Orthogonal polynomials

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'...

Author: Paul G. Nevai

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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 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 llste0.

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 conslo 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 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 gô' 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)-dto 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 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 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(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'î* 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

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 -1r

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
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

Î 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
0. r>O.

Then

Hence

!i q,t'r'ln

g=0,

[---7

I€t now f

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 oo

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

dt0.

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

â+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

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:eo

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

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

q6hâ

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: 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 ô ^ 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 c > o

beflxed' {henforeve:y e)0

I€t o, 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(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

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<xo 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.Ê

.,

/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 D0. 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

tr (w.x) 1r{xJ I1&- Ag(urx.)= -J-? u(x, n* holds unlfort].y

for x € [o,r]

Here ve also had

flxe 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)lcMe

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

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
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\-
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'\ Î 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 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
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*)

.

> EBJOrO€A0

l)-1,

\^

and O 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

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 - Î \ p'(w,x)x tu(x)

.

Ihus

2n + Because w(x) =

c

Î

\_, ni(",")*

q(x) lxl€

r

= zp2(w,r) "J_lr

- \t nltn,*)x dr,r(x) .

we obtatn

aw(x) = Î

^ \_, 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 Î 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

Î 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-.

=


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 -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

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 O for ponding orthonorsrsl Po\'nordal' It ls obvlous followsfrolo -r(x(l -1 <xl

)

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,

-ro)€r'11-r'r;' lnr,{oo,*) ls v(x)ax

fhen

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

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
Proof'

Tbeorem

o

o)

I'Ie have 2n

qb-(d0,"-1) 'att

=i ^r 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) 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)
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
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";.


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

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
_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 =

Ix

'

Then

EII

,

E, Ikn(e) 5 o(x+. * *1 -*l) - d(x- " * *2 -t$)

lx-xknl
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

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*'
- 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;

o for almost every x € (-l-,1),

bn(ao,*)

l;.0 t(*)

Then (nar
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

tîf -

l" lsl

for

r(n where C

every

Lema (e)

tloe8

tlot

depencl

f::p O

Buch

that for

'tn na:0.

If a 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 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

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

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(--)

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(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)_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) 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 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[

Orthogonal polynomials

Orthogonal polynomials

Orthogonal polynomials

Asymptotics for orthogonal polynomials

General orthogonal polynomials

Fourier Series in Orthogonal Polynomials

Symmetric Functions and Orthogonal Polynomials

Asymptotic approximations of orthogonal polynomials