Finite infinite range inequalities in the complex plane

Full text

(1)

FINITE-INFINITE-RANGE

INEQUALITIES

IN

THE COMPLEX PLANE

H.N. MHASKAR

Department

ofMathematics, CaliforniaStateUniversity,

Los Angeles, California,90032. U.S.A.

(Received April 23, 1990)

ABSTRACT. Let ECCbeclosed, wbeasuitableweight function on

E,

abeapositive Borelmeasure on E. Wediscussthe conditions onwanda whichensurethe existenceof afixed compact subset

K

of

E

withthefollowingproperty. Forany p, 0

<

p

_<

oo, there

existpositive constants cl,c2 dependingonlyon

E,

w,a and p such that foreveryinteger n

>

1 andeverypolynomial Pofdegreeatmostn,

f

[w"P[’da <

cl

exp(-c2n)

/K

[w"P[’da"

In

particular, we shall show that the support ofa certain extremal measure is, in some sense, the smallest set

K

which works. Theconditions on a are formulatedin termsof

certainlocalized Christoffel functionsrelatedtoa.

KEY

WORDS

AND

PHRASES.Finite-infiniterangeinequalities, orthogonal polynomials, weightedpolynomials, Nikolskii inequalities.

1980SUBJECT

CLASSIFICATION

CODE. 26C05, 31A15, 42C05, 41A17 1.

INTRODUCTION.

Let a

>

1 and

wo,(x)

:=

exp(-[x[),

x E R.

A

special caseofaninequality proved by G.Freud

[1]

isthefollowing. Thereexistpositive constantscl,c,

ca tepending

onlyon a such that foreveryintegern

>

1and polynomial

P

ofdegreeat most n,

l=l>c,

[w:(x)P(x)12dx

<

c2exp(-c3n)

l=l<c,

[w:(x)P(x)ldx"

(1.1)

An

inequality of the form

(1.1),

known as a

finite-infinite

range inequality, is of critical importance in the study ofweighted polynomial approximation on It.

In

the past few years, many mathematicianshave investigated inequalities ofthisform ingreatdetail

(el.

[3]

for

example). In

particular, Mhaskar and Saff

[4]

have obtained a highly generalized

and precise form of

(1.1),

using potentialtheoretic ideas. These ideas,in turn, have been extended to thecase when the underlyingsets aresubsets of thecomplex plane12rather than the real line. Instudying the finite-infinite range inequalitieson subsets of 12, an

immediateproblemwhichmust be solvedis todecide upon anaturalchoiceofameasure

(2)

real line.

In

particular, onehas to includethe

arc-length

measure onsufficientlysmooth curves aswell astheareameasure ondomainsandtheir combinations.

In

this paper, we defineone such class of ’natural’ measures and then extend the results of

[4]

to the complex domain. Our results will demonstrate the importance of Christoffel functions in this theory.

In

turn,weestimatethese Christoffelfunctionswhen the measure under consideration satisfies certain density conditions.

We

also describe certainapplications tothetheoryofextremalpolynomialsin thecomplex plane.

In

Section 2, we develop the necessary technical background including some of the definitions. Themainresults andapplicationswillbe described inSection 3. Theproofs

ofthenewresults will begivenin Section 4.

2. PREPARATORY INFORMATION.

We

say thatafunctionw" C--,

[0, c)

is admiaible

(or

anadmissibleweight

function)

if each ofthe followingconditions

(Wl),

(W2),

(W3)

holds.

(W1)

w isuppersemi-continuous.

{W2)

The set

E0(w)"=

{z "w(z) > 0}

hasnonzerocapacity.

(W3)

If

E0(w)

isunbounded,then

Izw(z)l

--,0 as

Izl

oo,zE

Eo(w).

Here and throughout this paper, the term ’capacity’ means the inner logarithmic capacity

(cf.

[13],

p.55). For

any set

A

C_

C,

itscapacity will be denoted by

cap(A). A

property is said tohold quasi-everywhere

(q.e.)

on aset

A

if the subset of

A

where it

does nothold isof capacity zero. Let

E

be aclosed subset ofC.

A

function w iscalled admiaible on

E

if the function wE that coincides with w on

E

and is zero on

C\E

is admissible. Typical examples of admissible weight functions are

(i)exp(-Izl"),

a

>

0, whichare admissible on every closed subset ofC having positive capacity and

(ii)

any

positive continuous functiononanycompact subsetofCwhichhas positive capacity.

