Harmonic analysis on the quantized Riemann sphere

Internat. J. Math. & Math. Sci. VOL. 16 NO. 2 (1993) 225-244

HARMONIC

ANALYSIS

ON

THE

QUANTIZED

RIEMANN

SPHERE

JAAKPEETREandGENKAIZHANG

Matematiska institutionen Stockholms universitet

Box6701 S-11385

STOCKHOLM

Sweden

(Received

Febr. 12,

1992)

225

ABSTRACT. We extend the spectral analysis of differential forms on the disk

(viewed

asthe non-Euclidean

plane)

inrecent work by J. Peetre L.

Peng

G.

Zhangtothe dual situationof

the Riemann sphere S

2. In

particular, wedetermine aconcreteorthogonalbase inthe relevant

Hilbert space

L,2($2),

where

-

isthe degreeof the form, asection ofacertain holomorphic

linebundle over the sphere S

2.

It turns out that the eigenvalue problem of thecorresponding invariant Laplaceanisequivalent toaninfinitesystemofonedimensionalSchr6dingeroperators. They correspond tothe

Morse

potential inthe caseof the disk.

In

the courseof thediscussion

mny

specialfunctions(hypergeometric functions, orthogonal polynomials

etc.)

come up. We

givealsoanapplicationto "Ha-plitz" theory.

KEYWORDS AND

PHRASES:

Pdemann sphere, quantization, reproducing kernel, invariant

Cauchy-Riemannoperator,invariantLaplacean, Morseoperator, Hankeloperator, hypergeomet-ticfunction, orthogonalpolynomial

1980 AMSSUBJECT

CLASSIFICATION:

30C40, 47B35

0. INTRODUCTION.

We

wish to extend the considerations in a recent paper

[10]

valid for the unit disk

D

or,

equivalently,theupperhalfplane

U (regarded

as amodel of the non-Euclidean

(or

Lobachevskii’)

plane)

tothecaseof theRiemannsphere

S

=.

Thatis,weshall pass to the dual symmetric space.

Let us briefly recapitulate themaincontents of

[10].

There westudied certain weighted

L

2-isPlanck’s spacesover

D

dependingon aparametera

>

-1

(in

the physical interpretation

E-constant)

and described their

orthogonal

decomposition under theaction of theMoebiusgroup

SU(1,1).

Thisdecomposition wasfound using acertain invariant Laplacean. Special attention

waspayed to the discrete spectrum.

In

particular, explicit orthogonal bases interms of hyper-geometricfunctionswerefoundineach of the irreduciblediscretepartsinthe decomposition

(each

isomorphic toaspace inthe holomorphicdiscrete

series).

The

orthogonal

baseswerethenused to study "Ha-plitz" typelinearoperators acting between these spaces

(boundedness,

compactness,

Schatten-von Neumann properties).

In

thispaperwe,

thus,

carryoverthesameprogramtothecaseof thesphere.

A

majordifference is now that the correspondinginvariant Laplacean has only discrete spectrum. Otherwise the

presentation is remarkablyparallel. Again lots of interesting specialfunctions (hypergeometric

and

other)

arise throughout the discussion, which in fact has been part of the motivation for

undertakingthisstudy. Wefurther elucidateseveral pointsinthe hyperboliccaseleftoverin

[10],

forinstancethe rather mysterious appearance of theMorse potentialintheparalleltreatmentin Weshould alsosaywhatweintendbythe word "quantization" (occurringinthe

title).

Weuse

itmainlyinthesenseofBerezin

[1] (cf. [8]).

Thusweviewquantizationas atheoryof deformation oflinebundles,and the relevantHilbertspaces,arenafor the operatortheory involved,arisereally

asspacesofsectionsof theselinebundles.

Mostof the results ofthispaper should extend rather easilytothecaseofP’

(complex

projective

space, which isthe compact counterpartof thecomplex unit ball

1).

Ofcourse, allthis isjust a

tiny portion ofavast program andweexpectsimilarresults tobe true for arbitrary Hermitean

226 J. PEETRE AND G. ZHANG

The plan of thepaper is asfollows.

In

Section 1weuncover the structure of the eigenspaces of theinvariantLaplacean,inparticularwefind anexplicitorthogonalbasis in them.

A

novelty isperhapsthat wedetermine thespectrumfactoring suitablefunctionsoftheLaplacean.

In

fact,

thesamethingcanbe doneonanyRiemann surfaceso wehave deferred the details of the proof toanappendix

(as

aSection

6).

In

Section 2weusetheorthogonalbasisfrom Section1 to write

down thereproducing kernels involved. Some applicationsin thespirit of Vilenkin’s greatbook

[12]

aregivenin Section3.

Many

moresuch applicationsarepossible! InSection 4someoperator theoryconsequencesof the previous discussionarebrieflyindicated. Lastly,in Section 5wereturn to thebasis vectors and consider the differential equationssatisfiedby them.

In

particular, we

writedown the "periodic"

analogue

of the

Morse

operator. 1.

THE ORTHOGONAL BASIS.

Weshall work withinafixed coordinate

neighborhood

withcoordinatez obtainedbydeleting

onepoint cx

("the

point at

infinity"). In

other words, S2 _{will be identified with the extended}

complex plane

:

CLJ

{cx}

(or

thecomplex projectiveline

I1).

Nearcx weuseinstead ofzthe

1 coordinate

Theessentialchangecompared to

[10]

isthat the factor1

-Izl

2,

everywhere, hastobereplaced by1

+

izl

(nd

X

z

by1

+

zt).

2

Thus,the invariant Cauchy-Riemannoperatorisgiven by

O

(1

+

Izl)

0

Similarly, theinvariant Laplaceanis

A,

--(1

+

Izl)

02

,9

0zO5

+

,(1

+

Izl)’

b-7

The relevant group is nowthe compact group

SU(2),

consistingof all 2 x 2 complex matrices

=(ac

db

)

such that ad- bc

l’

c

-’

d

’

and theactinwehavein mind is

U).

f(z

-

f((z))(t(z))_.

f(az

+

b

z

+

d

)(z

+

d)"

In

thisnotationwehave

DU

()

U(+2)D.

Noticethat, compared to

[10],

wehavechanged r,to -r,inthe last formula. Alsoweare going toassumethat r, isan integer

>

0. This is very convenient, as we are going towork with the

Hilbert space

L2’u(S2)

of functionswiththemetric

da(z)

Ilfll

If(z)12(1

+

Izl)"+

’

wheredaisthe_{normalizedarea}

measure, da(z)

dxdy.

da(z)

REMARK.

