Function spaces of harmonic and analytic functions in infinitely many variables

Function spaces of harmonic and analytic functions in infinitely

many variables

Citation for published version (APA):

Martens, F. J. L. (1987). Function spaces of harmonic and analytic functions in infinitely many variables. (EUT report. WSK, Dept. of Mathematics and Computing Science; Vol. 87-WSK-02). Technische Universiteit Eindhoven.

Document status and date: Published: 01/01/1987

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University of Technology

Netherlands

Department of Mathematics and Computing Science

Function spaces of harmonic

and analytic functions

in infintely many variables

by

EJ. L. Martens

AMS Subject Classifications: 81 B05, 46F05, 46F10

EUT Report 87-WSK-D2 ISSN 0167-9708

Coden: TEUEDE Maart 1987

Contents

O. Introduction

1. Preliminaries

2. Spaces of functions related to spaces of harmonic functions

3. A space of functions on a real separable infinite dimensional Hilbert space

4. A space of functions on the unit ball in IR£2

5. Special isomorphisms between

M,

the symmetric Fock space, the

IN

Bargmann space and (IR ,d~)

6. Characterization of

M

and related spaces

Acknowledgement References 1 3 15 33 56 61 79 96 97

Introduction

In this university report we study spaces of functions on subsets of Hilbert spaces. There are three main subjects in this paper.

The first subject is the theory of reproducing kernels. Most results on

re-~roducing kernels which we apply, have already been stated in Aronszajns paper [Ar]. We apply the theory of reproducing kernels to Hilbert spaces of harmonic functions.

The second subject is the harmonic or spherical harmonic function. As a guide and starting point we used the book Spherical Harmonics of Muller [MuJ.

The third subject is the representation of some special groups in the intro-duced spaces. The papers of KOno and Orihara [Ko] and [Or] served as a source of inspiration.

The purposes of this report are the introduction of spaces of functions of infinitely many variables, which are Hilbert spaces, analyticity spaces or trajectory spaces. These spaces are characterized both functional analytical-ly and anaanalytical-lyticalanalytical-ly.

Now we present a short summary of each section separately.

Section 1 deals with reproducing kernels, representations, analyticity on locally convex vector spaces and Gegenbauer polynomials.

In section 2 we introduce a Hilbert space ~ of harmonic functions on lRq and we study its reproducing kernel. We also consider the usual representa-tion of the orthogonal group on lRq in ~.

In section 3 we define a Hilbert space

M

of functions of an infinite number of variables, which contains all spaces A~, q E IN. The fact that the set of harmonic functions is dense in M is used to construct specific orthonormal bases and to prove some general properties of the elements of

M.

In section 4 we define a Hilbert space

N

of functions on B the unit ball in (U'2' We give results on this space

N

similar to the results of'the space

M

as defined in section 3.

In section 5 we consider isomorphisms between M and some well known spaces, such as the Pock space, the Bargmann space and a space of square integrable functions with respect to a Gaussian measure. For the latt.er type of space we refer to [Ko] and [OrJ.

In the Bargmann space of infinite order the harmonic 90lynomiAls are dense (Theorem 3.8). This is not true in a Bargmann space of finite order.

In section 6 characterizations of the space

M

and related spaces are present-ed. Now analyticity plays a more important role than harmoniticity. We have two kinds of characterizations: Estimates with respect to reproducing kernels or a combination of a growth condition and a (real) analyticity condition.

1. Preliminaries

In the first part of this section we present some general results on repro-ducing kernels and unitary representations of topological groups.

In the second part of this section we mention some recurrence relations of Gegenbauer polynomials.

In the third part we introduce the notion of analyticity on a locally convex topological vector space.

Finally we discuss an integral identity for c1-functions on a neighbourhood of the unit sphere Sq-l in IRq.

Reproducing kernels

In this section F denotes a class of functions from a set D into ¢ which establishes a complex Hilbert space with inner product (.,.) F and norm

n.

If",

Definition 1.1. The function K: D x D ~ t is called a reproducing kernel of F iff

(a) V D: K{"Y) E F

y£

f (y) •

We list some properties of reproducing kernels. These results can also be found in [ArJ or [YosJ.

(1.1) For

W

c

F

let Span(W) denote the linear span of

W,

i.e.

Span(W)

n

{ I

j=l The subspace Span({K(',y)

a.w.

J J

Y ED}) is dense in

F.

