J ^ J=O .t=v
-2 Iet us tiiviite this fornula by n 6.2.t2
amd
corolla4r 5-2.13 =
no,(*).612
n-k-l 4-n
and
let n*
We
obtaln frm Theorems 2p'
i"hat"
I rf-,{oo,*)l - lli r*-r(tlcr,x)nn('lo'x) + o(l) --f-"-r"?(da,x) r-x a(r-x-)
for almost everY x € A, that is
(16)
/-3 ,$i#
= n2n(ao,*)
for almost every x € a .
* n2o-r1oo,") - 2xPo-r(do,x)nn(cla,x) + o(r)
Ey Theoren
3'L'V,
and by the recurrence
fortula
OFT
l_41
HOGOI{AI, POLYNOMIAI,S
rrn laxp*-r(o,x) tl*(ea,x) - nfltu",") - l*-r(oo,x) no(ao,*)l unlfomlyfor x€A
Hence
forFtm6tevetXr x€A where ].1rno(1)=O unlforrn]-yfor x€A. .ll.s 6 c (-l-r1) ls arbltrary, the Theorem follows, I€t
uB
=
Since
note that the determlnent
!k(dd,x)
r*_r(dorx)
!**r(dc,x)
l*(dorx)
= 1(do,x)
ls a rather
lts positlvtty has been lnvestlgateti by Eeveral
) so far D*(dorx) has been confor the cl-essical wel8htB. Fbom Theorem 29 and (l7 ) we obtaln the fol-
autbora. ElaletEd
fauous e:qrreeslon,
(See Szegd, hoblena and exerclses.
lowlng
( - anti r c?'I(4r) d
corotlary 3t. I€t nlo.ber
N
fhe
J=o
6
c (-1,I)
Ihen there exlsts
= N(crrA) > o such that for eech k ) N, L(dorx) > 0 vhenever x €4.
example
of the Chebyshev po].ynoroiels
thows
that A
celurot be replaced
by [-1,1] in corollarv 3r. r c?'r(ao) ( o
corolrary 35. rf
then
J=o
tin
sup o'G) n.p
JG
e?n{an,il = ?
for aloost every x € supP(do) . hoof.
a
Ey TheoreEs
29 eJtd.6.2.rL
l"lm suD
o2(u,r) -n ' -t 3n Gr the other hancl tv (r5)
o'r,,^R '
n* for alnost every x € swp(do) pk-l(do,x) = a"k(do,x)
*
[(x2
-
:.pfl1oo,*)
. }}t#
*
o(1)]2
142
PAIIL
G. NEVAI
for al-nost every x € [-Ir].1, that 1s by lheoren 3.3.7, for x€supp(do). Hence
al.rnost every
t--
Iettlng n€
and.
^ x + o(rj '-' ' (r - xz)p:(da,*)
using lheoren 29 we obtain the Corollarlr.
Although the proof
of the folloarlng
fheorem
ls very slqr]-e lt ts
one
of
our strongest results. theoreu
3?. I€t 0 € S
Then
D
ll(do,x) ltn - n =O ns
for elmost every x € [-1rI]
Foof. ry Corollary
3.1.1, @
2,2,. \ -^,r \r | ,^ (rB) (r-x')p'(a,x) 1z( lp*(oa,x)l+ In*_r(do,x)l)z * 8". :
-c!,lfa")2n!(do,*). u J=k_I
ry
Ttreorell 9 antl 3eppo
levi's
theoren
r cY"(do)' p|(ao,*) < J=odd for alnost every x € [-lrf] Dtvldlng both sltlLes of (18) by n letting n+ o a.nd. then k* we finieh the proof. r€@a
38, Iet g^-(tlw,z)
be defined as
ln
Theorem
3,I.lt. rr i
first
c!,I1oo1<-
J=o
then (19
and.
'
q(dorz) = Itn p2,(d0rz)
)
n€ -'(z = e^') exlsts for each e € (0r2r)\[r) and the convergence Ls unLfom for 0 € r c (or2n) \{nl, tn particuJ-ar, ,p(aar"1e) 16 contlnuous on (0,2n) \[r] If
n1
r Jc;''(do) < - then (I!) exists for each lrl a r, the
nnlform
in the unit disc, q(furz) is analytlc for l, | < f
for lrl S r hoof.
.
Atryply Theorens
3,L.8, 27 and Bernstein's inequaIlty
convergence
is
and continuous
143
ORTHOGONAL POLYSOMIALS
lfi!,
1",(*ll
' tffi, 1"f,I? .,{*l I
=
(' : r)
*
LeI@a
39. I€t %n(dorz) be Es ln L]rm
n*
n
1
x Jc]'-(do)<-. r€t {-n
3'1'15'
zzn %n(dorz-') =
unlfortly for lzl < r - e, (e > Proof'
Tbeorem
o
o)
I'Ie have 2n
qb-(d0,"-1) 'att
=i ^r<*,$1 i_O
'zn'l
d
qy T:heoren Z/
JJ=OrIr....
for -1(x(tr
$r(ao,*)lsx
ffe[ce
-1
l(r - "2) zJ nr(a,L{)l1c for lrl : r, tr = orl1.,. r consequentfy
-t
'h )l lz"'rprn(&'z
c(r
S
- lzl2)-t '
n pln _l 1, -'+* ztor-1l 1zn-21 . x tli - #lY.-r, ["]-tzl12n-2J+r 'J J=o whlchconergesunifotrlyto o If theorem
j+0. I€t 0
€
lzl
M(0,1) antl ,i^ {'tt*) J=o "
p12n-zt+2, l:. - zk.1 ,J_r
-e' '
- ' rhen d0
can be
vritten
ln the foru dd(t) = cr'(t) at + dor(t) where o'
or(t) (20)
1s
is contlnuous snatpositlve in (-lr1), constant for -1" < t < I . I\:rther
ltu (sln0
r-rr"'rrr";;
p,,(do,cos0)
-
suPp(o')
= [-f'I]
a:rld
:
f?#F.$O't'zstn[(n+r)e - q(e)]l
o
s 6 rc (0,r), where q(s) = arscP(do,"10) , (see (I9)') rs
ln (O,l) a' caJt be calcuLate'l by the folnuJ-a prr*t(&,*)) 2 sin e - r.^/.r,1,elo) l2 = u.1l2rr1aorx) - pn-r(dor*) lv\q ; ofcoTTl =
contlnuous
=
PA1JI G. NEVAI
144
(x=cos0). rf
t J(,.,9,t(Uo)
then
i-n
fln fsin - e!-n'(ib,cos e ) - t(e) slnl(n+r)e-e(e)]i=o n-s
unlforo\r for 6 6 fornJ . Here '!(e) are continuous firnctlons on [Or:rJ
l9(e,"rg)
I *a
q(e) = arsq(df,,els)
.
r c!'r1a; < - '
hoof. Iet flrst
=
lrhen by r€@a 38 snil rheoreD
3'L'L't
J=w
l* rurlfomly
t"*opn(do,cos B )
for
0€
-
la(do,"10)
r c (o,n)
(r-:<'zh[fuil = le(dd,e10)f
Nov
| sht(n
+
l)e -
are'p(ao,els)lJ =
I'et us calcufate ltp(dct,"1e)l
We heve
[r-cos[2(k+]-)s-2arse(dct,eie)lJ + o(r)
*jo
to tltp{uorure)ln when nx the convergence Is unlform for 0 € t c (or") ry fheoren 5'2'5\ (x= cos0).
Ilere the
rlght
slcle convelges
= rirn lnr (1 "2) -t"+*-;T ns for aLtnost everv x € sugp(a) ' Hence r/tl1
o
lq(oo,"ls)
antl
#
f ='ffi
is equlvelent for aLnost every 0 q 1c (orr) . consequent\y to'l-l"G to a continuous functlon. $ Theorem 2!, Ot(x) > k > o for alrcst every Thus o'(eos 0) ana l9(do,"19) | are continuous and poslttve 0 € r c (-1,r) ln (o,r) and (21) holds for each e € (orrr) ' Hence (2o) hotd's rlniforrlv for 0 € r c (O,rr)
IYoE (2o) we obtain
r1 :-: l--n n ns n L(.ld,x) no'(x)Jl_x*
(D,\
uniformly for x €
A
c (-l-rl) .
Thus
lln
sr4r
ns
rl{ao,*) > o
NovI we vlL]fo} every x € (-1rl), tllat ls oJ nust be constant in (-I,1) ls constant ln (-1'I) Bhon that o ha6 no slngular corponent' Because cJ measurable. conseguently by lheoren 4'2'!+ ,a'v-l t" tb
145
OBTHOGONAL POTYNOMIAIS
rrn \ o'(i)^6Ja "* for every A c (-1rl-) ' ry (22),
do(t) = |tr [ o'(t)at ntnt*,t) -n' Ja we obtaln
[J6 -t.l
=
[Ja o'(t)dt
tbat 18 o,(t) a o for .1 < t < I . Flaally
,
we apply Theoren
3.3'7. If
-
^, ( - then lre use I€@a 38 and fheoren 3'l'15' t J c:'r(da) J=o d (20) does not colnclde v'lth Renark l+1. In genera,l, the function 9(S) ln l+'2'i+' Tor lnsts'nce' 1f B f(O) + O - [ vfrere f ls cteflned ln Definltlon ls the velgbt lntroducetl ln lfheoren 22, tben 9(g) I f(s) * e - e ' Ifwe knov that sup[)(O) = [-1r1] t]ien by Itreoree 1, o € S and by rheoren 20' Ihusbyf'heor€n\1, (l-l+)holds unlforulyfor x € tc q(e) f(9) * o -1. =
c (-1rI) lf the coniiltlons of fheoren
l+2. I€t o
€ M(or1)
Theoreu 20 are satlsfled'
*u
rilc!''{ac)<-'
lin pn(ttlrz) p(")-o-I =
1
- tv-?^tz -
then
e(do,
p(t)-I)
J.
unlfornly for lO(z)l )
R
> I where cp ls deflneci W (I9)'
analytic in the doEaln le(z)l > I
anat
tp(do,
p(')-r) is
vsrishes for z € sggp(do) \[-1,I]
3'1'1t' If x € slpp(do) \ [-1'r] then Hence }1n pn((b'x) = O that is by lbeoren 1.1.J, o has a Jrq) 8t x ll.s , .-1,) = v. t0(0r0(x/ p"*f.