Let

2(E)

denote theclass ofallunit positive Borelmeasuressupportedon E. Ifa E

.4(E),

thenitspotentialis definedby

U(a,z)

:=/E

lg(1/Iz

The

(w-modified)

energy ofaisdefinedby

Wedefine

X(w,

)

:=

fr

u(,

z)do(z).

V(w,E)

:=

inf{I(w,a)"

a

e .A4(E)},

zeC.

(2.1)

(2.2)

Q(z)

:=

log(liT(z)),

z

c.

(2.4)

For

integern

>_

0,

IIn

willdenote the class of allalgebraic polynomialsof

degree

at most

n. Finally, forany set

A

CCandacomplex valued functiong on

A,

wewrite

(3)

I111

sup

I()1.

(.)

zA

Ourstartingpoint is the following theoremproved in

[5]

in the case when

E

C_ R.

Theversionstated belowcanbeprovedinexactlythesamewayasin

[5]

withobviousand

minor modifications.

THEOREM2.1. Let

E

C C be closed, w be an admissible

function

on E.

(a)

The quantity

V(w, E) defined

in

(.3)

is

finite.

(b)

There exists a unique element :=

p(w,E)

E

.M(E)

such that

I(w,l)

Y(w,E).

This measure has

finite

logarithmic energy.

(c)

The setS :=

S(w, E)

:=

supp(l)

is a compactsubset

of

E0(w)

q

E,

and

cap(S) >

0.

Moreover,

Q

is bounded

from

above onS.

F:=

F(w,E):=

V(w,E)-

/Qdt,

u(,,

z)

+

Q(z)

>

F,

q.. o,E.

(2.6)

(2.7)

(e)

We have

U(t,,z)

+

Q(z) < F,

for

everyzCS.

(2.8)

In

particular, the equation

U(I,

z)

+

Q(z)

F

(2.9)

holds q.e. onS.

(f)

For any positive integern and

P

II,

if

Iw"(z)P(z)l <_

M

q.c. on

S,

(2.10)

then

IP(z)l <_

M

exp(-nU(#,

z)

+

nF),

In

particular,

(ILI

O)

implies that

zCC.

(2.11)

Iw"(z)P(z)l

<

M

q.e. onE.

(2.12)

()

S" :=

S’(w,E)

:=

{z

C

E" U(I,Z)

+

Q(z)

< F},

(2.13)

then,.9* D_

S

is acompact

set, Q

is boundedonS* and

for

any integern

>

0 and

P

II,,

IIw"PIl

IIw"PIIs..

(2.14)

In fact,

for

any compact subset

K

C_

E\S*,

there are positive constants cl,c2 depending

onlyonw,

E,

K

such that

for

anyintegern

>

1 and

P

II,,,

(4)

Theclassical notionsof capacity,transfinitediameter, Fekete andChebyshev polyno-mials,Fekete points andChebyshevconstantcanalso begeneralized.

([10]).

For example,

the

w-modified

capacity ofaclosed set

E

is definedby

cap(w,

E)

exp(-V(w,

E)).

(2.16)

The Fekete points arethe points

{z,,,

k 1,2,..-,n, n

2,3,...}

for which

}2/,,(,,-)

r,(w,

E):=

max

H

]zi

zt’lw(zj)w(z’)

zt,...,zn_E

l<_j<k<_n

II

)"

_<<:_<n

(2.17)

The

w-modified transfinite

diameterofthe set

E

is definedby

r(w, E)

inf

r,,(w E).

(2.18)

n>2

An

argumentsimilar tothatin

([2

l,

P.

161)

canbeused

(cf.

[10])

to show notonly that

cap(w,

E)

r(w,E),

(2.19)

but that for any set of Fekete points

{z,

k 1,2,.-.,n, continuous, compactly supportedfunctiong"

E

-

C,

n

2,3,...}

and any

lim

-1

g(,zk.)=/gag,

(2.20)

k=l

wherep is theequilibriummeasure defined in Theorem 2.1.

The notion ofthe Chebyshev constant can also beexplored in this generality. We

shall elaborateonthis in the context ofthe

LP

extremal polynomials.