The measure

(

+

[z[2)

’+

is sometimes named aftervarious authors:

Berezin,

Berg-man,

Dzhrabshyan,

ttarish-Chandra,

Kostant

_{etc. lom}

a highbrow point ofview we

should,

strictly_{speaking, consider the}

elements

_of

this space notasfunctions but as sectionsofan

Her-mitean

holomorphic

linebundleover ,5’2

(differential

formsof

degree

-).

In

_fancy

language,

_the

Picard group ofp1 isisomorphic

to/’,

sothe

holomorphic

line

bundles

over P1 are labeled bya

discreteparameter

,.

Ifweinterpret theformulaw az

+

b

cz

+

d as a

change

of coordinates on

P,

thefamily

{(cz

+

d)"}

givesthe transitionfunctions.

Thesubspaceof analyticfunctionin

L2,’($2)

willbedenoted by A2,_{’. Itisfinite}

dimensional

and consists _{precisely ofall polynomials of}

degree

_<

,,

so that dim A2," u

+

1. _{This gives}

theeigenspace of

A,

correspondingtothelowest

("zeroth")

_{eigenvalue0. To describe}

the other

eigenspaces we

must, following

_{the same pattern}

as in

[10],

first analyze the nullspace of the

(l

+

1)-st

power/)t+ of/).

We

_{denote thisspace}

HARMONIC ANALYSIS ON THE QUANTIZED RIEMANN SPHERE 227

eigenspace

A

’

(I

0, 1,2,...

),

withtheeigenvalue

l(v

+

1

+

1),

isobtained astheorthogonal complementof

Bt-1

in

’

0)

A

=Bt(Bt-1

(B-1

Thisfollows from thefollowing lemma.

LEMMA 1. Let D,

L

2’u+2

L

2’ _{be the adjoint of)}

),

_{consideredasanoperatorfrom}

L

2’ into

L

2,’+2,

that is,

D.

Thenwehave the factorization

D,,D,+2

...D,,+2tD

t+l

A(A -(u

+

2))(A

2(u

+

3))...

(A

--l(u

+

1

+

I)).

Its proofreliesonthe

following

identity:

/D

D+2/

(u

+

2).

Fordetails see the Appendix, where theproofis given in the context ofan arbitrary Riemann

surface equippedwithametric withpositive Gaussian curvature.

Nowwebeginto uncoverthestructure of the space

Bt.

LEMMA

2.

Every

globalsolutionof thedifferentialequation

Dt+l

f

0has the unique

represen-tation

()

where each gj

0

O,

1,...,

1)

is a polynomiMofdegree

<_

u

+

21.

However,

the -j highest

coecientsof9j aredetermined by9,...,9+1.

In

particular, thedimensionof

Bt

is

+

1)(

+

+).

PROOF: That every lutionof

Dt+

has the representation

(1)

with₉ entire in z follows in

[10]

relying the differential equation.

In

order to seewhich are the restrictionson ₉ at we

1

me

the ehgeof ablez Thenwefind

(1

Z

f=,()

,:o

1 z5

=,()

z"+’(

+z,)

,(

(

+

lz,)

3=0

(

+

I)

r=0 3mr

1

Forinstance,the l-th coecientintheoutersumis,up tosign,just

gt()z

+.

Itfoowsthatgt

mustbeapolynomialofdegr v

+

21. Similarly, the

(l-

1)-st

coefficient is, againup tosign, I- 1

)zU+21_

’-’(7-228 J, PEETRE AND G. ZHANG

Itfollowsthatgt- must beapolynomialof

degree

_<

v+2/-1 and that

llt(v+21)+.lt_(v+21-1)

0.

(We

usethe sign todenote Taylor

coefficients.)

Thegeneralstatement about thefunctions gj follows now readilybyinduction.

It

is nowlikewise clearthat

dimBt

=v

+

21+

1

+

(v

+

21-2

+

1)

+..-+

(v

+

2

+

1)

+

(v

+

0

+

1)

=(1

+

1)(v

+

1)

+

2(1+

1-1

+-.-+

1

+

0)

=(l+l)(v+l)+2.(l+l)/

=(l+l)(v+l+l).

| 2

Next,

wedecompose thespace

Bt

under theactionof the rotationgroup

SO(2) (the

isotropy

group of theorigin z

0).

Theelements ofthe nth spacein thisdecomposition

rhust

be of the form

{z[

2

(2)

f

=z"q(),

where q

q(t)

is apolynomial of

degree

_<

1. Ifn

<

0 it must vanish to the order -n at the

origin 0:

q(t)

O(t-").

Similarly, ifn

>

v it must vanish tothe order n- v at the point 1 1:

q(t)

O((1

t)"-u).

Thelaststatementfollows again by makingthesubstitution z

-Z

We then

get

(3)

f

’-"q(1

X/

I1

)"

Itfollows thatwemusthave

-I

<

n

<

+

v.

Let

usnowcompute thenormofafunction of the form

(2):

Izl

dd

:

/

ll

){

(:

/

{{)+

’{(1--77)

.(

+

)-+

r rdr

r2"lq(i

+

r){

(1

+

r

2)+

{q(t)12t,(1

t)r-,dt

{{ql{2.

Here

wehave introducedpolcoordinates, writingz re

i,

dput

{z{2

r2

1 1 2rdr

with 1 dt

1

+

lz{

2 ₁

+

_r2 _I

+ lzl

2 ₁

+

_r2’

(1

_4-

r2)

2"

Now

we orthogonalize the set of all polynomials in the metric

{{ql{. We get

then for each integer n aset of

orthogonal

polynomials

{qt,,(t)}.

Ifwekeepn fixed and vary instead weget

an

orthogonal

basisfor the l-theigenspace

A

of theoperator A,, oftheform

{e,,t(z)},

where -1

<

n

<

+

v. Itwillbeconvenient toset

{z[

2

e,,t(z)

z"qt,,(

1

4"Il

)"

In

particular,we seethatdim

A’"

v

+

21

+

1inagreementwith

Lemma

2. Namely,thislemma showsthat

dimA

’v=dimBt-dimBt_{

=(l+l)(v+l+l)-l(v+l)=v+(l+l)2-12-v+21+1.

REMARK. It is convenient to use also instead of the parameter

t

2t 1

{z{2

1

Izl

2

+

1;

then

1

+

t

2t, 1 t{

2(1

t), dt

dr. If ranges from 0to 1,thentl ranges from-1 to 1. Soif

wewrite

Q(tl)

q(t),

wegetthemetric

(apart

fromaconstantfactor 2

v+)

_’

)-

).