Use Ieroas 38, 39 and Theorem
h3. If su!F(do) = [-]r1l , then cx € S and bv leriDa 6'f'l8 o@) e(do, p(z)-r) = + o(v-Ido, p(")-r)-I .z^lz ^ fZ__= ^/2tt _r > r. lp(r)l
Renark
for
@
theoren
lrl+.
ret
o € M(o,l),
.E^
nt
tr
c!'t1aa) <
- . I€t s(: 0)
be RleDEntI
J=o
integrable on [-1,1] antl let #t
o"
bounded
on sugp(do) '
Itren
PAUI.
Il+6
(23)
-
Itj
,-t
vn(do*) = cp(ao,o) o(e,o)-' ,
p(")-t-r Iln p,,(dc*rz) --
(24)
nF
G. NEVAI
'
=
-# ZJr.
91ao,
n(z)-1) n(g, o(')-1)-r
-L
for lp(")l>r."a rln r-(db-,2)-t lo(");zn-z - 4(lp(z)l--I)lz--tl n, l'p(ao,p(")-rlt lo(u,p(")-r)l-2 ^Z D* II' a' for lp(")l > r. lr2' Proof. Lfuit relatlons (a3) a'nd (2\) f,ollow tmecliateJ;y fron fheorerns dlrect con6,L.2r,5.1.e5 and 6.L.29. Ttie LaBt statenent of the theorem is a sequence of (2\) and Theoremg 4'1'11 ard 6'L'/1'
I€t us note that rems. For exartrle, = o(t-2)
.
sone
[4] follovs from the Prevlous theo= Ttreorem l+0 under the cond'ition t!'tt*)
results of
case proved'
Case
-!€N/a;aa::f::Y-ji:.^
'',
..;
,'. "
':1..7
B,
1,
n;;
...,.j
Fowier Serles
RecsIL
that for e given trelght
cI
the welght
og
w8.6
deflaed
in Deflnl-
tton 6.I.3'
r.ema1. t€t sr+p(da) becoqract, 8lo,
(1)
lsn(tlr*,f,x)
-
rn(da,x) l,ir{oau,*) srr(o,fe,x) |
{riI{oc*,*) S llflldo., "E' for x €n ard Yt-L,2,...
rtt ttA-'
#e r,!'
rhen
s
i
[c,r(oo,e-1,x) cn(oc,s,*)
f
-
rj]U
.
Proof. I€t us denote the Lefb sicle ln (f) bV n(x) . Then I (e.x) K.(do,x,t)J do(t) f(t) n(*) [\(do,,x,t) s(t) = [J-@ Tgf"; ' ! 6 '\n\ug'"/ Ini")12 S
.
K(xt lrlt.e a'
where
(* ,u,^^ - nr - ^n(*'"] rNn[u,^,u/r (a^r.x.t)]2 ug\u/ d,r (t) ' '-, K(x)\ = )_. lKrr(ooUrxr!, I;@;lt Let us calculate tt(x) . We have I (dc.x) ^ nn{*u,*,t) xn(cb,x,t) dou(t) + x(x) = r!(oa*rx,x) - , f,ffi\__ n'g'
. Now
,r,!1oo,x).- .
ffi5_-d,-,*,t) ns'
'-6
dc's(t)
rh(d{i,x)
.-r Gn(d"s'x)-}l = r';'{oou,x)hJ-*;F
use Iheorera 5.2.3.
puttlng f = prr-f(fug) in (1) ve obtain 8r inequality help us ilerive asyurptotics for Prr-1(fu*rx) . Recall that Lrr, A:, 4, g etc' have been defined ln 6'2' Note that
which Eay
PAUL
LllB
Ttreorenz. I-et o€s,
G.
NEVAI
x€(-l'I)'
f€tt
s bee'bsolutelycontlnu\f "o leeufftclo'(x)>o'
I€t o'€B: vrth 'r(t)/t€l'I' ently snall nelghborhood of x then o tlxo lsn(turf,x) - srr(do", f lurx)) = (D\
ousnear x.
firnctlon of an arbltraly but flxed nel8hwhere 1o denotee tfre cbalactelistlc
q ls ebEolutely contlnuous Xn r'(e)' of x . If trc (-I,l)' and' ro lsasufflclently wftfr ur(t)/tei'1, o'(x)>o for x€rt o,€r|, t' lf ru ls the then (2) twlds rurlfornlry for x € snalJ. nelglrborhood of t, ftmctlon of a nelgirborhood of 11 '
bolhootl
characterlstlc
Pf,oof. Slnce o = (cl")e snd l€tmr.
6,2.\3t Renark 6.2.41we obtaln from Theorens 6'2'40'
I th8.t
(3)
),_(&r-rx)
[srr(tn,f,x)
sn(dcr'fs'x) I S
c llftl tF I'dOr2
is lowrdecl' thus fe € ,7^T . I€t us conslder for n = Lrlr... . Note that I now sr.(dcr.rfSrx) lle have qr e(tJ :g(x) f(t)(t - x) Kn(dc',x't)ddr(t)' t-x (r+) sn(alcxr,fg,x) - g(x) sr,(da",f,x) - J r (See is unifordry bound€tl for x € r* c to Slnce the aequence t lPn(do.r'") ll r€m' 5'2'29' )
and
- q(x)]2 lr(t) I.)'rs(t)r-x
l2at <
-
€r that the rlght 81'le ln (l+) tentls to t[equaHty ve obtah from Beaselrs b
n* . Fluther, by Theorero 5.2'43 l-(dc,rx)
r
tr;rT = 'Til
0
^,rr ' + u\;/
Hence
r-(ab-,x) ,,
= sn(do"rfrx) + '=. r S ,r(do"rfe,x) ).n(@rx,)
We
have furthe!
l(sn(ddr,f,x)l
o(|) srr(oa"'f'x)
s llrll*",, t;tl'{*",")
+
= 0(^6)
Thus slnce tu-(t)=dt {o" t€r r' \r( da' 'x) _IJ-# so(ch"'fgrx) = sn(tlc"'f'x) + o(r)
o(I) + o{r)
when
149
ORTHOGONAL POLYNOMIATS
To o-
localizetion princiPle (trleutt, $fv.'.),
we can apply fYeud's
t6r whlch
sr.(tlr"rfrx) = sn(do"r flbrx) + o(]) Itence bY (3)
tTj*
\)/
ffi
lsn(do,r,t)
- sn(clo",rlD,t)l
lFllocr,a
(or t € rr) where c coesnotalependon f.
Frttlnghere f-P
lnsteadof f vhere
P
ls a polynonial lr-ith lF-pllora a t, (. > o), and 88aln uslng F?eud's Localizatlon princlple for or, we obtaln that the }efb stde in (t ) is not greater than
Ce
'
NoL
let
e+o
.
3. I€t Eupp(dct) = l-1r11, r c (-1,1-), o be absolutel'y contlnuous on rt o'(t) = t for t € rr "1 - "o' .sul4)ose that there exists a polynonial n such that nz/(t, < f,I(-f,f), Iet t erL arid. let Iu be the characteristic fr.Dction of a sufficlently sloall neigtrborhooti of rI' fhen Theo"em
ltu lsn(dcrfrx) - srr(v, fru,x)J = o n€ r.rnlfornlyfor x€rr. proof.
We coulal
repeet Feud's argunent ($V.7.;, but bis proof can be slq)Ii-
First of a]-l flb e # tot b snal-L. I'urther, .r-lflu ( # atso and it is easy to see that tj.n [sn(v, f 1F,x) - v(x) srr(v,v-t f lu,x)J = 0 n€ uniforsJ-y for x € r, d.nce v is nlce on r. Ey I€@a 5.2.29 lxld, by Feutl's flecl.
He
requires, Doreov€r, that n21o'
l-ocal-lzatlon PrlnciPle
lin lsn(&,f,x) -
srr(do, f
lu,x)J = o
JI.*
unifond-y for x € rr.
qy theoren 5'2')+6
iffifr = v(x) + o(*) ).-(vrx)
Hence
lrr.,(vrx)
,-- ---r
-1*f }D,x) +
lu,x) = v(x) sn(v,v ^ q(-*F. Sn(v,v-.f (x € rr). Consequently'we have to shov that
(x € rr). 'l 0(..:-) ,,|lv-1-f . -Iulf,,'
G. NEVAI
PAI'L
v)o
,
tr-(v,x) . _1._ rr sn(vrv -f 1D'x) | lsn(do, f l6,x) - 1;6,a; e:qtresslon to O unlfortry f,or x € t, when n*' But thls (v.x) l, Kn(v'x't)]dtl 1 r(t) [Kn(dc,x,t) -
conrrcrges
I
s llrlblldo,2 trlr{oc,*)t#3
thls ls
^6
enough
Froro Theoreras
Theoren
tr,,(v,x)
!-t,
(,",.,t' e(t) - L.tL/z
ls not greater than 5.2.46 and I€@a 6'2')+l+' the latter enpresslon
cllrr^ile, and
equals
tr;
S
E tb
Ey I'heoren
,
l+. I€t
((v(x) * o(*ll ('(*)-1
for our
. o(*ll - l'/'=
Put?oses'
2 and 3 we obtai! the folloving
n2 1o'
o(r) llrtull*,'
equlconvergence
the con€ r,r1-frf ; wlth a suitable polynoniel n' If
2 are satlsfied then dltlons of the first part of Ttleorem 1in lsn(darfrx) - srr(v, florx)J = 0
part of fheoren 2 the convergence is un1lJx_orx*oJ (or = 1"r{o)) and b is sufflclently form for x € rr. Here ru =
end ulder
the
concll-tlons
of the
second
6D411.
po1yror0181 1o' e r'1 wltn a sultable ln r c (-1'1)' cl'(t) > C'> o for tr. I€t c be absolute\y continuous of € LI *rhe"e u is the nodul-us of contlnulty
corollary 5. ret sWp(oo) = [-1'1]'
n"
rt o' € Cl(r), o(o"rt)/t D 0,'. I€t f € t-d". Then I
Q
11:
sn(dCI,f,x) = f(x)
foraLrnogteverY x€r'
proof.
Use Theolen
\
and Carleson
[3]'
wilf investigate the L€besgue functlons nKn(tu,x) = \__ l*nt*,x,t)loo(t) is sure: 1 = I,Zi... , one trivlal thlng In the folloving
we
i
'l5 l
ORTHOGOML POLYNOMIAI,S
,, -l {,(ea,x) < };-(tu,x) [o(-) - o(--)i Hence
estlnatlng frrt
for t € [x-erx+e]
we obtein estinates
for L'
.
