A new proof of a theorem about generalized orthogonal polynomials

(1)

RESEARCH

NOTES

A NEW PROOF OF

A THEOREM ABOUT GENERALIZED

ORTHOGONAL POLYNOMIALS

A.McD. MERCER

Department

of MathematicsandStatistics Universityof

Guelph

Ontario, Canada

(Received November 21, 1989)

ABSTRACT.

In

thisnoteit isshown that afairlyrecent result ontheasymptoticdistribution of the zeros ofgeneralized polynomialscanbe deduced from an old theorem of

G. Polya,

usingaminimum oforthogonal polynomial

theory.

KEY

WORDS AND PHRASES.

Toeplitz matrices,

orthogonal

polynomials, asymptoticzero distribution.

MATHEMATICAL REVIEWS

SUBJECT

CLASSIFICATION.

15

A

18, 26

C

10.

1.

INTRODUCTION.

Let ix(x)

be a distribution function whichhas associated with it theunique

sequence

ofpolynomials

p,,(x)

(n

0,1,

2 satisfying

p.(x)-

,t.x

,...

(..

>0)

and

p,,,(x)p,,(x)dct(x)-6,,,,,,(m,n-O,

1,2

(1.1)

Thesepolynomialswillsatisfythe recurrence relation:

7.-1

]

xp.(x)-

./,t’.

""

/(x)+a.t,.(x)+

--/t,.-

(x)

1.2

p_-y_l-O,

po(x)-’o,

ct.

t,

’t.

>0(n-0,1,2

Conversely,

atheoremduetoFavard

[2]

states thatif asequenceofpolynomialssatisfies

(1.2)

then a distribution function

ct(x)

existssuch that

(1.1)

holds.

It

is alsoknownthat the function

t(x)

isessentially

unique

[3]

whenever the

sequences

t’/

’"

and

{a,, }

areboth boundedand that a

necessary

and sufficient conditionforthis to be the case is that the

suppor

of da,

iscompact.

Suppose

nowthat in

(1.2)

% a and Thenone writes a

_M(a,b)

and if b 0there

(2)

394 A. McD. MERCER

will be noloss ofgeneralityinsupposingthatttE

M(0,1)

[4],

and we shall take thistobethe case in all thatfollows.

We

nowhave

supp(da)

as acompactset and we willdenoteby

A

the smallestcompact intervalcontainingit.

To

elucidatethenatureof

supp(da)

wequotethefollowinglemma

[4].

LEMMA

A. Let

a

tEM(0,1)

then

supp(da)

canbewritten as

supp(da)

[-1, +1] LIB

where

B

is

enumerable and bounded and the

only

possiblelimitpointsof

B

are_1. Furthermore, if

X

isthe set of all of the zeros of all the

p,(x),

then all ofthelimitpointsof

Xbelong

to

supp(da).

Once

thepolynomials

p,(x)

have beenobtained,onecan take areal-valued function

f

definedon

supp(da)

andsuitablyrestricted, and form theToeplitzmatrix

H,(f)-ff(x)p,(x)p(x)aa(x)]

(i,j-0,1

n-l)

We

shalldenotetheeigenvalues ofthissymmetric matrix, takeninascending order,

by

xk.nff)(1

k n

).

The

following

important theorem,withslightweaker

hypotheses

than are usedhere, hasbeenproved by

P.

Nevai in

[4]

THEOREM

A. Let

a

tEM(0,1). Let

fbe

real-valued and

f

_

C(supp(da)). Let F

beacompact interval

containing/(supp(da)

and

suppose

that

FtE C(F).

Then,

as n oo

+1

I

,

F(f(t))dt

-’"F(xk"(f))

"

/1

NOTE. It

is not difficult toseethat ifm andMdenote the lower and

upper

bounds

offon

supp(da)

then

tn

xk.,(f)M

(1-:kn:n

-1,2

We complete

thisintroduction

by

quotingatheorem which we call Theorem

B. It

isaspecialcase of a theoremappearingin

[5]

(see

Section 5.2

there).

THEOREM

B. Let

ft

tE

C[-1,

+1]

bereal-valued and write

g(0)-

f(cos0).

Form

thesymmetric matrix

1

I

K,(f)

gl(O)cos(i -j)OdO

(i,j

O,

1 ,n

1)

Denoting

by

mland

M

thelowerand

upper

boundsof

f,

let

F1

C[ml,M1].

Thenasn owehave

F( ,,(f))

F,(g,(O))dO

k-1

In

this

.,,)

aretheeigenvalues of

K,,) (all

ofwhich lie in

[mt,M]).

The

purpose

ofthis note is togivean alternative

proof

ofTheorem

A,

showingthat it

may

bededuced from Theorem

B

usingtwoauxiliary lemmas.

2.

AUXILIARY

LEMMAS

LEMMA B. Let

t

tEM(0,1)

andlet/be a fixed

non-negative integer.

Let

f

_

C(supp(da)).

Then

ash

+1

f(x)p,(x)p,

/t(x)da(x)

-

lfft)

(3)

where

Ts(t)

isthe

Chebychev

polynomial.

LEMMA

C. Let A

[a,.j],

B

[bi.j]

(i,j 0,1,

...)

be Hermitian matrices whose principal

sectionshaveeigenvalues

and

For

all n let allof theseeigenvalueslie inacompactinterval

[m2,M2].

Let k(n

(n

1, 2 benon-negative integerssuch that

k(n

o(n)

and let

(k)--B(nk)ll

O

as

where

A(

k),

forexample,denotesthe matrix

[aid]

(i,j

k,k+ k+ n

1)

and

[1.

isanymatrixnorm. Then as n owehave

forany

F

E

C[m2,M2].