However,

wenote here thefollowingsimpleconsequenceof

(2.15)

and

(2.20).

PROPOSITION

2.2.

([I0])

integern and polynomial

P

E

II,,

If

K

C_

E

is compact, and

for

every sufficiently large

IIw"ell

IIw"PIIK,

(2.21)

then the supportset

S

defined

in Theorem .1 is asubset

of

K.

If

A

C_ C is a Borel set, a is apositive Borel measure on

A,

g

A

C is Borel

measurable,

thenwedefine

if0

<

p

<

oo,

(2.22)

ifp=oo.

(5)

isnot the usual one, butin thepresentcontext, thedistinctionwould be inconsequential. Also, the fact that

(2.22)

doesnot define anorm if 0

<

p

<

1 willbe irrelevant forour

purposes. Thus,wewill continue tocall the expressions defined in

(2.22)

’norms’

evenin

thiscase.

3. MAIN RESULTS.

Let

E

C_(3be

closed,

w beanadmissibleweight functionon

E,

anda beafixed positive Borelmeasure definedon E. Weshallassume that wiscontinuouson

E

and

[zw(z)]"

E

L"(a, E)

foreveryr, 0

<

r

<

oz.We alsocontinuethenotationdescribedinSection2.

In

thesequel,we assumethenotationandconditionsof Theorem2.1. The set

E

and theweightfunctionw beingfixedquantities,theywill not bementioned inthe notations. We also adopt the following convention concerning constants. The symbols c,0,c2,." will denote positive constants depending only on

w,E

and other explicitly prescribed

parameters. Theirvalue may not bethesamein differentoccurrencesof thestonesymbol,

evenwithinthesameformula. Constantsdenotedbycapital letterswill retain theirvalues aftertheyareintroduced.

DEFINITION3.1. Let

K

C_

E

be compact, 0

<

p

<

oo. We say that the

Lt’(a,E)

normof

(w-)

weightedpolynomials liven on

K

if thefollowing propertyholds. Forevery

e

>

0,thereexistsabounded Borel set AC

E

with

a(A) <

eand positive constants c,c2

(dependingonlyonw,

E,p,

e)

suchthat foreveryintegern

>

1 andpolynomial

P

E1-In,

.Iw"P]I,.,.E

<_

(1

+

c,

exp(-cn))llw"PIl,,.,-va

(3.1)

Theorem 2.1 shows that the supnormofweightedpolynomials liveson 8". Weshall show that for suitable measures a, the

L’(a,E)

norm also liveson $*.

As

mentioned in

the introduction, amajor issuehere is thedcfinition ofa’natural’ measure in this case.

We

shall show that thefollowingdefinitionwill work.

DEFINITION 3.2. The measure a will be called natural ifit satisfies each of the

followingconditions.

(M1)

aisa

regular

measureinthesensethatforany Borelset

A

C_

E,

a(A)=inf{a(O)’AC_O,

Oopen

(3.2)

sup{a(Ix’)"

g C_

A,

K

compact

(M2)

For anycompact subset

K

of

E, a(K)

<

oz and the restrictionofa to

K

hasfinite logarithmicenergy.

(M3)

ThereexistsanintegerN

>

0such that

Izl-Nda(z)

<

oz.

(3.3)

(6)

where

f

A,,(a,8,

z)

:= min

Ip(z)l

-z

/

Pq.IIn dEs(z)

IP(t)lda(t),

(3.4)

Es(z)

{t

6

E"

lt-

zl

_< t}.

(3.5)

Then,for every di

>

0andeverycompact set

K

C_

E,

/"

<

1.

(3.6)

lim sup

IIAX

(o,

,

z)llK

We

observethatnoneof the conditions in the above definition

depends

upon theweight

functionw. Whilethe conditions

(M1), (M2),

(M3)

abovearefairly

’natural’,

perhapsthe condition

(M4)

is not. The formula

(3.4)

defines alocalized Christoffel function for a.

A

good deal ofresearch in the theory of

orthogonal

polynomials isdevoted to obtaining

the properties of a from its Christoffel function

([7], [8]).

In

light ofthis

research,

the condition

(M4)

maybethought ofas adensitycondition for aonE. While it may not be very’natural’,it issatisfied in the importantcaseswhen

E

isa domain inCanda isthe twodimensionalLebesguemeasureon

E

andwhen

E

isasufficientlysmooth

cur,e

andais itsarc-lengthmeasure.