If e.g., a'(t):
C
>
O
then
({*,*) s.n. Tt ls rether surprlaing that lrelghts satlsfllng
ever tried to ilQrove thls eetiEate for
condltions (e,S., for o € S).
ltea.k
We
wil-L see that C!
be replaceil by o(n) ln nany cases. Flrst ne wj-1I flyrd condltlons for
can
=o
lin rn(do,x) ({*,*)
(6) If
nobody hes
.
D€
supp(da) ls corpact a,nd o has a
at x then
JUIEp
lim tnf rrr(da,x) t(iu",") > o(x + o) - o(x -
o)
n€
so
that (5)
cannot holct.
Iema 5. L€t € ) Or x
€
R.
Then
< 2[c(x + e) - o(x - e)]
ln(rlr,x) (t*,*)
2 ,-
n
$,'^'r"(oo,x) ,t\i!T' Y;(dc) u2 ^(uo)
*z# v-
koof.
we
- (d0)
+
,
+ 2(x
-
o,,-r(do))21
nl_r{0o,")l ta(*) - o(--)l
rl-r{oo,*) *
.
wlll- use the christoffet-Darboux arr(l the recurrence fortulas.
we
have
Kn(do,x)=[ \ ,-, S ]lrn(do,x,t)lo"(t)' lx-t'{<e lx-tl>e -Szt
rf(ao,*) -h'-''
\r
t2*ef
\
lt.
lx-tl>e lx-tlce Hele the ftrst lnteglal in the braces is not greater than [o(x+ e) -o(x-.)] trVrther
((oo,*,t) do(t) = rlr{ao,x) [o(x+ e) -o(x-
\ D
( \ f :4\l!T' € C
lx-t'{ >t
at - \ Yn(ocYJ
tr2"-r{oo,*) lr-r
* p2n1oo,*)l [o(-) - o(--)]
e)]'
?AIJL G. NEVAI
Nor tbe
flnal est{nFte follovs fron the recurrence fonnrfa'
Coro]-ls.ry
?. I€t
rn(ds,x)
dt*,*)
sugp(do) be eoupect, e >
1 efo(x
for x€A, n='J-r21...
e) - o(x - e)J
+ ce
then
-2Lrr(ao,*) tp|-r(do,x) +
{{ac'*)l
where C=C(dorA)
Use Let@a6 5 a-nd 3.3.1.
Foof, Theorem
(?) If
+
O a'ntl A flxed''
B. r€t o € M(0,1). rf s Is contlnuous 6g 1 6 [-l'rJ rlB trn(do,x) (i*,") =o.
then
n*
then (7) is satlsfied' un1-
o ls contlnuous on the closed 561 5c (-trl)
fonnJ.yfor x€IB.
Proof.
?
Use Corollary
Theoren 9.
and TheoreE
l+'f'Lt'
I€t o€M(orr), 1c[-1r1J, e)o. rf Js'1-eer'](r) = l1"t/t \(d''x)
(B)
tben
o
fo"
If 0 lscontinuouson t and. o'(t)>c>o t almost every t € r then (B) holds uniforoly for x € "1 "o '
foTeLmostevery x€r.
Proof. Stnce cr ls
in l-1'1]' the flrst part of B ana 5'3'3t' The second' part follows fron
al-nost everln'rhere continuous
the fheorem fol-lons fron Ttreorens Theoren B an(l Exa,nq)le 6.2.9'
.I
F}omfheorengonecarleasilyobtalnconvergencetheorerosforLi.€.Reca]ilthatallknownconvergencetheoreBsconcerntheclassll$wrrlchlscontalned ln Li+.
(See
e'g' Freud') I€t us
I€t us note that (B) ls
for
bad welghts,
to the reader'
for nlce $elghts better
e8-
be founcl.
tinates
can
fiheorem
l-o. tet
Then
good
l-eave the d'etails
be conrpact, rc suPP(tu)r
rin rnf n-'i ) q(do,x) < 1 l-
ll"s
s>
o, fo'J-t
e
*1";.
ORTHOGONAL POLYNOMIALS
foralmostevery x€r. hoof. I€t
us
put ln
o-tr(t*,*)
6 , = o-I/3. uslng I€ma 3.3.1, ve obtaln
Ierotnt
s rll
< z[nq(da,x)r-r o3 ;o{* * r, 3 ) -o(* Sr:mlng
for 1= ]-r2r...ru
* : n 5r({o,")
-'
3
)] + cl2o-r1ao,*) + ol-r1oo,x1
.
lte see that
<
Sl.nce C 1g aLrsost everJMhere dlfferentlable we obtsin froE IheoreB 5.3.35 that -F
lin
sr4r
ll-*
for aboet every x € r . Nov ve
,
n
r- n '({u,*) * tl=t
Hence
. -
the theoren follows.
wlIL be egain in the situation
I€@a IL. I€t o € s, 1 c (-t, l)'
G,
r + Etdt ' (see 5'2.
Then
\(dor,x) < c Logn uniforoJy for x € ", c "o Use Ie@a 6.2,29
Theorem
12. I€t o € S. I€t x € (-lrl),
€
{:,
(n>3)
.
Foof.
o'
)
arLd'
Ibeual, Uvrl+.
o be ebsolute\y contlnuous near x,
o'(x) > o. Ihen
1 .., n o(t) -,,2. -';' dtJ-J r..__JIt.! c[]og" * [\ \(dc,x)
(9)
(n>:)
;
on n. If r, c (-Ir1), 0 ls absolute\r contlnuous o' €Af,r, o'(t) > O for t € 11 then (9) holds wrifornly for
where c does not
near
alepentl
"1, vlth C lntiependent of x r, x€
a.nd.
n
.
PAUL G. NEVAI
that the corlespondlng Proof. Let us choo'e r (x € ro or tt c to; so sEaIL (o")U' Hence by g is borxrcled fron below and above ln [-1,1]' We have o = I€Ens
I rn(oo"x) + \(do,x) S c rn(do"x) {.r{ao,*) + c It{}(oor*) [c,.(oo",s-r,x) Gn(da"r8,x) - r]11/2 '
ry
Theoren
5'2'6 : c'
lrr(do",x) rlr{a'x) ry ora4le 6.2.9, rll(oor*) < cn' lrheo:en 6.2.38, Renark
Note that
lf
6'e'\1
and'
NoY
I€ma 6'2'29'
u:(t) = t llog t I then
J...,,r g9*
f"*' A vealcer
the theorem folLows fron I"ema II'
-
[tognJ2
verslon of TheoreB l2 was obtained' by lYeu'i' 0v'7'
13' I€t o € s, u(> o) e ll' w(> o) € lL' neas(u (a< -)' neas(w>o) >0, 1(9( -r uL/(r-s) e** "-t t"t O(p(o, P1q. If q(o andforevery feLl* llsn(ac,r)l[*,e S c lFlLdc,q
Theoren
for n = Lr2,.'.
(ro) andql
llith
C independent of n alld f
"61?tffi
If q=6 ancifor uf €1fu ils,r(oo,r)1lruo,n S
with clc(n,f)
.I )_, hoId.
c ilrfll*,n
for s = Lr1r.., then (10)
I
to,(r)
"61?1
(q=')'
then
't -P - t-l 'w(t) o'(t)dt < f"'(t)^/r t-. '-L t t [o'r.t u(t)E a'(t)dt J_r
> o) > o'
and
2r'(t)-r d'(t)d't <
'
<
-
'
ORTHOGONAL POLYNOMIAIS
Itoof.
For sluplicity we wtLL consider the case 1 lls -/ r-n\(ct.+,'.tt
(
( o. ry the conalitlons
g
s /dr.f
Thle neans that
(e)ll,tbdf,P (do)ll. l:. < n . llfp rr-!n\s/"&, lFrudorq . " lhll By H8laier's lnequallty thls ls equlvalent to sw tlbn(do)ll,r*,n lb,,(0").r-'lL-,n,1 . llD
nzl
where q' = S,/(q-)-). ry fheoren l+.2.8 tne latter eondltion is equlvalent to
sw lb-(da)11..^-<.wu,P -r,
,>I and
sup
n>_1 ' Now
the theoren fol.lows from fheoreros 7.31
anal
.
7.32.
Iet us note that nany speclal cases of Theorrem l-3 heve previously been known. We refer to Badkov [2] and to the llterature mentioned there. (In partlcular,
we
refer to
and Walnger.
works
of Askey,
Iluchenhal4)t, Ne1'maltr-Rudin, FoLLard, Steln
)
Corollary 14. There exists,au absolutely contlnuous o € s such that fron
(u) forlows E=2
s-up
n>I
llsn(dCI)ll.e _p ( o ";"'rd"
^ -+ Proof. F.lt o'(x) = ocpt-(f - x') " l. For p = l, o , (Ll) car never hold..
If 1 < p < o, then apply
Theoren 13'
Inequalltles
9.
prolnvestl8atlng the lebesgue functlons of lp€I8nge lnte4rolatlng cesses rre r,rlll have to be able to estlnqte the ecpresslon When
(r)
b-.o,?*l1..
\*(0")
'
llhe followhg result 1s very sfup1e'
I.e*r t. I€t su!p(do) be coq)act' Then t1n tln srrl, x, - \rn(dCI) = o(x + 0) - o(x e*O n+ lx-lor l< € for each x €]R'
0)
end y2€(x+etx+2e) hoof. ForevelT e>o r*ecanfilnd v1€(x-2e,x-e) y2' fhus ty uslng convergence theosuch that cr ls contlnuous at yt and we obtaln rens for nechanlcal quadrature processes (see e.g. Freud, $III'1' )
,!*'"
rin 6up r. \.(do) 1\'x-Yl n-* lx-x*l<e
dCI(t) S
o(x + 2e) - o(x
-
2e)
No$
l-et
or
gleater than o(x+e) - o(x-e) unfortunatel-y, 1t *6 not true that (1) is not ve obtaln the't o(x+2e) - o(x-2e). I?om the I'farkov-Stle].tJes lnequal-ltles
e+O .
t.
lx-x*r l < . where *1 =
nln xlm >x+€
\o
if
L.(d0)
S
o(*r) - o(*2)
{k: xn : x+e} ls not erpty an'l othenrl-se
2 ll8x \. or x' = -o. Sl4tpose that neither xt,!x-e 2' xtn noax xl {t: x* < r}) nor (k: x* > x-) ls eqrtY' ret - = rL <xr
anti
slrllar\Y x2 =
I
C, -
x* =lL nra, >x
'
Then
EII
,
E, Ikn(e) 5 o(x+. * *1 -*l) - d(x- " * *2 -t$)
lx-xknl<
€
a'BCl
x
ORTHOGONAI, POLYNOMIALS
I{ence }re see
that to estieate (1)
fur(tu) - \+r,n(do) I€@e
2. I€t
P€
we harrc
to
P""og,.