(1.2) Let the space F have a reproducing kernel K. Then for all x,y E D II K(' ,y)

I~ =

KCy,y)

K(x,y) K

(1.4) The space F has a reproducing kernel iff for each y E D the evaluation functional f ~ fey), f E

F,

is continuous.

(l.S) Let

F

have a reproducing kernel K. Then we have

(1.6) Let Fl be a closed subspace of F, let F have a reproducing kernel K on D x D and let P

1 be the orthogonal projection from

F

onto

Fl.

Then Fl has a reproducing kernel Kl with Kl (.,y) = Pi (K(.,y» and

Y E D, f E

F •

(1.7) Let Fl and F2 be two mutually orthogonal closed subspaces of the function space F with reproducing kernels Kl and K

2, respectively. Then F admits a reproducing kernel K with K = Kl + K

2.

(1.8) Let

F

have a reproducing kernel K and let _. (~) _{n n} be an orthonormal basis in

F.

The following equalities are valid:

I

I(j)n (x)

12

= K(x,x)

n

I

~n(Y)~n(x) = K(x,y)

n

x,y ED.

(1.9) Let (~) be an orthonormal basis in

F.

n n

The space

F

admits a reproducing kernel iff for all x,y € D the series

L

~ (y)~ (x) is pointwise convergent. _{n n}

n

(1.10) Let DO denote a set and let K denote a function from DO x DO into ~ with the property that for each n ~ 0, a

j ( €, Yj ( D, 1 $ j n

L

ajatK(Y~'Yj) ~

0 . j,t

Then there exists a vector space FO of functions on DO' which is a Hilbert space and which admits K as a reproducing kernel. This space FO is uniquely determined.

(1.11) The elements of a Hilbert space with reproducing kernel can be characterized with the aid of the reproducing kernel:

(1. 12)

Theorem 1.2. Let

F

denote a Hilbert space of functions from a set 0 into ¢ with reproducing kernel K •. Let f denote a function from D into C. Then f € F iff there exists c > 0 such that for each t E lN and all ct. c Il,

_Y

j 0 with J t

I

L

ctl(Y_j )

I

j=l Proof. *: Let t c lN, ct

- - -

j straightforward estimate t

I

L

ct/(Yj )

I

j=1 1 S j

s

t R.

_-

_~ s c(

_I

_{~ctjK(Yk'Yj) }} k, j=l E (;, _Yj € 0 where 1 :< j t ==

I (

I

u.K(.,y.) ,f)

FI

s j=l J ) t S IIf

IF

II

I

K{.,yj)II_F

s

t. Then we have the

-: Let

W

denote the Span({K(',y) l y e D}). The linear functional

m:

W

~ C is defined by t t m(

L

ct j K ( • , y j) ) =

L

ct. j=l j=l J R. c lN, ct. E 1:, _Yj f. D where 1 s j s t. If J t

L

j=1 a.K(',y.)

= a

I J ) then R.

I

j=l a.f(y.) ==

a

J )

because of (1.12), so the functional m is properly defined.

By assumption (1.12) and the properties of the re~roducing kernel of F it follows that for all w E

W

1m

(w)

I

~ ell w II •

The space

W

is dense in

F,

so by the Riesz' representation theorem there exists 9 E

F

such that for all w E

W

mew) == (w,g)F •

In particular for each y E D we have fey) = m(K(·,y» (K(· ,y) ,g)

This implies f == g, whence f E

F.

rJ

Representations

Let

G

be a group with identity element e.

The Banach algebra of bounded operators on a Hilbert space X will be denoted by

L(X)

and the identity on X by I or IX'

Definition 1.3. Let

G be a topological group, let X be a complex Hilbert

space and let IT denote a mapping from

G into

L(X).

The mapping IT is called a representation of the group

G in X iff

(a) (b) (c)

V

h

G:

TI(gh)

=

TI(g)IT(h) g, E II(e) == I

V X= 9 + IT(g)x, 9 EGis continuous at e •

XE

A representation IT is called unitary iff

(d)

_V

G: [IT(g)]

*

=

IT(g -1 ) . gE

Let

F

denote a Hilbert function space with reproducing kernel K: D x D ~~ <C

and let G denote a group of transformations on D. Then under certain condi-tions on K there exists a canonical unitary representation of

G

in

F.