In

Theorem 3.5,weshallgivesomevery

general

conditions,which aresometimeseasiertoverify, underwhich

(M4)

holds.

Moreover,

itwill beapparentlater

(cf.

Theorem

3.6)

that

(M4)

isexactly thekind of condition whichenables the

L(a, E)

normsto liveon afixedcompactset.

Wearefinallyinaposition toformulateourmainfinite-infinite-range inequality.

THEOREM

3.3.

Let E

CC be

closed,

w be an admissible weight

function

on

E

anda be a natural measure on

E

in the sense

of

Definition

3.. Further suppose thatw

is continuous as a

function

on

E

and wx is also admissible on

E

for

each

A

(0,1].

Let

0

<

p, r

<

oo, andS* beas

defined

in

(.13).

Then,

for

anye

>

O,

there exists a bounded Borel setA :=

A(p,

r,e,w,

E, a)

C_

E

with

a(

A

<

eand positive constants

c

c2 depending onlyupon p, r,e,w,

E,

a, such that

for

anyintegern

>

1 and anlpolynomialP tin,

Ilw"ell,,E\(S.,)

<_

exp(-c2n)llw"Pll,.,,,s.,a.

(3.7)

In

particular, each

of

he

L(a, E)

norms

of

thew-weighted polynomials lives onS*.

More-over,

if

K

is any compact set, 0

<

q

<

oo is a

fixed

number and the

Lq(a,

E)

norm

of

the

w-weighted polynomials lives on

K,

then

a(S\K)=

O.

tLEMARK. In

fact, it would be clear fromour proofthat thehypothesisin the last

statement ofTheorem 3.3 canbe substitutedbya slightly weaker hypothesis:

For

every

e

>

0, thereexists abounded Borel set A with

a(A) <

eand limsup sup

[[’wnp[[q’a’E\(KUA)

1In

Perl.

[Iw"Pllq,,Koa

0.

(3.8)

(7)

THEOREM 3.4.

positive numbers N, such that

lim

N

/" 1

and

for

0

<

p,r

<

c, anyintegern

>

0 and

P

Under the hypothesis

of

Theorem 3.3, there exists a sequence

of

(3.9)

(3.10)

Next,

wediscussthecondition

(M4)

intheDefinition 3.2.

THEOREM3.5.

Suppose

that

E

C_ C beaclosed regular

set,

i.e..for

every

R >

0,

everypointonthe boundary

of

En{z.lzl <_ R}

i. aregularpoint

for

the Dirichletproblem

for

each

of

the components

of

the complement

of

this set in the extended complexplane

to which this point belongs. Let a be a positive Borel measure on

E

which

satisfies

the conditions

(M1),

(Me), (MS)

in

Definition

S.C.

Suppose

that

for

every compact subset

K

C_

E,

there exist positive constants

L,c

depending only on

E,

K

and a such that

for

every6

>

O,

a(a,K,

8)

inf

a({t

E"

lt-

zl

<

})

>

c8

L.

(3.11)

fi.K

Thena

satisfies

also thecondition

(M)

in

Definition

S.l andhenceis anaturalmeasure.

Weobserve that thegeneralTheorem 3.3doesnotrequire anyregularityconditionson

E.

Thefollowingtheorem shows that the condition

(M4)

isactually necessary toachieve aninequality of the form

(3.10).

THEOREM

3.6.

Suppose

that

E

C_ C is a Borel

set,

a is apositiveBorelmeasure

supportedon

E

which

satisfies

conditions

(M1), (M), (MS)

of

Definition

3.e.

If,

for

any

compact set KC._

E,

the inequality

(S.lO)

holds with w 1 on

K,

thena also

satisfies

the condition

(M4)

and

hence,

is anaturalmeasure.

Next,

wegive someapplicationsofTheorem 3.3 to thetheory ofweighted extremal

polynomials.

Let

0

<

p

<

cxs,

E

and w be fixed as before, and a.be a positive Borel measure on E. Wedefinethe extremalerrors

e,(w,

E,p,

a)

and the extremal polynomials

T,(w,

E,p,

a;

z)

z"

+...