See
Then
S.
there exlsts a number
C
= C(dB) > 1 such that
c'^16,
and
ceroalsus,
$8.2.
I€ma 3. Iet o be an arbltrary welght, cbebyshev
the behavior of
.
2, _ < D-(da.x) -n'
for x€i-1,11
know
welght correspond.lng to a '
A
If
c sr4p(tlr). I€t rA
v^
logo' e f
1a;
^-^---C" r-. ^2rrt*) s \ "trttl o"1ty J-o' ^' x€A
for each fin with s sulteble
C
= C(cb)' > I
denote the
then
(n>1)
.
Proof. ret o*(t) = crlt; for t € a a.rra o*it; be constant otherrlse' I€t us transforn A into [-1,1]. He get a weight c** which satisfies the condltions of I€ma 2. Returnlng to A
r-, nl{*) -x€A s r-, "
x€A
t;1,(do*,*) "'-
we obtaln
\JA "ltt)a""tt) r'
5
s "cE JA--J-o" \ "lt.laot.t Theorem 4.
L€t supp(do) be coupact, A c
s
f \- ,?(rl*(r)
sugp(da),
.
vologo'€tl(a).
(do)-lc - 16s1 ag-L (n>1) :cKn' 8+frn' .6 € A wlth C lndepenalent of n and k - If At - Ao then fo" \r\*t holdls if elther 5 ot \*1 belongs to At-
rhen
(3)
* Proof. I€t v
denote the Chebyshev welght correspond.ing
be a natural integer and N =
r*(t) ls a :rn_, rtth r*(x)
=
t*1. fhen =
i(("",",t)rf(v*,x,x)
1. ry Lem" 3
(3)
to A(do) . Iet
rn
PAUI
I'B r-
(\)
(-
-'*ttl
*ttl Tben x€A'
ancl *=t(***\*f)'
Let \rxk+l€A
for x€A
< cE
G. NEVAI
I\uther
| 1 icrit** - \*r)-tl' rlght Calculating the integra} on the for J = Lt2r...rn {1th c, = c1(l(oo))' we obtaln roechanlcal quadrature forsul-a slde of (\) W the Gauss-Jacobl ln*(*rn(dcr))
t\ - \*rlt*
5 cE
tttfil2t lo(-) - a(--)l ,6
that is
1uz'
\-\+l
2mm
part of the Frttlng here n = t^6] the first 3.3,2
obtain the
we
f'heorem
'
secontl
;'
Theorem
folJ-ows' Using
part of the theoren'
c ret swp(dd) be conq>act, a supp(do)'
t.
I€80!a
e
Ttren
) 0' [cl']-"
e
*(a)'
(n ] 3)
xkn(dCI)-\*r,r(e)1cff
k' If A1 q Ao A where c cloes not depend on n and' x5*1 € \ ' (t) holds for either \ € \ o" - ..-€ t- -1, ! \a,^r that Theorem 5'3'13 and fron Lo'J
for
then
xkrl&.k+l €
Pnoof.
We
obtain from
r"* .lt*l
1
nA
\ "t'ttl
A ) I' for everY t(n wlth a suitable constant fle put n = [Ioe n ] ' Theorem I+ and finally
a"tt)
No1I
(nl2)
we repes"t the proof of
of tr ancl ! ls based on an 'dea I€t us note that the proof of Theoreros (See Szegd' 1s stTonger than that of ErdIs-Turan' Erd8s-llrran but our result a5.I1.
)
fheoren
*rn(.
)O a c supp(ocr), volo8 o' € L}( A)' "r ' 6. Let sulp(do) be conrpact, )cAo. Then (o(x + - 0(x x ". \n(do)
l*-t*rl< '
rfl
-' fl
ORTHOGONAL
unlfot:rlJ-y
?roof.
for
Uee
vt
(2)
L'9
POI,YNOMIALS
= i,.127,..v x € r, o<
€
< el where c f C(n,xre), c>
0'
antl Iheorem 4'
rct(e)
Lema?, Iet eupP(do) becoq)ect, e)0,
caocacsuIyIr(A)
then tlrere exlst8 a uunber N = N(erds,A) 6uch that
for x€r
o(x + 2e) - o(x
Lt*'<
t.-it"
-
2e)
antl n)N.
P..oof. Alply
I€!0EE
3.2,2, (2)
In the fo)-lorrlng
we 11111
(6)
and the Heine-Borel theolem'
elso
need'
estlnetes for
xtntr
It ls obvious thet (6) ls not greater than [cl(-) - a1-q1L/2' Ihe question nhether (5) nrv cowerge to O when n+6 seema to be more difflcult' U
B. Iet w be the liermlte lreight, that is w(x) = exp( -x2) for x €1R'
rhen
-l'v(x*r) , ,-i'
Inrr-r(*,"or) | 1 cn
for k = Lr2,...,n. Hence by old theoreus about quadrature sums -Ii nn d w(t)-at ' o r Ih'-tw) h -(v.r ,' kn')lccn-\ J-_ '-n-l_' n6 l]r We
lllLl
show
that this cannot
happen
if
supp(do) is compact and o
16
nlce in a certaln sense. Ior D(dor0) see Definltion 6'l-'t6' Lema
9. Let 0 € S.
Then
1ln 1nr n$
I 4"t*l
ln,-r(dcr,xo,)
t'-*
D(do,o)
KEI-
Foof. Iet n >- 1. ttren to-r(x) = Ln(do,Tn-1,x) Iet us dlvide both by *n-I and let x* . we obtein n
an'z < Yn-t(d0)
:
rL-(dCI) In.-.r (ao,n")
|
sides
PAIIL
150
Nolv
G. NEVAI
eIDIy I€@a 4.2'2. The following resu.l-t
ls a poor but very useff.rl a[alogue of
Theorern l+.2.8.
10. I€t o € S. fhen there exlsts a nunber b = D(do) > 0 such that Oc [-1,I] ls an arbltrary flnlte system of dlsJoint interval-s vlth
Ttreorep
lf
then
lol :a-b
llrr-r(oo,*or)lt o n+6 x* €tt^ \m(e)
rin 1nf x
(z)
Iet cfi = [-1,1]\n , Then 1"O it
?roof'
.
Rienann lntegrable
on [-]-r1l'
We
have
r
* k=I \"(e)
lno-r(ao,xo.,)l
=
n
:
,. < ra
xltrl
Nov t-
$
-- t .^ ;'(dc) lpn-t(d0,xktr) | + *t r"o(*rr,) Ln(dCI) brr-r(oo,"or) | I tcKII cu Aet 'ir t"nt **, t Lrrt oab2rr-rt oc, **r) 12' + t(o(r) -o( -I)) | lnn-r(oo,**.,) Li(dCI) By lFlnna 9 and fheoren 3'2'3
Let n-s . n{oo'o;
<
I
( lin lnf t-^ Lr(dc)lrn-r(ao,xor')l* f(or)-o(-r)) ? S/t-t2 nE co
atl2
\nt\/
Hence
(7)
Ttreoren
hold-s
lf
l"ol is
srnal-r'
11. Iet o € M(o,I), 1 c [-I,1]'
]-in
st4l
Iin inf t n+o x*€r
\n(dCI) lln-r(oc,**r) I
i\
u -n Jr
Then
:
'u,..- 6,,-r(oo,x*,) lt-1 "6- * tli*n+6:li x*€r
the lheoren follows fron
Theorern
3'2'3
and the lnequallty
r51
ORTHOGONAL POI,YNOMIALS
(oo)12-(dd.:<. -n-r - xn)<
t - L.En xkn
-
, Iheorem
12. I€t
6c[-l,L]. (B) lln lnf n-*
b,,-r(da,xo,)'
{i"
L,(dCI) b,,-r(ao,5o,)l '
,.j."
suprp(do)
= l-1",11, o'(x) > o for almost every x € (-l-,1),
bn(ao,*)
l;.0 t(*)
Then (nar<
x€A
lpo-r(ao,\ol
ll:$
to '6(t)-1v{t)at.
Foof. ry BetTl8teln's lnequalltY nax lPn(dcr,x) |
Ini(oc,t) I S f
for t €A.
- x€A lal "o
F\rrther, vn_r(.b)SYn(do) vo-r(o)
\n(da) po-l(e,lo,) = irn(u,*or)J-r '
*-,# llence the
left
slal,e
"
anal
fn (B) ls not lese than
S'T3"'*;.0
,,^(x*)-r
'
But
unf
v^(+--)-r-I(,r^(t)-I o * 1I JA a
r
n*-nxkr€A (
See
l!eu(l, 0rrr.9.
v(t)dt
)
13. I€t the.contlitions of Theoren 12 be satlsfled. Iet the t_,-n, lp-(oo,x) ll be unifonnfy bormdeal for x € A . fhen
CoroLLarV
sequence
x Lm'(aa) lp lirinf 'firt' )l>0. '-n-I'.(ac,x. €a n+o tC an I€t u6 note that the ?ollaczek neight satisfies the contiltlons of corollary Now
be
ue
v'lIL
dreal
to generallze the
wlth weigbtetl Bernsteln-MBrkov lnequelitles. our
foU-owing
I€@E!+. I€t J-(p(o,
a1e
result of Kha'filova [9].
a€lRrb€lR a,nd u beaJacoblwelgltt. lh; "-r11,,p S c"
lhnll..,,n
fhen
wlIL
a62
PAUI. G.
MVAI
and
("fiT * |lbrt
l"i(x)l t,A - * * *)*t
t
nax
l*ls
( cn,ma:c tl^n(x)l
t^if -
l" lsl
for
r(n where C
every
Lema (e)
tloe8
tlot
depencl
f::p<-. Ir. I€t e)-I, 12 I l.;f .t lp lt l" at
for evely
xt1
5
" * |lt t,6;= * |lbt
on rrn and n
.
Then
.