Proofs of

Lemma

B

aretobe found in

[4]

and

[6]

while

Lemma C

isaspecialcaseof a resultproved in

[7].

3.

MAIN RESULTS.

We

nowproceedto

prove

ourresult, namely,

THEOREM

1. Theorem

B

impliesTheorem

A.

PROOF.

@

C(supp(dct))

and

supp(dc)

isclosed so we can extend the definition of

f

toobtain

f/

C(A)

and this can be done in such awaythatthe boundsof

f/

arealso m and

M.

Next,

apolynomial qcanbe foundsothat

Sup[

/(x)-

q(x)[

is assmallaswepleaseand the boundsofqonA are also m and

M. We

shall

prove

thetheorem,in the firstinstance,forsuch apolynomial q. The virtueofworkingwith

qinsteadof theoriginal

.f

liesin thefact that thetwomatrices which

appear

in Theorems

A

and

B

will, then,each bebanded. Thismakes

Lemma

(2_easytoapply.

Sincethe bounds of

q

are the same asthosegiven originallyfor

.f,,

we note that

F

D

q(A). As

in Theorem

A

letF

C(F).

Now

inTheorem

B

take

f

tobeqand

F1

tobe

F.

In

the notation of that theorem m inf

q,M

sup qand since

[-1,

+1]

(Z

A

thenF

C[mI,MI]

(since

[ml,M]

C

[m,M]).

[-1,+1] [-1,+1]

We

deduce from Theorem

B

that

1

,

F(.kn(q))._

1

F(gE(0))d

0

where

g2(O)- q(cosO)

and in which

X,.(q)

aretheeigenvaluesof

K.(q)

-

g:(O)cos(i -j)oao

(3.1)

1_

q(t)Tli_l(t)

(4)

396 A. McD. MERCER

Wenext

compare

the infinite matrices

K(R)(q)

and

H(R)(q)

.[

q(x)p,(x)p(x)dct(x)]

(i, j-O, 1,2 and weremark, first,that each isbanded with bandwidth

2(degree q)

+

N

(say)

We

next notethat, accordingto

Lemma

B,

+1

q(x)p(x)p(x)da(x)--

q(x)T-il(X)

l

-x

---

0

(3.2)

as/-- for each fixed

-]1

"0,1,2

N. To

apply

Lemma

ewe

take

k(n)

[]

and the matrix norm toU

According

to

(3.2)

we willhave

IIK)(q)-H’)(q)ll

0 as n oo

andsoweconclude that

(3.3)

asn m.

From

(3.1)

and

(3.3)

wededucethat,asn

1/4k.lF(Xk,.(q))’-’

F(g2(O))dO

+1

.1

[

F(q(t))

dt

(3.4)

n

Jr-

/1-t:

which

completes

the

proof

ofTheorem 1 forthepolynomialcase.

We

can now extend

(3.4)

tothe

general

caseusingthecustomarytypeofapproximation argument.

Let

e>0begiven. Then,bythe uniformcontinuity of

F

on

F,

therewillbe6>0 such that

Ffn,)-F(ngl

<e whenever rI,

r121

<6

(lt,

rl

r)

With

f

C(supp(dct))

asgiven,wefind thepolynomial qaspreviouslydescribed so that

IL(t)-q(t)l

<

on

A

and m

q(t)M

on

A

We

nowhave

IF(f(t))-F(q(t))l

<e on

[-1,+1]

(3.5)

Next,

consider the matrix

I.--and let

I1" I1

denote the

spectral

norm

(=

max eigenvalues

I)"

Since

]f+(t)-q(t)l

<6 on

A

(5)

weget

[x,,,,(q)-x,,,,(f)[

[[H,,(q)-H,,(f)[[

(see

[8])

sup

Iq(t)-f(t)l

(see

[4])

sut,p(d)

suplq(t)-L(t)

<

k,l[F(x,nCq))-FCxk.n(]))]

<e

(3.6)

Theapproximations

(3.5)

and

(3.6),

withearbitrary,leadus,in the usualway,todeduce that

(3.4)

continues tohold whenreplaced by

f

C(supp(dct)).

ThiscompletestheproofofTheorem 1.

We

remark, finally,that if in Theorem

A

wetake the distribution functionwhichyieldsthenormalized

Chebychev

polynomials

Tn(x)}’,

thenanalysissimilar totheabove,but rathersimpler,leadstothe converse result thatTheorem

A

implies Theorem

B.

REFERENCES

1.

G.

Freud.

"Orthogonal

Polynomials,"

Pergamon

Press,

New

York,1971.

2.

J.

Favard.

Sur

lespolynomesdeTchebicheff,

C. R.

Acad. Sci., Paris,200

(1935),

pp. 2052-2055. 3.

T.S.

Chihara.

"An

Introductionto

Orthogonal

Polynomials,"

Gordon andBreach,

New

York,1978. 4.

P.G.

Nevai.

"Orthogonal

Polynomials,"

Mcm. Amer.

Math.

Soc.,

Vol. 18, 1979.

5.

U.

Grenander and

G.

Polya. "Toeplitz

Forms

and theirApplications,"Univ.of Cal.

Press,

1958. 6.

G.

Mitsis. "The AsymptoticDistributionof the

Zeros

of Generalized

Orthogonal

Polynomials,

Master’s

Project,Univ.of

Guelph,

1989.

7.

A. McD. Mercer. On

theAsymptoticDistribution oftheEigenvaluesof Certain Hermitian Matrices,

Jour.

Approx.

Theory

(to

appear).