{{QI{

{Q(t,){2(1

+

t:

(1

t,

HARMONIC ANALYSIS ON THE QUANTIZED RIEMANN SPHERE 229

lzl+

1

}’

whereasbefore -l

<

n

_<

+

u.

A

possiblyevenbetterparameterthan

t

ist2 1

t

1 2t 1

-Izl

IfwerealizeS asthestandard sphereinN (using stereographic projection onto

),

1/

Izl

"

itgetsanimmediategeometric meaning. Namely, ifwelet9,

ff

bepolar coordinates about the northpole,wecanwritez tan

:

e

’*.

Itfollowsthen that,indeed,

t

cos/9.

Itshouldbeclear thatourresult generalizes the classical Laplaceseries(expansioninspherical

harmonics).

This isthecase u

0,

when theelementsaregenuine functions,notsectionsofaline bundle. For instance,it iswell-known that the Ith

eigenvalue

of theLaplace-Beltrami operator equals

l(l

+

1),

which agreeswith ourformula.

Let

us studythe polynomials

LEMMA

3.

It"

properly normalized the polynomials qtn,

>_ max(--n,n-

v),

are given by a

Rodrigues’sformula:

(4)

qt,(t)

t-"(1

t)"-

(-

t(1

and

l!(n

+

l)!(v

n

+

1)!

IIq’"ll=

Iqt"(t)l=t"(1

t)-ndt

(2/+

v

+

1)(

+

l)!

Beforepassingto itsproof,letusintroducesome notation.

We

definethe "ascending" factorial

(or

Pochhammer

symbol)

(a)n

a(a

+

1)...(a

+

n

1)

and also the

"descending"

_factorial

(a)=

a(a

I)... (a

n

+

I)

(--1)n(--a)n.

d

Clearly

(-)"t

(a):t"-".

_{Recallfurther the usefulformula}

(a)n

(a),(-l)n-’(n

1

which willrepeatedlybe usedinwhatfollows.

Now

weproceed with the proof of

Lemma

3.

PROOF" By

Leibnitz’s formula

q,,(t)

t-"(1- t)

()(n

+

l)-_h(v

+

l-n)-x

h=0

X

t"+t-+h(--1)n(1

t)

-"+t-(5)

=(_l)()(n+l)_(v+l_n)t(l_t)t_h

h=0

soit isclear that qtn isapolynomial ofdegree

_<

I.

To

seethatit isexactlyof

degree

welookatthetop coefficient. Expandthe expression in

(3)

inapower series int. Thenweseethat for

large

the main contribution to thederivationcomes

from the term

d

’[t.+’(_

--+,t--]

t--n(--1)n--tn--(-d

1)

d

it21+

=(-1)tt-

(-

(-1)’t-(2/+

v)"

t+-t

(-1)’(/+

_v

+

1),t

t.

Thus thetopcoefficientis

#

0 and equals

(-1)t(/+

v

+

1)t.

Ifn

<

0,

wehave

0

(n

+

l)’[_

(n

+

l)(n

+

l-

1)...

(

+

l-

(l

+

n)...(n

+

l-

(l- h)

1)

_0,

provided l- h-1

>

+

n orh

<

-n 1. Thatis

qt,(t)

O(t-n).

Ifn

>

v,wefindinthesame way

qtn(t)

O((1

t)n-’).

230 J. PEETRE AND G. ZHANG

--n(

d

[tn.+l(

)--n]

(qtn,t 1

t)n-v(-)t

1 dr.

Ifj

<

we can integrate by parts j

+

1 times and each time the _{term integrated} out clearly

vanishes. Sotheintegralis0. Weconcludethat

(qtn,qn)

=0 for/ Tofind thenorm, weconsidertheintegral

d

t)v_,+t]t

(q,n,t

t)

()t[tn+t(1

dr.

This timewecanintegratebyparts times

oy

toproducethe

inteal

(-)/

+(

)-+

r(+++_++

(++

It

followsthat

=(/+

v

+

1)t/!(n

+

l)!(v-

n

+

l)!

l!(n

+

l)!(v

n

+

l)!

(v+2l+l)!

(y+21+l)(+l)!"

|

LEMMA

4. Ifn

>

0 it ispossibletoexpressqtnin termsof thehypergeometricfunction:

qt,(t)

(n

+

1), F(l

+

+

1,-l;n

+

1;t).

Werecall that

F(a,b;c;z)

2Fl(a,b;c;z)

(a),(b),

.=0

n!(c),

z.

PROOF: Werewritethe coefficient in

(4)

as

(In)(n

+

l)’[_(u

+

n)

_.(_l)n

(-l)n

1)..._.t. (_X)t,(n

(n

4- 4-

1),,,

h!

(n

+

1

(-/),(n

+

=(n+l)t

h!(n+l)h

It

follows that

qt.(t)

(n

+

1)t(1

t)’

F(n

v

l,-l;n

+

1;

_-).

Ontheotherhand,weknow,quitegenerally,that

([6],

bottom of page

8)

F(a,b;c;z)

(1

z

)-b

F(

b,

c a; c; Z

Sothattakinga n-u-

l,

b -1,c n

+

1,z wefind

qt,(t)=(n+l)tF(l+v+l,-l;n+l;t).

|

REMARK.

Ifv

_>

n weget instead

q,,,(t)

(-1)’(v

n

+

1)tF(/+

v

+

1,-l;u-n

+

1;1

-t).

Thisis infull

agreement

with

[6],

toplineofpage9.3

2.

THE

REPRODUCING

KERNEL.

BARMONIC ANALYSIS ON THE QUANTIZED RIEMANN SPHERE 231

(6)

I(,(z,

w)

E

"’(z)"’()ll"’ll

-="

On

theother

hand,

fromthe

SU(2)-invariance

itfollows thatwemust have

tt’t(z, w)

(1

4-

ztb)t’((i_

+

{l)(

x +

iwl)),.

where Xis afunction ofonevariable.

Todetermine_X it sufficesto takew 0. Then

(5)

gives

K,(, 0)

0()11011-/!,

l!,

n=0

(7)

e,,t(0)=

0,

n#0

Izl

z

(If

n

>

0

(6)

isobvious; if n

<

0 it follows from the fact that

e,,t(z)

z"qu,(i"

+

iz,

l

and

qt,,(t)

O(t-"),

sothat

e,,t(z)

O(Izl-).)

From

Lemm9._and

Lemm

₃

(with

n

0)

weget

Izl

l!(t,

+

l)!