.
np
[-
ln.(t) lp lt la at
.
r' be even, tha,t 18 tet rrr(x) = Crr(*2) . and. we have to show that n'(x) n' = 2xc'(x2) n' Proof. Let flrst
c1 \\- lx c;(xz
orf
)
lp
l*
l"
rr*
Ihen
( c nP \ '^ lo"t*'I lp l* I' o*
p+a-1
a-I
L lc;(")lpl*l t *Sc"pIJotlc-(x)lplxl2
Jo
p*l-r A + lc;(")lel*l ' *S.t lc;t*lfiva-"lpl*l 2 dx
But
.1 \
JO
Hence
(!)
,rrr(x)
= xcrr("2) . rn this case, ,rr,(x) = crr(x2) * z*2c;("2)
foLlorss from lemna 14 when
Drove that
a\
\
8rd
dx
lp l* I'
dx
<
c nP
1o.t*2)lp l*1"
dx
s
cnp\t l* o.{*')lp
1"2 c;{*2 )
(1 -I
rn is even. Let norrt rrn be
!j-
t- "rt-'t ln
l*
odd:
and we
wlrl
l' u*
l*1" o*
.
first inequality here follows fron the first part of the proof by putting there p + a lnstead of a . The seconcl inequality hes been proveil in corol]-ary The
5.3.26,
Hence
for evely r'
(p) holds if
r.
is either
even
or od.al. But then it also
wlth a possibly bigger constent C
From I€@as 14 a.nti
Ii follows
.
holcls
ORTHOGONAL POI,YNOMIATS
Theoremt5. Iet 1= tl>t2t..,
ro5
>tN= -1' fkt-1
Nf. w(t)- fl lt-tnl o k=L I€t 1(P(o. wbere C
fhenforeverlr
doeE
(-rStSr).
t(n
lh" t''l[,p I c" lhnll*,n not depend on n and n. .
f7. I€t a €lR. Ihen there exlsts a nleber
Ier@a
every
for k= Lrzr,.,rll-,
e
= e(a) > O
Buch
that for
'tn na:<^ I'r"(x)''| < c nu , r"t t lx la ln-(x) 'n ] l*lSf ' n' - *S l*lSr
(10) lrlth c=c(a). koof. Iet first
0.
a=
Then (10) follows fron le@a,s 6.2.101 6.3.5 and
6.3.22 applted for the I€gendre welght. If (I0) holds for a = O then it
hoJ.tlefor a>0.
If a
(11)
nar.
-'
Inrr(x)
l'.ls*
|I c
t!ar(
f:l"lsl
In"(x)
a]-so
lqrJ-iesthat
|
rith a posslbly new e ) O atril c. Iet w be deflned ty w(t) = ltl-a fo" -1 < t < 1, supp(v) = l-1,f1. Rrttlng fn (U) Kn(w,xrx) rrr(x) insteaat of rro(x) we obtain ftoE Le're 5.3.f9 nar(
l*l<3
Inn(")lnt-"Sc
'|-
Hence
(1) follows for
a
Iema 18. I€t a € lR. |
Sr
( o
, tlnrr(x)l l*l"tn. l"lci
al-so.
Then
na:r tl,rj(x)l l*
nax
3<
(l"l * |l"t
g
co
&ax (l,ro(x)l (lxl * *)"1 l* l5z
for every r(n . Proof.
Repeat the proof
U Iemas
of
I€mnr 15 anai use Lemes
14 and 18 we obtatn the followlng
I4
anal 17.
rb4
PAUL G. I{EVAI
TheoreBl-9. I€t \€tR w,r(x)
for l"ltr.
=
for k=1r2r...;I,I2 IrtAr...,tN-I
*nt tl* - r, l * k=2 ^'*;/
(^,[- * !)t\ -
lcr (Cn
tlax Ilrrr(x)l wo(x)) l*
for every rro where C ls we
Ir\ r.m (vr+x+;/, 1.,2\
Then
lx
Now
,_1.,
inalependent
lsr of n and. rt_ . n
retuln to estlnatlng the iuetance betlreen trro consecutlve zero6 of
orbhogonal lo]{moEla].s. Tbeoren
2o. I€t srap(itr) be conrpact,
a
c sr4ry(alo), t* € Ao, f > _I. I€t
a be absolutely continuous Ln A lrith
0'(t)
-l.-r"lr
(t€^).
fhen
for 1*e5cao
1r(ac) -\*1,n(dCI) -*
koof, & Le*" 3.3.2 we can sullpose that both an and 1*r,o First lre sha].l sholr thet 4
be1on8
to
\rr-\*r,n(cn-l We
have the foAlovlng possiblJ_ities.
(12)
**l
t <:<,-,(:{,(t-+l _ K+l A_
n
(13) t*.\*r-
or
t"-*S"**r<*nSt* *l tu -: * -;<\*rSt*.*.t**],
or xo*rct*-*S\Sr*,
or \*r<,aSt"-*, (1t) l1*rSt*-*.r"<\St*+* or ."-*S\*rSr **
(16) \*rSt*-*..*.*st.
ro)
ORTHOGONAL POLYNOMIAI,S
In aIL
cases (L2)-( 15) we,n111 use Theoren 6,3.21
tr
do(t) s \sn(dd) +
s,Jld'
the estieate
\*r,rr(do)
(k = r,2,...,n-r)
\+1,n
whlch follolrs from tbe }S.rkov-stleltJes
lnequalltles' In the fLrst
c&se
of
(13)
we obtaln
= .
(*r - t*)f*l Hence
S
c["-r-r - *r** - t*)f]
. tn the flrst cage of (tl+) we have a < t* * c| * F ' .x.l+l , -*rf*' - t )' + (\*r -' cfi(x1 (rqtB[+r- t^)'*- - (xo,. - t )'''<
x.
r-
_
*r
" t)
a,!d
slnce f+]- > O.
Tbe other
possibllitles
rnair
be treated" s{m{}ar\y' To estlrate
fron below let us renark as Erdds-turan ttld that \ - lc+f r = (*u - \*r,n) $ rfi{u",**l where ***t S **:i ** . lle bave
rf,t*,*) Uslng Theorerns 19 snal
6'3'2rt
: Ikr(dCI) rn*(do,x)
we obteln
lf; rfuto,"l15
cn
fron below for xO - xO*t untformly for xlmrx e 5 c ao' Hence the egtlnate fol]-olls. b>0 2I. I€t surrp(dcr) be coq)act, A(e) = [cr'crJ' a ) -1' I€t o be absolute\y continuous 1n [cr-DrcrJ and ]et o'(t) - ("a - t)t cos 0* for f,or t € [cr-b,crJ. i€t xkn(do) = ]t"r*"r) * ]{"r'cl) *n+Irn="1-' lfhen and x*=c2r where oSe*1r k=orlr...rn+l
Iheoren
t**r-'*-* for rLlrr.c€ Ac (c^-6rc.l
PAIIL
G. NEVAI
hoof. t{e calr assu&e lrithout loss of generality that A(do) = I-J-rlj anA 'l DS t . (concernlng the second. assumptlon see e.g. neua, otU.r.) We obtaln rmeillately from Theoree 5.3.a7 aoa Me.rkov-stleltJes. rnequalitieg tbat
6r=o(*)
antl eil=o(n).
Nowwewtrlehowthat eil=o(n).
ret n>n
be flxed. Then by the Gauss-Jacobl necbanlcel- quadreture fornu-l-e (l- -
nl (t eE - t)xL(dc,t)do(r) =
x*)I.(do)
)_r_
(r - xh)r2h(dCI,xh) k=xr
Ln(da)
.
Hence
(r -xtn)\.tt(tu)
and.
: (1-xr.)
= (r consequentl-y
x
bi-
m
ri(do,x*)\*(dCI) -
- x2n) rr_n(dCI)
1 _ *r,. >
hrttlng here
,
m
ir
t? (a.x^h\*
(r _ xrr) Ir
(1 - xrr).g]-tl(dc,xrln)r]!(do) =
t
\r(dCI) l
I- d0) _;ffij: (
.
= Nn lrhere I,I ls blg but fixea we obtaln from rheoren 6.3.2?
ths.t
r-*t :](r-*rrr
lf only n ls big enough. Hence ei] = o(n). To prove gt*f - t = O(rr-1) for \ e a c (f-S,tj we wlIL use the inequellties (u) "J,_n x (1 +x )L (do).-J-I tn (rrt)e(r) I r (r-+a-_)\--(oa) (Lr,...,n) k=1+1 -lgl r(ll' S \ ls alvays true ff A(do) c [-IrlJ. We shAt.l not prove (17), it can be provear ln the sarne lrey that F?eud proves the l{Erkov-stleltJes lnequal-ltles ln whlch
bls book. Iron (f7) ek+l
-
and Theoren sk
6J.q \te set for x* €
A
: sup ! -J=1urn2*3* * -'os",islffi s
s1t'243t]
is of order f sfnce 2a+J > 1. The estinate [e**r_eOJ-l O(n) for = xn € A follows froe lenma r-r+ and rheoren 6,3.27 in the same way es we obtalned which
the estlnates fron belov ln Theoren Theoren
=-1,'
22. I€t
O<e.Kll-
w
be as
rr)
Then
in
20.
Theoren
15, 5or(w) = cos g*
(x* - 1, xn+lrn =
1trJ1
ORTIIOGONAL POLYNOI,IIALS
! a "k+I,n -o-kn- n for k = OrIr...rn. Proof.
Use theorems 20 antl
2l'
Ibe followlng lnequellties rl]-t play a f\IidrneDtaf role ln lnvestlgatlon8
of
DeaJx coBvergence
of lnter?oLatlon
processes'
23. Iet drArt*rf and \ be as in rB S clD . Thea for eacb t(m 8JItl n fbeoren
r
*!ot € \
thecrem
20' I€t I( p < -
a'nd'
Inr(1-)le 6,ta"l < c to l',(t)lp u"(t)
wbere C=C(orAlrPrcl).
Proof. Iet
w=
Ioor wlth
rr,(v, 5( \m(dd) -
sr44r(w) do)
)-
- a'
t*o(v,n, *or(
since nScln'
for x*(ao)€5cao
Then
by ltreoren 5'3'2t
do )
) - i'rr*r(w,r, x*r( clr) )
Hence
In.(x*r) lp r,orta"l < c
!o In'(t)P
*ttl
tlefor x*(do) € ar. !\rrther ve can suppose that t* € Ar' I€t i = J(n) be * fine
r(
t-
€A-
In,(x*r)le iortu"l
.