H,,

by the formula

en(w,E,p,a)

:= min

IIw"PII.,p,E

=:

IIw"T.(w,E,p,)II,,E

(3.12)

PIIn

When p o, then the extremalpolynomials arethe modified Chebyshev polynomials. For convenience,weomit thementionof p andafrom thenotationfortheextremalerrors

and polynomialsinthiscase. Thus,forexample,wewrite

en(w, E)

:=

n(w, E,

o,

a).

The ideasin

[5]

and

[11]

canthen beused to showthat

,,lirnoo

e,,-/"

(w,

E)

=:

cheb(w,

E)

exp(-F),

(3.13)

where the constant

F

:=

F(w,E)

isdefinined in

(2.6).

Whenp 2, then the extremal

(8)

Theorem 3.3 and the resultsin

[6],

weget the following

THEOREM

3.7. Let

E,

w anda be as in Theorem

;.;,

0

<

p

<

oo. Then

lim

e,/"(w,E,p,a)

cheb(w,E),

(3.14)

Tn(w, E,p,

a)

andg is anycompactlysupportedcontinous

function

on

E,

harmonic in the two dimensionalinterior

of

E,

then

Moregeneraltheorems similar to those in

[6]

with the supnormreplaced bytheLrnorxns canalso beproved,but weomit these details intheinterest of brevity. Itisveryey to

sthat even inthe simplecewhen

E

{z

.Izl

1},

w

x

on and a isthe area meure, thehypothesis that9 beharmonic intheinteriorof

E

cannot bedropped. The

equation

(3.13)

isin

fact,

anequation for the balayagesofthe limitingmeures for the

zeros and theequilibrium meure

.

We do not wish to elaborate on this further here. Theinterested readermayrefer to

[6].

4.

PROOFS.

The prfofTheorem 3.3 will be given in several stages. The first stage is toestimate

the Christoffel functions associated with the weights

w2"da (cf.

(4.1)

for a

definition.)

Next,

weshallusetheseestimatestoproveaNikolskii-type inequalityto comparedifferent normsofapolynomial. This inequality will be used toprovethat thenorxns actuallylive

on afixed compactset. This set will then be pruned to the desired set. We begin this

progrby recallingcertainfacts about the Christoffel functions.

If

A

C is aBorel

se,

v is apositive Borel meureon

A

having infinitely my

points in its support and such that for each integer m O,

f

Izldv

<

,

then the Christoffelfunction is defined follows. Forany z C and integern 1,

inf

IP(z)l

-

[

[P[dv.

(4.1)

A,(v,A,z)

JA

Thefollowinglemmasummarizes someofthe important properties of the Christoffel

func-tionwhiweshMl needin thesequel. Let

A

and v be above, d

{p,(v,A,z)

H,}

be thesequenceof orthonoized polynomials,

p,(,a,)p,(,A,:)a()

=,m,

,,m O,1,....

(4.2)

(.)

LEMMA

4.1.

[121

(b)

I

B

C_

A

then

With the notation as inthe previousparagraph,

(9)

A.(v,B,z)<_A.(v,A,z),

z_C,

n=l,2,....

(4.4)

Wenow resume thenotationsandconventionsadoptedin Sections2and 3.

LEMMA

4.2. Leta be a naturalmeasure on a closed set

E,

K

C.

E

be compact.

Then there ezists asequence

of

positive numbers

{ti

n}

such that

tin

J,

0 as n oo and

1/,

imooll’(,,$,,z)ll

1.

(4.5)

PROOF. In viewof

(4.3),

it isenoughto show that

Ills

limsupll,l(a,6.,z)l1

<

1

(4.6)

foraproperly chosensequence

{ti.}.

Usingthe condition

(M4),

foranyinteger k

>

1,we

may choosethesmallest integernksuchthat

IlA=’(,l/k,z)ll<

<_ (1

+

l/k)

n,

n>_nk.

(4.7)

Withthe sequence

{ti,}

definedby

ti,,:=l/k,

n,<n<nk+,

k=l,2,-..,

and ti, := 1 for 1

_<

n

_<

nk,it isreadily seenfrom

(4.7)

that

(4.6)

issatisfied.

LEMMA

4.3. such that

There ezists asequence

{M,

M,,(w,E,a)}

of

positive numbers

IIw"(z)A(w"d,r,E,z)ll

<_

M,

(4.8)

and

lim

M/"

1.