(1. 13) (1. 14)

(1. 15)

( 1. 16)

Theorem 1.4~ Let G denote a topological group of transformations on O. Let F have a reproducing kernel K with the properties:

(a) _V

R € G V x,y€ 0: K(Rx,Ry)

=

K(x,y)

(b) Vy€O: The F-valued mapping R + K(-,Ry), REG is continuous at I D. Then the mapping IT: G+

L(F)

defined by

X E 0, f E F, REG is a unitary representation of G in F.

Proof. Put W = Span({K(',y) lYE oJ).

For each REG the operator 'IT (R): W + W is defined by

t

IT(R) (

I

(),K(·,y.)) =

J ] Cl.K(-,Ry.), J ] R, ?: 0, Cl. E

e,

y.

€ 0 •

J J

It is easy to see that

-1 [1T(R)f](x) = feR xl, 1T(RS)f = 1T(R)1T(S)f, fEW X E 0, f €

W,

REG R,S E G, fEW -1 ( 1T (R)f , g)

F

=

(f, IT (R ) g) F' f,g E

W,

R E:

G .

We only prove the last statement. Let f,g E

W

with

k

f =

I

Cl.K(-,a.) and

j=1 J J

Then for each R E: G

(1T(R)f,g)F "" g -1 (f,lT(R )g)F' m

=

_I

B,Q,K(' ,b,Q,)

.

1=1

The statements (1.14)-(1.16) imply for all REG, fEW: Ihr (R) f If

=

II f If .

The set

W

is dense in

F.

Let Ti(R) denote the unique unitary extension of

1T (R) to

F.

Let REG, let f E F and let XED. Then we have [1f (R)

fJ

(x) (n(R)f,K(o,x»F -1

=

(f,1T(R )K(-,x»F -1 (f,K(o,R x»F f(R-1X) • In the sequel we denote IT(R) by IT(R).

For all R,S E

G

it follows from (1.14) to (1.16)

II (RS) II(R)IT(S)

Let f E

F.

Finally we want to prove that R + IT(R)f is continuous at 1 0,

E

Let £ > O. We take 9 E W such that II f - 9 II <

'4 "

Let

m

9

I

j=l

CX,K( ,y,)

J J with cxj E ~, CX, ~ J 0, v, - J ED.

Let U_j be an open neighbourhood of 10 such that for all R E U

j

m

Put U = n U

j < Then U is an open neighbourhood of 1D"

j=l For R E U we have m

~

t

+

I

I

CX]'

I

II K ( • , RYJ') - K ( • , y J,)

n

F+

:i

< £ " 0 j=l

Let V be a subspace of the Hilbert space X and let A E L(X) be such that A (V) c V.

This subspace V is called A-invariant.

The restriction of

A

to

V

will be denoted by

Alv'

It is clear that

Alv

E

L(V).

*

If also A (V) c V then V is s.aid to reduce A. If U is a unitary operator then each V-invariant subspace is also reducing.

Definition 1.5. Let

n

denote a unitary representation of a topological group

G in the Hilbert space

X.

The representation

n

is called irreducible iff each closed subspace

V

of

X,

which is TI(g)-invariant for all g E

G,

equals {a} or

X.

Lemma 1.6. Schur's lemma.

Let IT be a irreducible unitary representation of a topological group

G

in the Hilbert space

X.

Let

A

E

L(X)

be such that for all g E

G

the operators A and

neg)

commute.

Then there exists

A

E ~ such that

A = AI .

• See [so], I, Satz 2.3.

-Theorem 1.7. Gurevi~.

Let IT be a unitary representation of a compact topological group

G in

the separable Hilbert space

X.

Then there exist finite dimensional invariant subsoaces _{' - n}

X ,

n E IN, of

X such that

(a) (b)

X

1

X

n m X = (9 X n n n

:F

m

(c) g + IT (g)

Ix '

g E

G,

is an irreducible unitary representation of

G

n

in X for all n E IN . n

(1.17)

( 1.18)

(1.19)

Gegenbauer polynomials

For q 2 2 and n ;:::

°

we define the function pq: :IF -+ lR by

n

Lemma 1.8. Let q ~ 2.