*?i-t'
Observe
that Inr(1or)lp
< In^(*)ln -
fot *k*lro S * S xk-frr,'
n
^\_f.o
l'r'(t)lp-r l"'(t)l L^ ^k+r' n
dt
Thus by the Markov-StleltJes lnequalitles we obteln
:Lllll t&r k< J-I 2 \ In-(x)lp dc(x) s--Ja't"'
+p
It-. lnr(t)lp-I l^i(t)l * t^-"'^ L. \m(dCI) '\*r,. "k"€\ K<
J.I
at '
PAIJL G. NEVAI
r6B
atfi.6.3.25
By Theor€Ds 20
r *,r -i lt - t l' 1f k<J-I and *I"r€\'
\m(dc)
fot \*lrnStS\-r,r, .
f'
consequent\v
--k-1. n -
I'r'(t) lY-- ln;(t) ldt S \rn(da) L t '. ^ "k+Irn 'f. "tn k<J-l
s
r*{ t l*.tttl!
p-I
> o is chosen so thet \(e) c ao' Ef Ienna 15 ve obtaln
e
,
x'xnl€'"n k<J-t The sun
ln'(t)lP lt - t*lratlP r,.n.
ar( e)
\(e)
where
l
f. (J tqer lt -!- n*rfor.iT
In*(1rr)lp lo,rtool S c T
So
l^,,t,lp u"(t)
'
k > J+l can be estlmated slrnilarly'
for t1o € \r
24. T-et c,rc}tarl^ antl A be as ln Ttleorem 2t' I€t f > -l - a' * Then m
Theoren
for each rh where Theorem
*
C
= C(o,PrArfrc ) '
2t. I€t w be as in anal mScn'
f:p<-
Theoren
16, o e {
Ihen
n-a Inr(*or)lp'(*ur) rr*(w)
-ileI
for every n, where Theoletn6
applJ.catlon
2\
anti
C
2!
Ue a Jacobl neight'
!
c
w(t) S-- l'{m(t)lp ''(t)
d't
= C(vrurP) ca'n
be proved' by the
of the prevlous results,
we
sa'rne methocl
as TheoreD
23' As an
wiIL ?rove two theorens'
25. 1"1 ., - "("rb) be the Folle.czek nnelght deflned. 'n Definltlon is 6.2J2. I€t I ! P j @r P I 2' Then the sequence of operators [srr(w)] not unifonrlY bor:ndeit in {
Theoreu
Proof.
(f = O)
By Corollary 13 an'I T'heoleB 23
inf \^t ll.(",t) n.* J-l
Ltn
pl2.
I-et l(p(e,
for l(q(o,
L59
OIIIAL P0LYNOI'1IALS
OHTHOG
lY
w(t)dt >
o
SuItIloBetbat
sup lls-(w)lln>I " q -i{-(-. Ilren rre obtab exactly
ln the
sa'ne way
aa
ln
llheoren B'L3 that
a "tE ",.tplb-(rl)ll"-<-
n>]anal
e"p
D-I
( lb-(")l[ -, L', "v
p' ls greater tharr 2 thls canxot where pi = 5ft . Slnce eltber p or p = o' the Tbeorem fol]-owB from olcl rehalpen by Theoren ?.31' For p = I or sults.
(See
e.g. trteud, renarke on Chapter IV'
llxecEetr
ff.
lheoren 26 reroalns valid'
by the velglrt ciefinetl ln
proof.
b(a.ruple
we replace the Pollaczek welght there
5'2'Il+'
By Koroust theorem (see trYeud, EI'?'
forrnly bounded on eech 6 c (-frf)'
Deflnltion
lf
if
The
28.
suPP(v) =
)
Now
) the correspontllng
system
we can repeat the proof
of
Is uni-
Theoren 26'
(w welght w ls called a generallzetl Jacobt welght
[-1,1] w(t)
€ GJ)
antl *'
F
f, * t) = q(t) (t - t) l^ l\ - tl'k ir K=Z
I
N
ca(>o) ls contlnuous -1 (k= L'zr...rN), It t2 t "' t tN-It -l' of contlnuitv of on l-r,11 ana ur(o)/o € r,1(o,r) where o ls the mo'lulus w*-GJ' Henceif w€GJ then e. If n*-w where $€GJ thenwesrite e,lso w - GJ. For w - GJr "n ls defined by - 2f-' 2f. N-I ,'t- - tl * (^/T;E - *) N !)''t * (d[*)'k 'n(t) = J,
where frt
(-1
StSl;
n
=L,2'...).
u0
PAIJL G. NEVAI
IfjIfirf. 29. I€t
lt € GJ.
Then
(^ffT + |l for -1<xS1, Proof.
n=1121...
See nadkov
t^,n.
c(nrx)
[2].
Ihen
ll,r-r(oa,**) | where x* = xor(oo)
- tr - d)
b,r-r(a"*,**)
|
.
Proof. In course of the proof of p. .
converges
I
where c
I€ a 30. I€t o € S, g = v-2.
The erqrression
. * |) *n(x) {(w,x) 5 c
in the brachets
to l- when ns
Theoren
aloes
lr'2'3
we have 6hown that
Yn-r(dos) Y*r(tu) n-l
, .vn-r(do)
;@J-
not
on k
depend
and
l.
'flt."r lr'2'2 it I€loee,
W
.
31' Iet w € GJ. Then z , i\ /F-l*-,=\ wr(xor) li-t(w,xo.,) -^/1 - *k" - (^A - *k" t t) (v/r + xkn + ;l Proof. If w € GJ then v-2w € GJ' Hence by I€mas 29 snd 30
Theoreu
vr(x*) 3y fheoren 22 the
p;_l(}t,xkn) < c(VI -
_.,,=-+=). (vr + rtn * ;)
"k"
rlgbt sitle here ls - /;A'
n/
The converse inequallty
follovs fron nn-r(w,x*,)-1 =
-,$i
and fron Theorens 6.3.28, 19r 22 and' Le@a
for ior(w).
Theoren 19 and l€ma 29
al\y,
22 shows that
Theoren
r-6i-1.
A - tr -
(^/]
ls
\,(w) 29'
used
- *r* * ;)
ni(w,1o,)
Theorem
5'3'eB glves us
bounds
to estinate lln(vr*or) I ' E[n(^/r +
xh + ;J
32. Let o be an arbltrarlr welgtit' I€t x*(dr) = -r xn*rrn(tlr) = --' rhen (b0, " ' 4on(dsrx) =o arit ,n+r,n((b,x) =0. *t \*t,rr(uo) 5x5xor(ao) 'n)'
Ie@a
VT
ORTHOGONAI, POLYNOMIALS
[*(da,x) * rk*l,Ir(&,*) : r [)].
Proof.
See Erdds-Turen
Theoren
33. I€t w € GJ.
(ts)
.
trn(w,x) n2n(w,")
Then
-
- *")2 t"ffi, . |l-t tvf-
n(x
* |l-t
for -1 S * S I where x* is the zero of Prr(wrx) whlch ls closest to x ' koof.
22 8nd 5.3'28
BY Theorens
rfu(v,x) 5 l*,(w)
ft*,*)
5c r for -1 S *: I wbere k ls the lndex of x* ln (18)' zurther' lf k = ,nd tSxSl or k=n and -rSx1\ thenbvL€m832, lfl(*'") 2r' (k > 1) or x* and x**, (k < n)' OthenrlEe x ls betveen either x*-, *d
\
I.€t for stuPliclty,
ry fheorens 22, 3f
\
xk-f'
l-et@s 32
h-t,n(t,*) + t*(v'x)
>
r'
and 5.3.28 |
an
SxS
[hen by
tr-r, rr{")
Prr-1(w,xtr-1, rr) I
obvlously fu-r,rr(*r*) > o, ,0*(vrx) >
= -sign lo_1(wr1rr)
.
o
- 5rrt"l |
x*)
I
anci slgn pn-r(v'xn-1rn)
=
Hence
,k_l,rr("r*) < c x*(v,x) Consequently
nrr-r("'
ln a}l posslble
.
cases
rfi(",") - r
.
The Theoren foILowE nov froro Theole!0s 22, J1 anti 6 1 '28'
corolLary 34. I€t 1I € GJ.
Then
-1 (w.I) rt "' o -n' '
-.r
and
llrr(*,-r)l - n'tltu. hoof,. Inthiscasereither Renark
35. Let ora
8,ntl
k=l
or k=n'
c2 be as ln Theoren2l' Then prr(tlr'cr) 2 Ctt" tot
PAIJL G. NEVAI
n = Lrzr.,. and
c I c(n). This fo]-lowe imedlate\y ftom I'l:eorens 2L, 5.3.4
'where
tr -r2 frolo trn'-( tkr, x ) =.t_ U(dc) 5n(da,x) .&t
35, Iet su!p(do) be compact. L€t there exist 8n intellJal t the sequence ( lprr(aa,x) ll 16 unlfornfy borrrded for x € r ' Then .(e) . ^ Yn-ft0. rtulnf I€@E
sucb tbat
n' "*ilt
n.s
koof. ry Theore4 ?.t, r c 6utrtp(o') c supp(do) Iet r = lct,crJ srti a'nd be tlefineti bv "r S \,n 5 = ror(n), % = %(n) \*r,o' req)ectively wtrete xn*trr. = -@r *Or, = -' ry Lema 3'2'2 S "Z a \-fro \rr Ilm x- - = c'| antl lIn x, -, = c2. We cen suppose that d is continuous at ' n* 2'n*- \r" If not ne can replace r by a sroaller interval' Lt u, < k S to,-' *a "r. ", rhen bY I€@a 32 x. , +:qrq- , + x,-
q.-r.,rr(*'l#) * rtrr(o'Jff) : r'
thatls (da)+L (da)l u L\_1,n,*, z *r_r - -\: -orr-r(*)rr, '.t(n.*'. T1dc-r where
C
= sr& nex [P-(oo,x) n>-o
xer
.U Letting n+-
we
|
Hence
+u 'n-'(*) \ :''"'? -q1Ef
obtain
lrl
<
[f ^ ltu
lnr
n+6
(tu) '. ,'*.*, F )'' *],^
y_ ,(do) n
#*f 'n
\
dCI(t)
.
3?. r€t 6wp(dd) c l-1,1J, 6u!p(de) c [-I,rJ, r c (-r,r) an'l ret da('b) = dF(t) for t € t, s be absolute\v contl-nuous cr = [-l,I]\t. ln cr and let exist tr'to polynolals rt ancl ra such that and Inr(x) rrr(o,*)lS r ror x € cr antl n = L,2,"' nLa'/A' e d(c.) Theoren
ihen
lpn(d€,x)l S ctlpn(ao,x) | + hn-I(dCI,x) ll antl n = L,2,"' ' unlfor:nly for x € "t - "o
ORIHOGOMI, POLYNOMIALS
Proof.