-,

h’,(z,O)

=l!F(l +

v

+

1,-/;

1;

..)

((2/+

v.+

’)

+

v)!

=(2/+

t,_4-

1)-F(l

+

r,4-1,-l;1;

1 4-

Izl

)

Thuswedraw thefollowingconclusion.

TItEOREM 1. Thereproducing kernelin

A

’

is

(8)

Kt(z,

w)

(r,

+

21

+

1)(1

4-

zff)’F(l +

u

+

1,-l;

1;

Iz-

wl

(I

+

Izl)(1

+

Iwl

)

In

particular

(1

0),

thereproducing kernelin

A

2’_is

K0(z,

w)

(u

+

1)(1

+

zt)"

1.

|

REM,RK.Notealso thatifu 0then(writingasbefore z tan

e

i)

Kt(z,O)

(2/+ 1)(1

+

zff)F(l

+

1,-/;

1;sin2

})=

(2/+

1)Pt(cos 0)

is the spherical functiononS

SU(2)/SO(2),

where

Pt

aretheLegendre polynomials.

Write

/

Izl

can writethis as

3.

APPLICATIONS TO SPECIAL

FUNCTIONS.

Now

wecanplaysomegamesinthestyleof Vilenkin’s book

[12].

Chap.

II,

dealingwith

SU(2),

isespecially relevant forus.

Let

us usethe aboveformula

(8)

inconjunction with relation

(6).

Theresult istheformula

Izl

)qtn(

Iwl

,,=_tz"a’"q’(1

/

Izl

1/

Iwl

I-

1

)o

=(v

+

21

+

1)(1

+ zff)F(l

+

u

+

1,-l;1;

(1

+

Izl)(X

+

I=,1 )

Izl

Iwl

2

s and let _3’be the

angle

between the vectors z andw. Then we

232 J. PEETRE AND G. ZHANG

.=_((

)(1

))

q"()q"()/ll"ll=

-(++

+((_(_;)

"

x

(

+ +

,-;

;-(

+

(

(

Wemayview the above the ourier expansion of the fetion to thefight. Thweconclude that

(

+

2

+

)

(

+

(

)(

)

x

f(l

+

+

1,-l;

1;-(t

+

s 2ts

2(1

t)s(1

s)cos;))e-’"d7

1

t)(1

s)

q’"(t)a’"()/llq’"ll

for n

.

0 else

This isamtiplicationthrem.

Nextweinvokethe

trsvectt

(s

[4], [15]).

Itisquestion of thefollowingbi]inedifferenti expression"

Ifwe let

fl

transform withweight vl and

f2

transform with weight ix2, then the transvectant

7i(fl,

f2)

transformswithweightix1

+

ix

+

2s.

2,, _A2,+2/

We cannowexhibit anisomorphism between the spaces

A

and which respects the

SV(2)-action

(an

"intertwining"

map).

Namely, wetakein

(8)

ix -ix-

21,

v2 1,

:f

gE

A

’’+2t

.[

K_(z,z)

(1

+ Izl2)

-This givesanelement

f

E

A

’u_definedby

Toseethat themap g

f

indeed isanisomorphismwetakeg z

"+t,

-1

_<

n

_<

v

+

I. This

givesus,apart froma

factor,

back thebasisvector e,t

(use

Lemma

4):

e,t(z).

I

Let

{tnm()}

be thematrixofaction

,

onthe space in termsof thebasis

{z"+t},

-l

<

n

<

n-

l,

i. e.

n+l n+! n+l

z

IIIz

t..,()z"+llllz’+ll

The functions

tnrn()

areexpressiblein termsofLegendrefunctions

(cf.

[12]).

Usingthe

isomor-2,

HARMONIC ANALYSIS ON THE QUANTIZED RIEMANN SPHERE 233 n+l

u

"+

,,,111.,11

,

m=--l

Ifwespellout this indetail, weobtainarelationinvolvinghypergeometricpolynomials. 4.

HARMONIC

ANALYSIS OF OPERATORS.

However,

the principal goalof all this business is operator theory. Now we say afew words what operatorconsequences canbe drawn fromourresults.

Weare interestedinlinear operators actingbetweeneigenspaces

A

for different values of andv

(or

complex conjugatesof such

spaces). In

the formercasewearedealingwith

"generalized

Toeplitz

operators",

inthe lattercasewith

"generalized

Hankel forms"

(we

identify anoperator

’’"’

).

So from a2,’ say, into

A

’

withthecorrespondingbilinear formontheproduct t,

A

’

itmightbe justified tousewith Nikol’skii’ the ackronym "Ha-plitz" asaunifyingconcept.

As

the spaces involved all are finite dimensional, everything blows down to linear algebra

(matrix theory)

or- ona more sophisticatedlevel- representation theory for compact groups.

So,

at least in principle, the problem is in a way trivial. We remark however that non-trivial analysisproblems perhapsariseifweallow theparameters

and/or

vtendtoinfinity. Ifwerecall

the isomorphism

A

’

A

2,+21

(Section

3),

_{we see also that we have essentially the classical}

problem connected with the determination of

Clebsch-Gordan

coefficients in disguise

(see

e. g.

[12],

SofChap.

3).

We beginwith the observation thatitis,inprinciple, superfluoustoconsiderconjugatespaces.

Indeed,

thereisacanonicalisomorphism from

A

2’ onto

A2,,

which isgiven by

f(z)-+

g(z)=

5’f()

andwhich intertwineswiththe groupaction.

Let uscheck theintertwiningproperty: If

f

isreplacedby

f

(az

+

b

cz

+

d

)(cz +

d)",

then

g(z)

gets replaced by

1

"f-,+

d

(7

+

d)"

Z

cz

+ d)"

f

k.a/

(cz

+

d)"

g

)(cz +

d)".

|

Soinawaythe distinctionbetween Hankel and Toeplitz fadesaway

Let

usnowlook at linearoperatorsfrom

A

2’’’into

A2’.

Assume,

to.fix

the ideas, that

v[__J_<

v

Then the simplest

SU(2)-invariant

operatorfrom

Az’’

into

A

z, isthe Toeplitztype operator

TB

withkernel

(0)

T(z, ,)

f(z)(

+

where the "symbol"

B

is apolynomial ofdegree

<

which transforms accordingto the

repre-(Ifv

v’

sentation

U

(t) _{that is, like aform ofdegree}

-3" weget aToeplitz operator exactly

(multiplication by the

symbol).)

It will be convenient torefer to the number -t as the weight.