Then the following statements are valid. (a) pq is a polynomial of degree n .

n

(b)

°

I

(c) 1 , n E IN •

Proof. See

[MuJ,

p. 16-18.

n

:f

m

In [MOS], § 5.3 the Gegenbauer polynomials cA of degree n, n

~

0, and order

n

o

A, A ;:::

-~, are introduced as special cases of Jacobi polynomials. These poly-nomials satisfy 1

f

cA (t) cA (t) (1 _ t2)A-~dt =

° ,

n m n ;. m -1

""

I

CA(t)rn = 1

,

n=O n (1 - 2tr + r2)A A ;. 0 I t E [-l,lJ,

Irl

< 1

""

I

CO(t)rn -log (1 - 2tr + r ) 2 n=O n Lemma 1. 9. F ix q ;:: 3. Then pq = r(n + l)r(q - 2) c~q-l n r{n + q - 2) n n ~ 0 p2 =:; ~ cO n 2 n ' n 2: 1 1

(1. 20)

(1.21)

Proof. 14' rom Lemma 1. 8 (b) and re lation ( 1. 17) we obtain a q E lR such that

n

~rom Lemma 1.S(c) and relation (1.18) it follows (aq}-l

=

c~q-l(l)

=

r(n + q - 2)

n n r(n+ 1) ( q - 2 ) '

The proof for p2 runs similarly.

n

We present two recurrence relations for the pq,s. By pq we mean (Pq) '.

n n n

Lemma 1.10. Let q ~ 2 and n ~ O. Define P~1 (t) = O. Then we have

(2n + q - 2)t pq(t)

n (n + q -

2)P~+1

(t) + n

P~-l

(t)

and

(2n + q -

2)P~(t)

=

(n(~; ~)2) P~+l

(t) - (n + : _ 3)

P~-l

(t)

Proof. From [MOS], § 5.3, we have

2(n +

A)tC~(t)

= (n +

1)C~+1

(t) + (n + 2A -

1)C~_1

(t) and

With the aid of Lemma 1.8 we get the desired results.

Lemma 1.11. Let (t) be a Cauchy sequence in IR with limit t. q q

Then for all n ~ 0

Proof. Since pq - 1 we have lim P6(t ) = 1.

0 _{q-l-<X> q} Assume for 0 ::; j ::; n lim P~(t ) t j q-)<X) ] q Because of Lemma 1.10

o

t E IR

o

pq 1 (t ) n+ q (2n + q - 2) t pCJ (t ) (n + q - 2) q n q (n + n q - 2) pq l{t ) n- q whence lim pq l{t ) n+ q q~ n+l = t

1<' or q,m € IN and n ::>: m define the function Aq+1: [-1,1] -l>

m.

by

n,m (1.22) Aq+1 (t) n,m Lemma 1.12. Let q,m,n 1,n2 t IN with nl,n2 ~ m. Then (a) + 1 (b) Aq+1 (t)

rem

+ ( _ _ _ _ _ ) 1:1(1 _ t2)m/2 • m,m

_r

(m +

!)!;

Proof.

(a) See [MU], Lemmas 14 and 15 and pages 3 and 4.

(b) Because of Lemma 1.10 the coefficient of tm in pq+1 is positive.

m

By (1.22) there exists a > 0 such that

Aq+1(t)

m,m

Since -1 we have Analyticity q + 1

rem

+ - 2 - ) 1:1 ( )

r

(m +

!)

!;/r

(m +

The notion of analyticity will be generalized.

fEfini tion 1. 13. Let W denote an open subset of

«:q.

The function f: W -+ C is called analytic on W iff for all w € W there

exists an open neighbourhood N of w such that for all Z E N

w w

f (z)

where the series is absolutely convergent.

k

(z - w ) q q q

fEfinition 1. 14. Let V denote a complex (real) locally convex topological vector space and let W denote an open subset of V' •

The function f: W -+ it is called analytic on W iff f is continuous on W and for all w ( Wand v (' '/ there exist an open neighbourhood _. M _w,v of

o

in it such that

A

-+ f(w +

AV)

is analytic on M (is extendible to an

w,v analytic function on M ) .

w,v

If f is analytic on an open subset W in

c

q because of definition 1.13, then certainly f is analytic because of definition 1.14.

Theorem 1.15. Hartogs theorem.

If u is a complex valued function defined on an open set ~ c Cq and u is analytic in each variable z. when the other variables are given

ar-J

bitrary values, then u is analytic in Q.

Proof. See [HoJ, theorem 2.2.8.

An integral identity

[J

Let sq-l and sq denote the unit sphere and the

o~en

unit ball in mq, respec-tively.

q-l

The usual surface measure on S will b~denoted by dO q_1 .