We
can sultpose that neither
p(*) pn(ap,*) = 5_, ott)
qytheconclitlons ^l\ fla"=1 J-].
x € r, c ro.
has zeros
^
1 (
J-I
!n\w,
Krr**r(ckr,x,t)
u/
e(t)
.
)uF.t"c1 ()ao-["ct tlue. t".
fhen
p(t) pn(de,t) Ko*r*1(do,x,t) dF(t)12
{t
in "o. I€t
*"n",
degF=n'
P=n{2,
,2
nor
f1
"ct
s c I lpn*r(do,*) |
*
<
l]'^
lprr***1(ds,x)
^1 \_r_
"'rtt)
aett)
and
t\vct p(t)
:
pn(ag,t) Kn***r(do,x,t) e(t)12
c[ lpn+n(.i],*) | * In,,***r(oo,*)
ll2 5""
F\rtherfor n>m ^t \,_r_ o(tl pn(d8,t) Ko*r*r(dc,x,t) de(t)
f
S
.l$f$.
f
!"" nituu,.)u't.lur'
=
'I
n+B
tn*(ao,x) lo(oo,t)J de(t)
= \ - o(t) lrr(oe,t) . I
.
consequently
^1 l\J_1 e(t) !n(dP,t) Kn,.,1(do,x,t) dF(t) | S
s But
c ;' X=n-Itr
r1
lp,(,b,x) 'A
| ([.--I
e(t)2
f (uo,.t de(t)]2
.
1
\
p(t)2 nfiuo,tl dg(t) =
\Jr
Thus we have proved
e1t12
=
pfl1ao,t) oo1tl o
*\
01t12 pfl1ocr,t)
de(t): c.
'""
that for x € r. c ro n+n+1
r ll*(0o,")l . In.(as,*)lcc" k=n-E v- . (rh) S Ie-" 35 the sequence tffofl 'n'
ls
bountled
froe below by a posltive
stant. thus by Iema 3.3.1- entl by the reqrrrence
formufe
con-
PAIII
ln*(a,x)l whenever Renark
nfl,
38.
n-k=O(1)
| + lp,r_r(ao,x) lJ and x€A nhere C*=C*(ln-klrorA).
S
.*t
Inn(oo,x)
Theoren 37 becoDres u6efu1
with resul-ts ln
G. NEVAI
lf
we eoubine
it wtth Korous'
a.nd
Freud.rs end Geronimus' books.
r ls of the forrn iarll where lrl < f and 11 c (erIJ or t, c l-)_ra) respectlve\y. Theoren 39. Theoren 3T renains valid.
Proof.
theorem
The sarue as
that of
lf
Theoren iZ.
or
[-J_raJ
10. Iagrange Inte4)olation Elrst $e nlIL
consl-der the I€besgue
functlon lrr(dcrx) of
lagra.nge
lnter-
polatlon correspondlng to o whlch ls deflneci by n
to(da,x) = E lrkn(do,x)l
.
Ihe egtlEate
(1) shor.s
S {l(dCI,x) [o(-) - o(--)]
{to,*) that lf
o
at x then
h&6 a Ju4l
l2(aa,*) n
o(-1,- o(--) <, -c(x + u7 -E(iTT)
but ln general (1) ls not a very strong result. Our atm rrtll be 16 r.Eaove (I), IF@e
1. I€t supp(cf:) be coupact antl
e>
O,
fhen
).n(&,x) {{aa,*) < e,
for x€A,
a=L,2,..,
Proof.
Repeat the proof
Theoren
2. I€t d
€
r, _ \tl(do) * Ln(
of Ie-" 8.6
ancl app\y I€!@e
3,3.1.
M(orI). If cl ls continuous at x € [-Ir1]
(2)
then
rr$ rn(dcr,x) {{ao,*) = o . n.s
If ct ls continuous on the closetl set 8c (-lrt)
then (2)
hoJ-ds
unlforroly
for x €8. koof. AFpIy Theoren l+.1.11,
I€!ma6
Theoren3. I€t o€M(orI),
1c[-]rIJ.
9.1, 9.7
If
and
L
Jq'l-€€Ll(r)
wlths@e e)o
then
(3) foral-nostevery x€r.
lr^ n-!/2 r*^(dc,x) = o
n.s
If ot(t)>c>O I7q
foralnostevery t€r
and o
PAUI, G. I\EVAI
L76
c ro ' ls contlnuous on r then (3) is satlsfled unifonoly for x € r, Proof. Renark
llheorem
8'9'
for L19*' 4' Uslng fheoreB 3 we can eeslly obtaln convergence theorens
In the folloltlrg
If
ln the proof of
Repeat the leasonlng
we can esttnoate
we
vl}l lnvestigate {{ao,x) defined *)ltri L;(do,x)* = ; t- t{{*,*) ' " t=l "
*, I1(erx)
by
we can also estfuste the Lebesgue function
of the
of Ia€range interpolating polyno!0la1s' vhlch we d'enote by can also estlmate slnce obvrously f,n(do,x) S L;(ddrx)' Moreover' we ?.-(ao,*), -Ir' the convergence rate of the strong (crl) neans: (Cr})
mean6
on\tlo,f,x)
ttt*l - h(dd, frx) : *,i. lf=I
I
in c(a(o)). i€t Xk(f) tienote the best approxinatlon of f bv r*_, o,,(do,r,x) a Hence
by Jacksonrs theorem
*
j, tt .
to(oc,x)J Ek(f)
Then
.
a ro *r2 r j 3'1"' ut'r/z t'
!
t;nJ*1 o t"':-< s T;(dc,x) -n'-' 'n''(r,do,x) rrhere (lJR denotes the R-th
r. l€t e r,I(r; . lnren
Ttleor|em
sufP(e)
modu]-us
of
smoothness
of f
beconxpact, rcsr44r(do), e )O tlc
and [o']-ee
+
rin sup n--rJ q(tlarx) < n* fora].nosteverl;x€r.If0€I,iI)x}ando'(t)>c>oforl*-tl on r and o'(t)>c>o sna.ll.then(l+)hotcts. If o€Ltpl r, c ro ' then (l+) ls satisfled untforn'ly for x €
('+)
for t€r
part of the Theorem' hoof. For slu;)Uclty let us prove the first sntt e= n-r/3. Ihen uy Theorero 9'5 arrd L€@a t x€rrcto
Iet
ORIHOGONAL POIYNOMIALS
1'7'7
l_12
\(oo,x) {tuo,")
* "o-5) - o(x - *-5)l * croi \(e,x) rl{o,*)
< 2[o(x
l-T;' *. .2 r 5 r 12n {{oc,*)' n#;r k= Since O 18 al&ost
that (l)
everXrwhere
I T kj;or, *
"t
-TIJ)
tllfferentlable
-tL 3)l + c,n -f''
-o(*-"k
we
.
1
3
)-r '-crzr x / ' ''o*1(
obtaln fron Tlheoren 5.J.25
for ahoet every x € rr. But trc ro ls arbltrarlr. Iet ue note that the second. pa].t of theoren t ls not nev. (See Fteutl, hoJ-als
SoEe uDsolveal
Renark
problens.
6. App\yllg
)
TheoreD
of regrange lnterpolatlon coro1la^:rJr
7. If
5,
cotrvergence
polynoroJ'als
of (Crl) antl stlong (Cr1)
for f € ffp]
BEatrs
can be proved.
supp(dcl) ls coqract anti ;g'1-e € f,I(r) wlth Bone € >
then
o
-* rtnlnf n rL(do,x)
foral-nosteve4; x€r. T'heorem
8. Iet
cr €
S arid x € [-1r1J. lhen Inn(o,x) | S\
for n = Lrzr...
fr
o(oc,o)-r Ln(tu,x)
.
hoof. lfe obtaln fron I€mF 5.f.19 and froE the ineguallty
between the
arith-
metic and geometrlc neans that Yn-r(do)
f\:rther
r(4,01-r =r"-' J?
.
(,b' n h lpn-t(oo'xo') pn-l(do,t) < do(t) 2 ; L-(dd) y*fatT J-, 'n-r(t) k=1 nn' l" - "r.rl for -l ( x ( 1, Hence the Theorem fol,Lows. ^n-2 1l
=
^1
9. In gene"al, fheoren B carurot be iuproved.. Iet fhen v€S and Remark
w
- GJ vtth f, > O.
r;k=l u"r"lHdi2
|
Kn'
(1 -*r,",)-,
PAlJt
1?B
which
ls
bounded
G.
IIEVAI
by Theorens l+,23, 63.28 and.9.22,
Deflnltlon lo.
I€t xon(do) - -, *rr*l,n(&) = --, .e*((brx) = 4o*rrn(tbrx) = 0. I€t n = n(nrx) be
i.(ao,x) -k=1*
n
= r tfn,
11, Iet o
=
l4-(ac,x)l
.
ro+1-
ilelght. Then I.n(do,x) - r-in(&,x) for x €n and n = l.r2r.., . I€@a
be an arbltrarT
Foof. ry Ie@a 9.32 in{a",*) +LSLn(ab,x). On
the other hanal, since ,nn(darx) >
tr(da,x) =i,r(acr,x) * t -
0
anal 4r*lrrr(&r*) > O, we have
;- /t*o(do,x)
2in(dd,x)
Recall that cJ ard wn have been deflnetl in Definltion !,28. I'heoreE
12. I€t $ € GJ. Then nlx - x{l
(5) l.(wrx) - r--:::=* "l-21 4L for -I S * S I hoof.
. \r
,.,, I-
II +;.v I t' l" +. -
.wn(rJ(^,/f-t' r__::_____-------:1*
x *i nl*_rF"ff4
lj
.. /--\/.' -x wn., {x)t ^/J.
W a loEg calculatlon fron
9.3I, 9.33 antl Lerma ll-. For the case when w ls a Jacobl weight
fheorero
.-
n
-
rrhere *J denotes the zero of pn(v) closest to x
Tbe lheoreu follows
tl
.
Theoreus 6.3.28, 9.22t
l2
hes been proved by
[10]. fhe lntegrat on the rlglrt hand side of (t) fs a rather standaral one, lt can easi\y be est{nq.ted but the flnal- forsula is so coryllcatecl that we
l{atanson
wlI1 oolt lt.