We shall assume that

<

v

+

’. (This

will be explainedin a more

general

context below, see

(17).)

In

order for the kernelin

(10)

to have the right transformation properties, that is, to be an

element of the tensorproduct

A

’

(R)

A

,’ wemusthave

(11)

v

+

v’.

Indeed, thekernel

(1

+

z@)"

behavesasanelement of

A2’"

(R)

A

2,’

(of

_"biweight"

(-v’,-v’)).

Soit isclearthat

Ts(z,

w)

has therightbehavior inthevariablew,thisevenirrespective of how

234 J. PEETRE AND G. ZHANG

Asageneralizationof

(10),

wemay,following theprocedurein

[15],

consider for an integers

theoperators

T(

’) with kernel

(12)

T(t)(z,

w)= (1

+

zt)"

z

wherewe

agn

sume that thesymbol

B

trfformswithweight -t. If a 0,

(12)

reduces to

(10).

In

other words:

TB

T

-’).

The connection betwn a, s d in

generM

will be uncoverednow.

As

there is no derengagionin the w vable it is

agn

tfiviM ha we have the

correc

w

behavior. Werequire that theproduct withbracketsin

(12)

beofweight 1

.

This gives the

relation

(13)

s-t-t/’=l-o or

o+s=t+v’+l.

Ifweapply "Bol’s lemma"

(see

[4])

itfollows that, after the differentation carried out,wehave

anexpression whoseweigth is 1

+

o.

For

theresult to have the requiredweight -t/wethenget

anequation which wemaywrite as

(14)

s--t/=l+a or a-s=-v-1.

Thus, eliminatingsbetween

(13)

and

(14),

we

get

(15)

t=v-v’+20

or a=

t-(-,/)

which isanalogoustothecorrespondingrelationin

[15].

4 _Clearly,

(15)

_impliesthat

(16)

>

v-

v’

(a

must beapositive

number),

whichgeneralizes

(11),

and further that =_ v t/’mod2.

From

(16)

wecanlikewise determines:

s=l+v+a=l+v+

t-(v-v’)

1

+

v+v’

+t

2 2

sobothsandaarenowexpressedin terms of t.s Ifa

0,

(15)

reduces to

(11)

and thens 1+t/.

Besides

(16)

thereisone moreinequality imposedont. Namely:

(17)

<

v+v’.

Toseethis,weobserve thatifwedifferentiatethe factor

(1

+

ztO)

’-"

he

exponent goesdown to

v’

s a. Fromthisweobtain

v’

a

>

0,which is in viewof

(15)

thesameas

(17).

|

Thus

altogether

wehave

((16)

+

(17))

(18)

t/-

-<’-<

+

Thisagrees with theclassicalfact that

A2,,,

@

A

u’

AZ,t

wherewesum overprecisely theindices occuringin

(1)

with the sameparityasthe difference

v-t/’

(see

[12],

_page

177).

Thuswehave madethisdecomposition rather explicit.

Note

that the

rangeofais theinterval

[0,

v’].

Let us indicateone application of the preceding connected withClebsch-Gordan coefficients.

Wetake the symbol to beamonomial,

B(z)

z

(k

0,1,...,

t),

andapply thecorresponding

operator toamonomial. Weobserve first that the kernelsatisfies

HARMONIC ANALYSIS ON THE QUANTIZED RIEMANN SPHERE 235

Itfollows that theoperatorintertwines withthe

SO(2)-action:

t, ),/,,7,(

where

U#f(z)=

f(e’z).

This again implies thatmonomialsare mappedontomonomials. More

precisely,wehave

n’

,.z

=Ct(n,n’)z"

with n=n +k-a,

where the numbers

Ct(n,n’)

are our version of the Clebsch-Gordan coefficients.

(We

do not

indicateinthenotation thedependenceon vand

v’.)

Forageneralsymbolthisgives

t)(n,n’)

Ct(n,n’)(n

n’

+

a),

wherewe presentlyuse the hat todenoteFouriercoefficients. Now ageneral operatorcan,by

theabove,bedecomposedintoasumof Toeplitz typeoperators:

It followsthatwehave thefollowingexpression for the Hilbert-Schmidt

(S2-)norm:

n=0n’=0

In

particular, we obtain the following orthogonalityrelation

(cf.

[9]

for asimilar result in the hyperbolic

case):

=const

C,(n,n’)C,,(n,n’)llz"ll2llz"’ll2,-

o

if

#

t’.

The numbers

Ct(n,

n’)

canalso beexpressedintermsof the

generalizel

hypergeometric

func-tion

3F2"

PROPOSITION 1.

Ct(n,n’)

const.

3F2(-a,v’

s,-n’;-v’,

1

+

k

a;-1).

PROOF: Wehave

(see

(12))

Tz

,

(1

q-z@

zz

zk(1

+

w"

(q

+

iw12),+2

.,

a(w)

0.=

(1;-(’

1;

-+

x

.,

da(w)

236 J. PEETRE AND G. ZHANG

Theintegral gives acontributiononlyif j

+

h

n’. In

thiscase itreducesto

(z

/

lwl)-’+

r(1

-

r2)

"’+2

(1

+

p2)"+:

B(.’

+

..’-

’

+

)

r(.’

+

)r(.’

’

+

z)

r(’

+

2)

n’!(v’-

n’)!

(v’

+

1)!

Thereforeweobtain

n’!(u’-(u,+1)!n’)!

(-

1)3

(-a)3j!

(- 1)-3(-k)’-3 (- 1)3

(u’

s)3

(n’-(u’j)!(u’

-j)!-n’)!

C,(,

’)

(_I)

n’

(-a)3(.--l)’(-k)-(

’-

*)3

u’!(n’)’f

(.’+ )!

₃₌₀

(

+

_

),

(-:)’(-).

(_:),

(-b(.’- *),(--’),

(-:)(-)

u +1

F(-,

s,-n;-u’,l

+

k-a;-1)

andweedone. ]

FinMly, duplicatingwhat isdonein

[10]

d

[14]

in thehyperbolicce, we say afewwords

about

SU(2)-invadt

Ha-plitz operators from

A

," _into

A’"

(it

_{isthus the speciMce}

t’

_0, u of the

generM

situation).

Such operatorcanbedefinedbythebilinearfoa

C

da(z)

(H(’)

f,g>

f(z)B(z)D’g(z)(:

+

(Thus, B(z)

is thesymbol of theoperator

H(’s).)