1~~~~j~j~ Fo~ a C'-function on a neighbourhood of sq the normal d~rivative in a point I; E sq"1 wi 11 be denoted by

d f (0 v

Theorem 1.16. Let f and 9 be continuously differentiable functions on a neighbourhood of the closed unit ball Bq• Let j c IN with 1 ~ j ~ q. Then we have the following identity

a

[ax.

fJ(I;)g J f ( ~)

_ax.

9

J (

~) dO' 1 (~) + q-J +

J

l;.[a _{J v} f(l;)g + f ( t;) a v 9 ( t;)

J

do q-1 (l;) + Proof. Put sq-1 + (q - 1)

J

t;.f(t.:)g{t;)dO _{J q-}1(1;;) • q-l S

[~.

fJ( 9 J () f (I;) [-" - 9

J (

l;) do 1 ( l;) • ox. q-]

Straightforward calculation yields

M =

I

[d

_ax.

(f. -g) ] ( 1;) d a q-l ( t;) J q-l S

I

k=l q-l S B

f

-3 -

()

[ 3 -a -(f.g) ] (x) xkdx -~ x. q J + q

I

[ax. (f.g) J(x)dx

a

-

=

J (Gauss theorem)

J

q-l S B

f

~-a

ar ()

-(

f . g)

-

(x) x. Jdx + x. ~ k q J

f

a

-(q-l) [ax.(f.g)](X)dX J

s·[a

f(l;)g{t;) + f(l;)a g(t;)JdO I(!;) +

J v v

q-+ (q - 1)

J

l;jf(t;)9(I;)do

q_I (1;) • sq-1

2. Spaces of functions related to spaces of harmonic functions

In this section we introduce spaces of fUnctions which are restrictions of harmonic functions to sq-l or are harmonic functions on IRq. Most results are extensions of results in [Mu] and [EG].

We use the following notations where we take q £ lN, q > 1, fixed. q-1 S (a,r) dx dO' _q-1 2rrq / 2 0' q-l

_r

_(s.) 2 f.:,

r

a

2 q 2 j=l ox. J

a

_q,v

₌

r

x. l. j=1 Harm(W) (

...

) II • II

a

ax. _l.

, the sphere with centre a and radius r in IRq , the unit sphere in IRq

, the open ball with centre a and radius r in IRq

the open unit ball in IRq

the usual Lebesgue measure in IRq

, the usual (q-1)-dimensional surface measure

q-l on S

Note that dx

q-l , the total surface measure of S

, the q-dimensional Laplacian

, the q-dimensional normal derivative , the space of complex valued functions f

which are harmonic on a neighbourhood of each point of W c IRq

Note that f € Harm (sq) implies

f E Harm(Sq(O,l + Ell for some E > O.

, the compact group of orthogonal transforma-tions from IR q onto IR q

, subgroup of O(IRq) consisting of all rota-tions R(det(R}

=

1)

, the inner product on IRq , the norm on IRq •

(2. 1)

q-1 q-l

Consider the Hilbert space L

2(S )

=

L2(S ,dOq_t) of square integrable

q-l

functions on S with the natural inner product

f ( ~) g ( ~) do 1 ( t;;) ,

q-We denote the corresponding norm by

II. II

1 •

q sq- q-1

For each R E O(IR ) we define the operator L

R: L2(S ) Then we have: - LRS LRLS - LIf

=

f , q R, S € 0 (IR ). q-l f € L 2(S ).

The mapping R + LRf is continuous - The operator (L

R)* is equal to

L

T the surface measure do 1

q-R

q-l for each f E L

2(S ).

because of the rotational invariance of

q q-1

So R + LR is a unitary representation of O(IR ) in L

2(S ).

Since O(IRq) is a compact group, theorem 1.7 yields the existence of mutual-ly orthogonal finite dimensional subspaces Vq n

~

0, of L (sq-1) such that

n' 2

- L

2(Sq-1) = $

V

q_•

n=O n

The subspaces V~, n ~ 0 remain invariant under the operators L

R, R € O(IR q

), and the representation R + LRI is irreducible.

V

q n

In [Vi], Chapter IX, § 3 the following has been shown:

The subspaces

V

q can be selected in such a way that for each f E

V

q there

n n

exists a unique harmonic homogeneous polynomial p of degree n in q variables (i.e. p(AX)

=

Anp(X) and [6~](X) = 0, A € IR, x E IRq) such that

f(~) = p(~) ,

This leads to the following definition:

Definition 2.1. Fix q € IN and n ~ O.