(4.9)

We

note that in

(4.8),

the integer n appears several times. This notation will be consistently used. Thus,for instance, if k

>

1isanother integer, then

IIw"(z)h:(w"da,

E,z)ll

<

M,.

PROOF OF

LEMMA

4.3. Without loss ofgenerality, we may assume that

E

C

supp(w). Also,

it isobviousthatweneed todefine

M,

onlyforlargevalue ofn.

For

ti

>

0, set

w(ti)

sup{IQ(z)-

Q(t)l

z

e E,t e

S’,I

1

<-

}.

(4.10)

Sincewis continuouson

E

andboundedfrombelowbyapositive constantonthe compact

set

S*,

itis easytoseethat

w(6)

0asti

--

0+. Let

din

be the sequence defined by

Lemma

4.2 for the compact setS* and n be large

enough

so that

w(tirt)

<

oo. Let z E ,9* and

(10)

>

wZn(z)[P(z)l

exp(-2n(n))An(a,n,z)

(4.11)

where

A,(a,

6,,z)is

definedin

(3.4).

Let

M,

(n

4-

1)exp(2nw(6,))llA(a,

6,,z)lls

(4.12)

Then

(4.11)

and

(4.3)

yield that forz

Iw"(z)p(w"da,

E,z)l <_ M.l(n

+

1),

k 0,...,n.

(4.13)

inview ofTheorem

2.1(g),

the inequality

(4.13)

persists forzE

E.

Using

(4.3)

again,we

get

(4.8).

Theestimates

(4.8)

and

(4.3)

implythat lim inf

M/n

>

1.

Since

w(6n)

0asn oo,

(4.5)

and

(4.12)imply

(4.9).

I

Lemma

4.3completesthefirststepinourproofofTheorem 3.3. Thefollowinglemma gives theNikolskii-typeinequalities ’inonedirection’. This will beenoughforourpurposes.

LEMMA

4.4. Letn

>

1 be an integer,

P

IIn,

0

<

p

<

r

<

ex. Letk be the least positive integerwith2k

>_

p, and

{llln}

be the sequence

defined

in

Lemma

4.3.

Then,

(4.14)

PROOF.

we

may write

(cf.

[12])

Pk(z)

f

Pk(t)K"(z’t)wi"(t)da(t)

where,

inthisproof only,

(4.15)

Kn,(Z, t)

pt(w

2"’da,

E, z)p,(w

2"’da, E, t).

(4.16)

UsingCauchy-Schwartzinequalityin

(4.15),

andtakingintoaccount

(4.3), (4.8)

weget

(4.17)

Usingaconvexityargument repeatedly,weget first

(4.18)

and then

(4.14).

I

(11)

LEMMA

4.5. Let0

<

p

<

c. Then there exists

R

>

0 such that, with

K

z

E

lzl

<_ n},

we have,

for

ewyintegern

>

1 and

P

(4.19)

where cl,c2 arepositive constants depending onlyonp,a,w andE.

PROOF.

Let k be the least positiveintegerfor which2k

>_

pand n

>

N

+

1, where

Nisdefinedin condition

(M3)

inDefinition3.2. Sincew"/("+I+t"/pJ} isadmissible, there

exists apositivenumber

A

such

that,

forz

E,

(4.20)

(Here,

weusedLemma4.4 in the lastinequality.)

Hence,

if

B > A,

IP(z)lPw"P(z)da(z)

<-

A:’BNA"Ik"B-n

Iz

IzI-Nda(z).

(4.21)

[>_B,z.E I>_B,z_E

In

viewof

(4.9),

it isnoweasy to deduce

(4.19).

PROOF OF THEOREM

3.3.

Let

e

>

0 and

K

be the compact set introduced in

Lemma4.5. Without loss ofgenerality, wemayassume that S* C_ K. Since a isregular, thereexists aboundedopen set

V

such that ,5"* C_

U

and

a(cl(V)\S*)

<

e, where

cl(U)

denotes the closure of

U.

In

view ofTheorem

2.1(g),

there exists a positive constant c,

independent ofn and

P,

such that

Iw(z)P(z)l

<

exp(--cn)llwellE,

UsingLemma4.4and

(4.9),

thisleadsto

z

I(\U.