If V is the isomorphism from

A

2’"+2 _onto 2,

A

displayedinSection3,thenitis easy toverify

(for

details ofthiscomputationsee

[14])

that

(H’S))*V,

which is thusanoperatorfrom A2,+2t _to

itself,is oneof the Toeplitzlikeoperators

T(

t)

considered earlier in thisSection.

5.

DIFFERENTIAL EQUATIONS

SATISFIED

BY THE

BASIS VECTORS.

In

Section1weconstructed thebasis

{e,a()}

*"q"(i

I1

)

where -l

_<

n

_<

+

v,in

thespace

L,’(S).

Now wewrite down variousordinary differential equations connected with

these functions.

First ofall,as the polynomials

qm(),

fornfixed, areorthogonalin the metric

Ilqll

Iq(t)l=t"(1

it is _{easy to see, by} general principles governing orthogonal polynomials, that they satisfy the differential equation

,,d

)_,,+adq]

-t-n(1-

t)

-

tnq-’(1-t

HARMONIC ANALYSIS ON THE QUANTIZED RIEMANN SPHERE 237

(19)

-t(1-

t)--

d q

[!n

+

1)(1-t)-(v-n+

1)tfl-

dq

=l(v+l+

1)q. n+l --(u+2)t

This is ofcoursejust aspecialcaseofthehypergeometricequation inslight disguise.

On the other hand, we know by their very construction that thefunctions

e,t(z)

satisfy the

partialdifferential equation

-(14-

Izl2)2OOf

4-v(1 4-

Izl2)0f

l(v

-4- 4-

)f

(notation:

0 0 0

zz’

0

).

Of course, separating variables,thishastolead backtothesameequation

(1).

Letusverify this!

Izl

Differentiatingtherelation

f(z)=

z"q(

1

+

i1

yields

Z Zn+l

--znqt(1

+

112)

q’(x

+

Izl)

’

OOf=q"

znIZl2

z"

(1

+

Il)

+

q’

l-

+

1)(1

+

I1)

(1

+

I=lp"

Addingup and using

Izl

1

1

-4-Izi

2’1- t= ₁

+

Izl

wefind

-(1

4-

Izl2)2Obf

4-

u(1

+ IzlZ):Of

I-t(1

t)q"

4-

(v

4-

2)tq’

(n

+

1)q’lz".

whichapparently yields

(19).

|

Let usreturn foramoment tothe "hyperbolic" case

(disk

or

halfplane).

Actuallyevenbefore the paper

[10]

waseverconceived, thesameeigenvalue problemhad been studied

(unknown

to

us at the

time!)

in

[3]

in a different context

(in

connection with the

Feynman

integral "with affine kinetic

variables")

and thereit had essentially been reduced toasingle onedimensional

SchrSdinger

(or

Sturm-Liouville)

equation, namely with the Morseoperator

dy2 t-

D(e

-2 e

-)

(D

a

constant)

in

L2(A).

Recall that this operator first appeared asa phenomenological model foradiatomic

molecule

[7].

6

Indeed,

this isintimatelyrelatedtothe "continuous" orthogonalbasis constructed in

[10]7,

viz.

where

Pt(t)

are"certain orthogonal polynomials considered by Romanovski

[11].

They are

(see

[11])

orthogonalwithrespect totheweigth

t+Ze-/,

aregiven byaRodriguesformula

pt(t)

tt+2e/t

(

d

)

[t2t-’-2e-1/t

and satisfy anordinarydifferentialequation:

(20)

t2P

’’ +

(1

at)P’

l(l-

er

1)P.

It is very curious that essentially the same differential equation in twoslightly different guises appearedinthesame year 1929s madin twocompletelydifferent fields: instatistics, respectively

inquantum theory.

Let

usformallyworkouttheconnection.

In

multiplicativelanguagethe Morseoperator isgiven by

(see

[7])

:

-[2" + 2’ +

(-2

+

v

+

)]

+

₂

238 J. PEETRE AND G. ZHANG

+1 +1

2 "2

=/(a+l-/)

(/=0,1,...,[a+2

1]).

Write

Differentiatingwefind

(()

P()(’e

-,,

=p,,()1/4(-4e-

+

p,()(_

.a(-a

+

1/2(--2)e-

+

+

P’()(-(-

+

(-)e-+

+

P()((y

-

1)(

-

-,

-+

)e

-[p,,g

{-4

p,

=,

,,a

+

()(-(3

)- +-=)+

+

P()((

1)

-=

-’

+

)]-.

Put or{

.

It follows that

(we

omit the factor

{

e-

d write

P

etc. instead of

--2

="

P"t=

+

P’(-2(

)t

+

)

+

P((3

)-

3t-’

+

at

),

1--1

{’

P’t

+

P(3

),

--2 --1

(-{’+{+a)=P(-at

+

t

+

).

Hence

[P"t

+

P’(-2(

1)t

+

1

2t)+

+t

+-

-t

+-;t-

+)1

+

P((-

1)

-t-’

-2

t-1

--2

a 2a+4a

+

1.]

a

+

1)2

P

P"t

2

+

P’(1

at)

+

P.

4

+

2

[P"

+

(1

at)P’].

Itfollows that

(20)

isinddaconsequence of theeigenvMueequation

It is now natM to k the question: What ia the analogue

o[

the Morae operator

]or

the aphere?

Toswerthisquestionwemustfirstdigressalittle.

DIGRESSION: DUCTION OF

A SELF-ADJOINT

SECONDORDER OINARY

DIFFER-ENTIAL

EQUATIONS

TO NODAL FORM.

Considera-adjointeigenvMueequationof the form

u_w_d

wa

dq]

+bq=q

in

L2(I,

wdx),

where

I

issomeintervM C dwisapositiveweight. Furtheore,wesume

that thecoecient a is positiveand, similly, that bisreM. So the operator is foy

seg-adjoint dweexpectit tobe sebounded from belowon astable domMn. Wewish toreduce

itto thenormM

-f" +

in

Le(J,

dx),

where J issome other intevM

(the

ce a w

1).

We will thi of V the

potentiM

Wethen have

HARMONIC ANALYSIS ON THE QUANTIZED RIEMANN SPHERE 239

Ifwerequire this to coincide with

wemust have

rnw

dx

d(x)

’dx,

whichgivesthefollowingcondition.

CONDITION I.

[m2w-

I’11

o

m-w1/2

(There

are twocases, according towhether isa sensepreservingdiffeomorphismora sense

reversingdiffeomorphism.) Derivation yields

f,,

q

m+frn’,

d

(waq’)

f"’2mwa

+

f’(’m)’

f’

d-

wa

+

’m’tw

+

fm"wa +

(wa)’(f’’m

+

fm’).