Let ~ denote the space of harmonic homogeneous polynomials of degree n

n

in q variables.

Put dq dim(~) dim(Vq).

Lemma 2.2. Let q ~ 2.

The numbers dq can be explicitly calculated. We have

n

dq = (2n + q - 2)f(n + q - 2)/(r(n + l)r(q - 1»

n

dq

_o

1

_.

Proof. See [Mu], pp. 3 and 4.

n ~ 1

Since

V

q is finite dimensional, it follows from remark (1.9) that

V

q has a

n n

reproducing kernel. The following lemma yields an explicit formula for this reproducing kernel.

Lemma 2.3. Let {eq .

I

1 ~ j ~ dq } be any orthonormal basis of ~.

n,] n n Then we have dq n

-I

j=l

Here the polynomial pq is defined by (1.17).

n

Proof. See [Mu], theorem 2.

Corollary 2.4. Let f E

V

q•

n

Then

If(e;) I

Proof. See property (1.5).

q-l

~,n E S .

Let Dq denote the orthogonal prOjection from L (sq-t) onto

li

q•

n 2 n

Corollary 2.5. Let f E L

2(Sq-l).

For each

n

E sq-l we have the equality

o

o

Proof. See property (1.6).

o

In the following lemma we present the generating function of the polynomials pq.

n

Lemma 2.6. Let t,t € IR with It I < 1 and It I ~ 1. Then we have

L

n=O

For fixed t the series is uniformly convergent in t ,

ITI

~ 1. Proof. From Lemmas 1.9 and 2.2 it follows

q q (2n + q - 2) c~q-l

d·P =

-n n (q - 2) n

Relation (1.17) gives the wanted result. IJ

Corollary 2.7. Let t f IR with It

I

< 1. Then we have

L

n=O

1 +t

(1 - t) q-1

Proof. Take t 1 in Lemma 2.6.

o

Now we are in a position to link to each f € L (sq-l) a harmonic function

~f

2

on a ball Bq(O,r) where r ~ 1 depends on f. q-l

Definition 2.8. Let f E L

2(S ). Then

Let

r = sup{p ~ 1

(2.2)

We remark that the series in (2.2) is absolutely and uniformly convergent on each ball Bq(O,p) with p < r. This is a consequence of the definition of r and the corollaries 2.4 and 2.7.

q-1

Theorem 2.9. Let f E L

2(S ) and let r be as in definition 2.8. (a) The function Hqf is harmonic on Bq(O,r) .

(b) If there exist 5 > 1 and a harmonic function g on Bq(O,s) such that

g(~)

f(~)

for

~

E sq-l, then

(i) r ~ 5

(ii) [Hqf] (x)

=

9 (x) , X E B (0,5). q

(c) For y € Bq the following integral representation holds:

[Hqf](y) =

~

J

q-l q-1 S 1 - II Y Ir f (

~)

do (

~

) •

"~

_ y

f

q-l

Proof. (a) and (b): See [Gr], Lemma 2.1. (c): Take y E Bq, Y ~ O.

Then we have

""

dq

J

::

I

~"Ylr

f(~)P~(II~

_II

.~)dOq_l (~)

n=O q-1 S q-l (Corollary 2.5) 1

J

1 - Ill' Ir f (~) do 1 (~)

=

₀ q-l (1 +

lIy

It - 2(y.E;:»Q/2 Q-q-l S (Lemma 2.6) -=---.::...~ f(~)da 1 (~) .

II

~

-

y

IF

q-=

Moreover

Let n ~ 1.

[Iflf]

(x) [Hqf] (0) So we see that for

Let f €

V

q• n = _{II x IPf (II: Ir}

=

0 . all 11 € S q-l f(~)dcr 1 (~) . q-Then

We remark that the mapping

Ifl:

L

2(sq-l) + Harm(B q

) is injective.

- -1

Next we consider the spaces Harm (Bq), Ifl(L

2(Sq » and Harm(B

q

). The first and the second are contained in the latter. So we get

o

This triple of spaces fits in a functional analytic setting which we describe now.