(4.22)

(4.23)

wherecisanother positive constant. Itfollows from

(4.19)

and

(4.23)

that

(4.24)

forsomepositive constantsc,c2.Since

a(cl(U)\S*)

<

e,wehave provedthat the

L(a,E)

normof theweightedpolynomialsliveson

Itisnowsimpletoseethat theestimate

(4.14)

canbe extendend to

0

<

p,r

<_

oo,

(4.25)

where

N,,

isasequence of positive numbers with lim,,_oo

N/"

1. Ifwenowuse

(4.24)

withrinplaceofp, andtake

(4.25)

intoaccount,thenitis easytoprove

(3.7).

Toprove

(12)

with

a(A) <

esuch that

(3.8)

holds.

In

viewof

(4.25),

appliedonce to

E\(K

U

A)

and onceto

E,

we seethat for sufficientlylargeintegern and

P

(5l’In,

IJw’pJJE

_-JJw’PJJK,.

Proposition 2.2 then yields that ,9 C_ KUA. Since

a(A)

<

e and t was arbitrary, this provesthat

a(S\K)

O.

I

Weobserve that theproofof Theorem 3.3 includesaproofof Theorem 3.4. Theproofs

ofTheorems 3.5 and 3.6usethefollowinglemma.

LEMMA

4.6. Let

E

C_ C be a Borel

set,

t be a measure supported on

E,

for

every compact set

K

C_

E

and 1

> O,

there exist a seqence

of

positive numbers

Ln

:=

L,

(,,

K, E,

r)

with

limsupL/n

_<

1,

(4.26)

and the

Christoffel functions defined

in

(4.1)

satisfy

Affl(,,If,z)

_< L,,

z (5

K,

(4.27)

where

{z

lz-

tl

<_

,

for

some

K}.

(4.28)

Then

,

satisfies

the condition

(M4)

in

Definition

REMARK. In

[9],

a measure awithcompact support

A

issaid tobecompletely regular if

1/n lim

IIP(a,A

z)l,

a 1.

When a is not compactly supported, but the restriction of a to every compact set is completelyregularin this sense,then theLemmashows that asatisfies condition

(M4)

of

Definition3.2.

PROOF.

Let

>

0,and

K

C_

E

beafixedcompact set. Withthenotationasin

(3.5),

wefindpointszl,.-,z, in

K

such that

K

C_

U

Et/,l(z,).

ThehypothesesofLemma4.6 k=l

applied to each

E/4(zt,)

showsthat foranye

>

0, thereexists anintegerN such that

Al(v,E/2(zt,),z)

< (1

+

e)

n,

z(5

Et/,(z,),

k 1,...,m, n

>_

N.

(4.29)

Weobserve that N depends onlyon

K,

tiand v.

Now,

ifz (5

K

isarbitrary, then

Et(z)

E/2(zt,)

forsome k, 1

<

k

<

m. Lemma

4.1(b)

then implies that

A(v,,E(z),z)

< (1

+

e) ’,

zeK,

n>N.

(4.30)

Thisgives

(3.6)

withu inplaceofa asrequired.

I

Itis convenient to prove Theorem3.6 first.

PROOF

OF THEOREM3.6. The hypothesis of Theorem 3.6 imply,withr cx and

(13)

A=(a,K,z)

<_

cN/2

z

.

K,

(4.30)

In

viewofLemma4.1

(b),

thisimplies

(4.27)

withainplaceofv.Lemma4.6 thenCOlnpletes theproof. 1

In

the proofofTheorem 3.5, weneed another lemma toestimatethe derivative ofa

polynomialin terms ofits maximummodulusonIf.Lemma4.7ismostprobablynot new, butweinclude aproof becauseit isverysinple.

LEMMA

4.7.

Let K

bearegular

(in

thesensedescribed in the statement

of

Theorem

8.5),

compact

set,

and letu be its classicalequilibriummeasure

(4 [18],

P.

55).

We

set

u(z)

,,(If,

z):=

f

,og(a/iz

tl)du(t),

z

e

c.

(4.31)

Thenu is continuous on C. LetV be the convexhull

oI

the set

{z

Iz

1

for

some

e K},

and

(6)

:=

max{]u(z)- u(t)]

z,t fi

V,

]z

t]

6}.

(4.32)

If

n is anypositive integer, PE

H,,

and0

<

6

<

1 is arbitrary, then

IP’(z)l <

6-’

exp

(n((6)-

u(z)-

logcap(K)))llPIl,,.