Thus wegetasecond condition.

CONDITION II.

(’)a

1 or

’

4- 4.

f

-’-r.

In

thishypotheses

(I

+

II),

actually,the coefficient of

f’

vanishes:

(qo’rn)’wa

+

p’m’wa

+

(wa)’’m

0

:

(division

with

’rnwa)

(o’rn)’

4---I---

m’

(wa)’

p’

m m wa

p’rn2wa

const

and the latterisalogicalconsequence of

&

II.

Now

wegivealookat the coefficient of

f.

This gives, finally,

V=-(m’wa)’

+b

| mw

EXAMPLE.The multiplicativeMorse operator

-(z’)

+

be.

In

this case a x

2,

w 1.

(The

exact form of b is not ofinterest to us at the

moment.)

ConditionIIgives

’

1,

log

x,whileCondition givesm

x’}

so

x-1/2

f(log

x),

which isthe transformationonp. 89 of

[31

(see

formula

(3.12)).

Thuswefind

v

-(-

"

+b=-

(-

+b=-,+b,

so weobtainthe "additive" Morse operator.

Afterthisdigressionwereturntothecaseof thesphere. Itis convenienttoputp v- n

(and

forget the

v),

sothe equationthat interests us is

d

-t-"(1

t)-

t,,+(

1

t)+

dq

l(n

+

p

+

+

1)q.

Thusitisacasewitha

t(1

t),

b 0,w

t"(1

t)p. FromCondition

II

weget 1

(’)

(1

t)

This timewechoose the minussign:

240 J. PEETRE AND G. ZHANG

0

dt sin

.

cos2₃

dO dO.

0(1

cos

1

Thus the parameter 0 is the same asin Section I. We are definitely on the right track! From

Condition we nowget

m

x/’t(1

t)t"(1

t)v

t+(1

t)+1/4

Logarithmicdifferentationnowyields

1 1

m’=m.

-(+1/4)y+(l+)]-=-1 1

+

(

+

i)

"+-

t)

+’}+

(

+

g)

t’+’+(1

t)+’

(1

Thuswefind

"-

t)+x-

t’+-

t)-m’wa

-(-ff

+

)t

(1

+

(

+

-)

(1

(m’w)’

(

+

1(

1/4)t--l(

t)+,-_

-(

+

1(

+

1

1/4ItS-t(1

)t-1/4(1-t)-1/4

+(+.)(.+1--"

+-1/4

-(

+

1/4(

(1

-whence finally

(see

boxed

formula)

mw

t’-(1- t)-1/4

V

(m’wa)’

.

1

mw

=("

+

i)("

’)

+

(

+

i)(

i)]--

+

n_3 3

-[(

._

+

)(

)

+

(

+

1/4)(

)]

1

4

+

l’--t

_3 3 3

-[-7

.+.--+-.+.-1

1

The constant term here canberewrittenasfollows:

Thusweendup with theSchrSdinger operator

d2f

(n4_

:_1

1(.__1

)

--+

tan

2+cot2-i

)(p-1/2)+1/2

f=l(n+p+l+l)/

which isthesought "periodic" analogueof theMorsepotential.

Wemay summarizethe abovediscussionasfollows.

PROPOSITION 2. The eigenvMue problem for the invariant Laplace operator for -v-forms on

S2_{is equivadent, in a sensemade precisein theforegoing, toan infinitesystem of synchronous}

Schrfd/nger

(or

Sturm-Liouville)

eigenva/ueequationson the

interva/.(0,

r).

|

HARMONIC ANALYSIS ON THE QUANTIZED RIEMANN SPHERE 241

which thus accounts for the

"degeneracy".

If we were to use the disk realization, we would certianlyagain get infinitelymanyequations.

(One

hasonly tosubstitute,inthe aboveformula,

thetrigonometricfunctions bytheirhyperbolic

("alcoholic")

counterparts tanh and coth

.)

We donot knowifanyofthishas any bearingtophysicswhatsoever.

6.

APPENDIX.

FACTORIZATIONS

OF

CERTAIN DIFFERENTIAL OPERATORS

ON

A

RIEMANN SURFACE.

Wecontinuethe computationin

[10],

remarkinSection3.

As there, welet

X

denoteaRiemannsurfaceequippedwithanHermiteanmetric,in termsof

alocalcoordinate zgiven byds2

(z)ldzl

.

We

_consider t,-formslocally given inacoordinate neighborhood

(with

coordinate

z)

by

f

f(z)(dz)".

Ifwemakeachangeof coordinate

(z

X(z))

then these coefficients experience the change

f(z)

,--+

f(X(z))(X’(z))", 9(z)

g(x(z))lx’(z)l

2.

(Notice

that theparametervplaysinthis Section thesamerrleas

v/2

in

(the

rest

of)

[10],

while compared tothe restofthe present paper it is the sameas

-v/2.)

We

denoteby

L2’"(X)

the

spaceofsquareintegrable

v-forms,

that is,

f

E

L2’(S2)

if andonlyif

fx

0 0

Letus write0

z’

(Wirtinger

operators).

TheinvariantCauchy-Riemannoperator

isgivenby

D

-’.

Wefurtherset

D,,

-0

+

(r,

1)

Og

g

It may be viewed asthe adjoint of/)

=/),,

regarded

asanoperatorfrom

L

2,u _into

L2’-1. In

particular, it maps

L

2,-’ _into

L

_2,.

(In

[10]

weinterpreted

D

asthe "metric" connection on

the sheaf of all holomorphic

v-forms.)

Thecorresponding Laplaceoperatorisdefinedby A

A,,

D,,/).

Written out it is

A

--g-’

0

+

r’g-2(Og).

In

[10]

we

had,

for some reason, written the factors in adifferent order. Wenow explain this

discrepancy better.

Itisconvenient to set e

(=

01ogg). In

thisnotation

Du

-0

+

(t,

1)e.

Furthermore, g

welet

g -2g

-

te

-2g

-

t(-)

-2g-1

t0(log

g).

bethe Gaussian curvature. Thenwehavethefollowinglemma

(in

a

lbcal

coordinate

neighbor-hood).

LEMMA.

[0,/)]

--e/),

[/),

e]

-K.

PROOF" Itis clearthat

[o,D]

[o,-’]

0(-’)

--(0)

-D.

Kg,

wefind

Onthe other

hand,

asby definition

e

--K.

I

[,]

[-’,]

-’(0)

COROLLARY.

/D+,

D/

+

K.