We
wiLL for:uulate only one particuLar cese
&s
'l'7
ORTHOGONAL POLYNOMIALS
coxo]-lary
13. Let
1r €
0
GJ. Ihen l-r
for -t.
I
for -L '1-
2
for _L Lt'
-d '
t-
l.(w,r) - { rogn -11
I ln
Before flndlng necesEary eonclltions
'r*e for
t, . -*
mean boundedness
of Iagrange tn-
terpolatlon plocesaes, let us nake sone rmarks. If we define tn(e)
by
tn(do)f = Ln(&rf) then the norn of tn(do) as e napplng frm some Lq (o < q < -) is never bornded, f nust alvays be bounaleal ln A(do)o. To avoid eoupllcetion, rrhlch we caturot solve at the present tfuoer ire wll-I
ls
on A(do)
bountled
and we
vill wrlte f € t-(A(e))
&ssume
where lFll-
that f
=
An iul}ortant dlfference betseen Fourier su$s and Ia€renge
= sr{p lf(t)|. t €a( do)
lnte4rolatlon polynonlals is that tn+I(dorf) - Ln(dorf) ls not proportlonal to
pn(do)
and.
If
we
wrlte
introduce the notatlon
for 1(kSn-l
I.n(d0,f,x) = -';^ .- pk(do,x) k=0 k-1
Lr(do,frx) = t a.pr(do,x) rrr^ k=o d J then
Ln(dorfrx) - Ln.n_1(@,frx) = an_l pn_f(al}rx) where
obvlously n
(n\ +l* a'n-1 - s 1 -"1tn'\ no_r(dorxor) g-"1 'lrn'*' fheoreru l]+,
I€t either o € S or o satlsfy the conalitlons of, Corollary
9.13.
Let F be an erbitrary weight. Let us conslder the fol]owlng three conditions. (i) s'rplll(dn)ll - (-, n>_f " L-( o( oo ) ) 'l'* laI,
;g llL,n-r(do)
llr-(o(*)
and
;I?
lbn_l(dd)lluu,n.
-
)r.&
.
18O
PATJL C. NEVAI
lrhere p € (0r-) is glven.
Then each
pelr of (i)-(flr)
iqrJ.ies the thirit one.
hoof. App\y I€Ma 9.9 ana Coroua.r:f 9.13. The
following fheorem 1s one of our Eain
Theorenl). I€t o€s, o(p(-
!esu-l_ts.
w(>o)
Thenfron
1!o
foll-owe
p
(5)
!t
,o,
zw(t)at
(t),ltll
v_L
<
-
.
Proof. I€t 6 = b(dd) > O be tiefined by Theoren 9.10. I€t r c [_1,1J wlth Itl = 3. Thenwe can fintla systen o= [t1,r2] w.ith ,rrr= fr, Inl >a-o and dist(rr0) > O euch that (9.t) ls satisfieat, Let f be a ftrnction on [-1,f] whlch setlsfies the conilitions lfll_ = f and r(fur) = rn(1r) sirr[%-r(do,fu,) (fu, _ a)] rrhere B is the center of r , Of course f depende on n, r, q
and
o.
We
have
lEr
Since
Ln( do,
r ) lf,
nS
l*-1rl 1z for x€r, \o€n
lli.n
{ao) llr_*"n
veobtaln
,.^ lkn(do) In"_.,(ao,xu.)ls, rr-r _;t"(11.' rul"'t *hr€n ' l rn-l\u/ lE_(do)il -' f,--{
llr" nn(ar)lL,n
Letting
n+@ we get
fron lem" 4.2.A anil
Theorem
9.IO that
rlm inf llr n (ac)ll n.* " r -n' ''wrP Hence
by Theoren 7.32
S" ,*'1.y
rE-
E,(t)at < -.
this lnequallty holds foI every r c f-lrll > O it al"so hotd.s if r = [-J_rIJ . Slnce
with l"l
D
'2
anai b=D(do)>
uslng the results of sectr-ons T and 9 we ca.n prove slnilar theorens We restrict ourselves to the follovlng
o F S.
when
ORTHOGONAL
Iheor:en
16. I€t
POLYNOMIAI"S
1B].
6upp(do)
= [-I,Ij, o'(x) > 0 fs1 arnngf, every x € [_l,lj interval r c f_lrl] such-that tbe sequence {lpo(Orr)l} lsunlforalyboundetlfor x€,r. I€t rr(>O)elr(_frf). If O
1et there exist
a.n
sJco
,, r-1_rrr;*{ ( ll5 llt-(do)ll
Ilm oup then
ltu
sr4r
n*
If p>2
and (7) hotds then
lb.ta"lll, ntY -<"
o
.
(6) is sattsfled.
hoof. $ the conclitlons (llr" no(u)lln,nt is bounated. I€t cr [-1,1]\r = I€t f be defined. by lbll_ = 1 ajrd r(1-) = 1"(\rr) Btsr pn_l(tu,\D) . Then
lE"" nn(o)ll*,n
r-
t(a"l _ '- tmer -sr
Non
In,r_r(oo,xo,)l y_
S
(do)
fridI
lirn(tu)lf _-,n
.
'-\,r
the fheore&. folLows fron Fheoren./.J1, Corollary 9.13 anti l€r@a 9.36.
17. Let us ssy thet o Just bel0ngs to s (o € .rs) lf o € s but for every e ) o, Js'l-ev f, r,1. Dre.uple: o,(x) err;t(-(t_"t)-u] (o
18. I€t elther o € JS or o be e Forls.czek weight or 0 be deflned by I o'(t) = q(t) e:cpt-(r - tz) d ], q(> 0) 6 Lipl, supp(do) = [-lrrj and. o ts absorute\r continuous. Then for every p > 2, there exists a functlon f € Cf_I,l_] such that corolr-ar:r
11o sup
ns
hoof.
Use fheorens
\t u_I
lrf.l - Lh(.b,f,t)lpo,(t)at > o.
I5, 15 and Banach_Steinhaus,
theoreE.
r8a
PAIJI, G. NEVAI
I€t us retrark that CorotLary 18 gives a uore or J"ess compJ-ete answer to I€t us recall tbat Turart Tr:ran,s problen ancl verifies Askeyrs contrecture (llJ). askett
lf there exlsts a velght o with
clusion of Corollary 18 holas and Askey
= [-frl] such that the conconJecturetl that the Pollaczek $ei8ht sr4tp(dd)
solves furan' s problen. Theoree
l-9. I€t
sequence
.l(
c [-1-,1J, 0 ( p ( o, w(> 0) € L-(-frl). be glven. If for every f € cl-lrl]
supp(ocr)
t2 (...
u'\ ^1 It+6
"-I
lr," (do,f,x)
r(x)
Let the
lP w(x)ox = o
K
then
Iln sup lE- taolll - . < -. "k L *\; k*-
/R\
*l-r..
ff p I l-, then the Theoren follows from Bs.nacb-Steinhausr theorem. O < p < 1. Iet us clefine the functionals (fu: C[-1r1] +lR blr
,q(r) = ^1 \_,
l"*l*,t,x) -
r(x)lp w(x)ax
.
rhen q(r + e) S q(r) * en(e), cnn( rr) = Irlp ,fu(t), +n(r) 2 o ane *Ottl = O for every f,g € C. S\IDose there exists a subsequence ilr_ \.. %. .,. such that (f) cr = sup pk --j J rc lEl[
we choose J3
t Je so that ,fur(rr) < r for every i ) Ji '
contlnulng tbis process we bull"d up two sequenees iitl;I
fx
e
c,
llrrll. s
r,,fu. (tr) llt, Jx
Let us choose e subsequence J\t
so that
(fn) S l for n = r'z'"',t-r' oL * Jrrt ... such that j, ";;: t and
*u,fu.
'
and tfrl;I
oRTHOGONAL POLYNOI,IIAI"S
d
r
c{ (r d,'nu'n+r'Iv ";'1":1
Let f=
for B=L,22...l
F\.uthelfor n>l .- i,+\) > 9kJ.0n ' n-P "l
(f, )-
,_1 -i
tms]*Jnn ,ir
-Jru "-p
x-"rl-l fr. Er -L\ v
; -Jl, ln. q. *r& (f, ) - c.', Jrr {-or*r ";}
Hence
(tu (r) Zhl-P -2. -J x^ - "nt Iettlng 6x
we obtaln
tTj*
c{L(f) =
-.
The contraallctlon shovs that
1T
sr4) ,q(r) < -
"np ktlFl[sr
whlch
ls equivalent to (8).
rbea f€c.
REFMffCES
Sooks
Ahlezer, N. I., Theory of AlDroxlFAtion, fb. UDgar PubL Co.,
Ileud, G., Orthogonal Polynonlals, Geronlnus,
Ios Angeles, 1!!8.
G.,
heas,
New
york,
York, 1!!5.
1%1.
L. Ya., Orthogonal FolynolulaLs, Consultants llureau, New York,
Grenander, U. a.nd Szeg6, Szeg8,
Perga.Don
New
G., Toeplitz
Forms antl
fheir Applications,
1961.
Berkeley,
O"thogona,l Po\monials, AMS, New York, 1!67.
Perlodica]-s
I,lea,n convergence of orthogonal series snd Iagrange interpolatlon, Acta. l"Eth. Acaal. Scl. Hungar, 23G912), 7L-6,
t1l Askey, R., t2l t3]
Bedkov, V., Convergence ln Eean and almost ever-Jmhere of Fourier series in orthogonal po\momia16, l"{,at. Sbornlk %(L31 )G97\), 229-262. Carleson, L., Gr convexgence and grou'th of partlal sums of tr'or:r1er setles, Acta trath. we965), Br-rfl.
14] c&se, K. M., Orthogonal- polynouials revlsited, In "fheory and Alpllcatlon of Speclal t\:nctions", eal. R. A. Askey, Acaatemlc hess, I%r,2894&. t:] Erdtls, P.
and
furan, P., on interpotatlon. III,
Anna,ls
of
lr,lath. \l-(191+0),
,ro-r>3.
t6J trleud, G., ilb'er ttle Konvergenz ales Herolte-Fejerschen Interpol-ationsverfahrens, Acta. tbth. sci. Itungaf. l!r]+), l-o9-128. nl t8l
, tber elne nasse lr€rsngescher IntetToletlonsverfatrrens, Stutlla 3EI. uetl. Hunsar. 3(l-958), 2\9-2)r. , Or l{errlte-FeJer lnter?olation processes, Studia Scl.
l(:9tz),
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