(4.33)

PROOF.

The fact thatuis continuous isvellknown.

(cf.

[13])

The Bernstein-Walsh

inequality

([14],

P.77)

yields

IP()I

<

exp(-nu()-

log

cap(K))IIPIIK,

e

C.

Ifz

e

c

and

F

:=

{’l

zl

6},

then 1

J(r

P((:)

d.

(4.35)

P’(z)

((,_

z)

Theestimate

(4.33)

follows easily from

(4.35),

(4.34)

and

(4.32).

I

We

are nowinaposition to prove Theorem 3.5.

PROOF OF THEOREM3.5. Since

E

is regular,every compact

sabset

of

E

is

con-rained inaregularcompact subset ofE.

In

viewofLemma4.1

(b),

it isthereforeenough

to verify

(3.6)

only in the case when

K

is regular. Let

K

be afixed, regularcompact subset of

E,

andweadoptthenotationinLemma4.7. Letn

>_

1 be anyinteger,

P

E

II,

and

IIPIIK

1.Let z0 6gbefoundsothat

(4.34)

IP(zo)l-

IIPII

1.

(4.36)

Let

e,

(2n)-’

exp(-2ni2(1/n)).

Then,forz 6

V,

Iz

z01

_<

e,

(_< l/n),

Lemma

4.7yieldswith

1In

inplaceof

6,

(4.37)

IP(z)-P(zo)l _<

e,

max{IP’()l"

I-zol

_< ,} _<

1/2exp(-nu(zo)-log

cap(K)).

(4.37)

(14)

IP(z)l _> 1/2,

z E

V,

Iz

zol

G e,.

(4.38)

Since% --,0asn oo,wemaychoosensolargethat

%

<

r/.

We

integrate both sides of

(4.38)

withrespect toaand use

(3.11)

toget

:lP(z)ld(z)

>_ (o, K, .)IIPIIK >

cLIIpII,W.

(4.39)

Since

L/,

-4 asn-4oo,wehave

shown

that asatisfiesthe

hypotheses

of

Lemma

4.6.

In

viewof that

Lemma,

the proofis nowcolnplete.

Theorem3.7follows fromTheorem3.4andthe known results about the asymptotically

extremalpolynomialsin the supnorm sense.

(cf.

(a.a)

and

[11]).

ACKNOWLEDGEMENT.

would like to thank ProfessorE.B.Saff and ProfessorV.Totik

for theirfriendlycriticismofa part of the present work.

REFERENCES

1.

FREUD,

G. Onpolynomial approximation withrespect to general weights, Lecture

Notes

inMath., Vol. 3OO, SpringerVerlsg, BerlinandNew York, 1074, pp. 149-179. 2.

LANDKOF,

N. S.

Foundationsof modern potential

theory,

Springer

Verlag,

Berlin

and

New York,

1972.

3.

LUBINSKY,

D. S. and

SAFF,

E. B.

Strong

asymptotics for extremal polynomials

associated with weightson

R,

Lecture Notes

in

Math.,

Vol. 1305, Springer

Verlag,

Berlin andNew York, 1988.

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MHASKAR,

H.

N.

and

SAFF,

E.

B.Where does the

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109-124.

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Amer.

Math.

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308

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MHASKAR,

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MHASKAR,

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NEVAI,

P. Orthogonal polynomials, Mem.

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NEVAI,

P.Gza Freud, orthogonalpolynomials and Christoffel functions,a casestudy,

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SAFF,

E. B. Orthogonal polynomials from a complex perspective, in "Orthogonal

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KluwerAcademicPubl., Dordrecht,

1990, 363-393.

10.

SAFF,

E.

B., TOTIK,

V. and

MHASKAR,

H. N. WeightedPolynomials and

poten-tials inthe complex plane, Manuscript.

11.

STAHL,

H.

R. A

noteon atheoremby H.N. Mhas and

E.B.

Saff, Lecture Notesin

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New

York, 1987, pp. 176-179.

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SZEG,

G. Orthogonal polynomials,

Amer.

Math. Soc. Colloq. Publ., Vol. 23,

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Math.

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Providence,

R.I.,

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TSUJI,

M. Potentialtheoryinmodernfunctiontheory, 2nded., Chelsea, New York, 1958.

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WALSH, J.

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