PROOF: We get

/)D+,

b(-O

+

t,e)

-0/)

+ [O,/)] +

t,e/

+ t,[/),

e]

242 J. PEETRE AND G. ZHANG

Fromnow on we assumethat wehaveametricwith constant curvatureK. Weclaimthatone can define recursively polynomials

Or(T)

(where

the letter

T

stands foran

indeterminate)

such

that

D,,D,,_I

D,,_D

+a

Ot(A).

Indeed,if 0 this isjust thedefinitionofA

A,

with

O0(T)

T.

Assume that Ot_l isalreadydefinedandmultiply thecorrespondingrelation

(l-

1 insteadof

l!)

with Afrom the right. This gives:

D,,D,_I

...D,_(#_I)DD,D

O-I(A)

A.

Consider the

operator/)tD.

Wefindusing the corollary repeatedly

(the

first timewe applyit with vreplacedbyv-1,the secondtimebyv-2,andso

forth)"

As,

summingthe arithmetic series,

(u

1)

+

(u

2) +...

+

(v

l)

i(2,-0,

itfollows that

In

other

words,

we can take

K)Ot-I(T)

Or(T)

(T

+

l(2v

1

-l)y

or, solvingthis recursion with initial condition

O0(T)

T,

O,(T)

T(T +

(2v-

2)-)

(T +

2(2v

3)-)...

(T +

l(2v

1

l)-).

Altogether,

wehavenowestablished thefollowingresult.

THEOREM 2. Wehave the following factorization

D,,D,_...

D,_,/)

TM

A(A +

(2

2)-)

(A +

2(2v

3)-)...

(A

+./(2v

_I

-l)).

_| Weconcludewithseveralremarks.

REMARK. Introducingthegradedvector spaceL

]e

L

2’u,

where the summation isoversome remainder class of the indices u mod

Z,

we can regard the operator /) as an endomorphism of

L and,

similarly, we can define an endomorphism

D

on

L

extending

D,

L

2’"

L2’"+1.

Thenone canget a moreelegantformulation of the above results.

In

particular, we can write

D,D,_...D,,_D

+

_{more compactlyjust} as D+/)

+.

This is in line with howit is donein cohomology theory.

REMARK.

To someextent thepurported generalityofanarbitrary Riemann surfaceis illusory,

asby theuniformizationtheoremone can reduce oneselftothe situationofa simply connected

manifold,that is,eitherthe disk,the diskorthe("parabolic") plane. Onecanthen alwaysassume

that themetric isgiven by

1

g-(1

+

K--)

2"

HARMONIC ANALYSIS ON THE QUANTIZED RIEMANN SPHERE 243

bb32J

the

(second)

relation inthe lemma canbe written

[/,

e0]

1, Putting

co(z)= 5/(1+

,,

Thus theoperators /) and

(multiplication

by) e0 formally span an

algebra

isomorphic to the

"Weyl

algebra"

(see [2]).

Wedonot know what the deeper consequences ofthisobservationare, ifany.

Footnotes. Note that numbers on the left corner of the _{following footnotes have} been cited in the text.

Forthis casesee[13].

2Foraunifiedtreatmentonecould put oneselfin thegeneralsituationofaRiemannianmanifold

withconstant Gaussian curvature. Cf.Appendix.

3Ifoneoftheparametersa, bofthatformulaisanegativeinteger, then the coefficient of thesecondterm tothe right vanishes,so get

r()r(c b)_F(a,b;a

+

_b

+

_{1; z).}

F(a,b;c;z)

r(-a)r(

2

4Therewewrote,forsomereason, 2tin place oft.

Sln[15]theparameteranalogousto wasused tolabeltheoperator. Inretrospect, seethat thiswasperhaps not themostnatural choice.

6Wearegreatfulto Thierry Paul foracquantinguswiththe contents of[3],in particular, the r61e of theMorse operator,duringamemorable "workshop"attheMittag-LefflerInstitute(fall ’90).

7In[10]thereisalso givena"discrete"orthogonalbasis, whichistheonewhichisclosesttothe basis encountered in thepresentpaper(Section1). Inviewofwhatisdone in[9]onemay also ask whatistheanalogueof theMorse operatorforastrip.

SThisisalso the year of theweddingofthe parentsof theseniorofthe two presentauthors!

REFERENCES

1.

BEREZIN,

F.

A.,

Generalconceptofquantization. Commun. Math. Phys.40

(1975),

153-174.

2.

BJ(RK,

J.-E.,

Rings

of

differential

operators. North

Holland, Amsterdam,

1979.

3.

DAUBECHIES, I.,

KLAUDER,

J.

and

PAUL,

T.,

Wiener measuresforpath integralswith a.ffinekinetic variables.

J.

Math. Phys. 28

(1987),

85-102.

4.

GUSTAFSSON,

B.and

PEETRE, J.,

Notesonprojective structuresoncomplex manifolds.

Nagoya

Math.

J.

116

(1989),

63-88.

5.

HELGASON, S.,

Groups

and geometric analysis. Academic

Press,

ondon,

1984.

6.

MAGNUS,

W. and

OBERHETTINGER, F.,

Formulas and theorems

for

the

functions

of

mathematicalphysics.

Chelsea,

New York, 1949.

7. P.

MORSE,

Diatomic molecules according tothe wave mechanics. II. Vibrational levels. Phys. Rev. 34

(1929),

57-64.

8.

PEETRE, J.,

The Berezin transform and Ha-plitz operators. J.

Oper.

Theory 24

(1990),

165-168.

9.

PEETRE, J.,

Orthogonal

polynomials arisingin connection with Hankel forms ofhigher weight. Bull. Sci. Math.

(2)

116

(1992),4.

10.

PEETRE, J., PENG,

L. and

ZHANG, G., A

weighted Plancherel

formula

I. The case

of

the unitdisk. Applications to Hankeloperators. Technicalreport, StockholmUniversity, 1990.

11.

ROMANOVSKI,

M.

V.,

Sur quelquesclasses nouvelles depolyn6mes orthogonaux.

C. R.

Acad. Sci. Paris 188

(1929),

1023-1025.

12.

VILENKIN,

N.

JA.,

Fonctions spciales et thorie de la representations des groupes.

Dunod,

Paris, 1969.

13.

ZHANG, G., A

weighted Plancherel

formula

II. Thecase

of

theball. TechnicalreportNo. 9, Mittag-Leffier Institute,

1990/91.

14.

ZHANG,

G.,

Hankeloperators between Moebius invariant subspaces. Math. Scand.

(to

appear).