In [EG] the analyticity space

S

B and the trajectory space

T

Bare

intro-Y, Y,

duced for Y a Hilbert space and B a self-adjoint operator. Here

I

tB

{f € Y 3

t>0: f E D(e )}

and

They constitute the triple

As we shall see

for a suitable self-adjoint operator

A.

Definition 2.10. The self-adjoint operator A in L (sq-l) is defined by

2

Af

I

f E D(A)

n=O

The operator A has the following properties:

2 ~

(2.3) - The operator A is equal to -~(q - 1)1 + {~(q - 1) I + ~LB} where ~B de-q-l

notes the Laplace-Beltrami operator of the sphere S • (2.4) - For each f E D(A) with ~f E Harm (Bq)

Af (i;) [d ~fJ(i;)

q,v

because

a

h = nh for all h E Hq.

q,v n

(2.5) - Fix t > O.

For all f € D(e(log t)A)

Remark:

D(e(log t)A) =

From (2.5) it follows

The Space

T

1 can be represented by Harm(F3q). To this en.d we extend q-L 2(S ),A the operator

H

q: L 2(Sq-l)

~

Harm(B q₎ tension will be denoted by

gq.

to an operator on

T

The

ex-L

2(sq-l),A"

For each F €

T

we define

q-1

< L2 (5 ) ,A

x [F (-log II x ID] ( 9 )

[~FJ<O) = [F(1)](0) , 1 > 0 • We observe that F (-log II x ID E S q-1 •

The function gqF is harmonic on Bq•

Further gq is a linear bijection from

T

1 onto Harm (Bq) • Also this

q-is a consequence of (2.5). L2 (S ),A

If moreover F E L

2(sq-l) (i.e. F(t) e-tAf for some f E L2(Sq-l» then

'if1F

Ff!f.

Next we introduce some operators in L (sq-l) which will be needed later on.

2

Definition 2.11. We define the operators

P

k and

~,

1

~

k

~

q, in L2(Sq-l) by

Lemma 2.12. Let f E ~ and let k E 1N with 1 ~ k $ q. m

The following relations hold true:

~~f

=

(2m

+

q - 2)Dq (0 f)

J\. m-l -X

Proof. Fix ~ E sq-l.

Applying corollary 2.5 and lemma 1.10 we get

[~fJ (t;) o q-l

f

q-l S q ~,.P (n.t;)f(n)do len) J\. m q-l:" [(m +q - 2) • q "k - (m + 1) P m+l (n. t;) (m +q m · q -3) P_m-1 (n.1;;) J(2m +q fen) -2) dO_{q _1} (n) ~(m +q -2) pq ( •• ~) _ m dY k (m + 1) m+l (m +q p q ( •• t;) ] (11) f ( n) do 1 (n) . 3) m-l 2m +q -2

To this end we remark that

[ a

(JIlf) ] (nl

q,v

=

m (EfIf) (n) = mf (n)

and also that

[-f-(Hqf) ](n)

₌

[Pkf] (n)

.

Y_k So we get cr q-l

f

nkf(n)

a

[m+q-2 pq (e.l;) _ _ m_ pq (-.l;)](n) dcr len) + q,v m + 1 m+l m+q-3 m-l (2m+q-2) q-q-l S dq +_m_

J

[(m+ q -2) Pmq+1(n.l;) - m pq (n.l;)]' cr 1 (m+l) (m+q-3) m-l q- q-l S Now we have dq m

J

[(m+q -2) oq Ll

=G

m :; 1 P_m+1 (n.l;) q q-l n kf (n) m eq (m+q-3) Pm-l (n.l;)](n.~) (2m+q-2) dcr_{q _1} (n) • S

Applying Lemma 1.10 twice yields

dq

=~

J

q-l q-l S whence :iI: ... em + q -

n,

(2m +q -2) (m +1) dq m

f

pq (T'!. t;;) [ ... ('''f(n) + (2m.+q) 11kf (nl Jdcr q_1 (11) + cr m+1 k . q-l sq-l

(2.6) dq m m (2m+q-2)(m+q-3) (1q_l

J

P~-l (n.~)

[-(Pkf) (n) +2nk f (n) ]dO"q_l (n). sq-l S' P f E:

V

q and ~ + q - 2 dq l.nce k m-l (2Q, + q - 2) (9, + 1 ) R,

=

1 d q R,

~

0 we get (2Q. +q) H1 ' + ___ 1 _ _ _ Pkf) --(-2m--.-.:;;=-q----:-

D~_l (~f)

Application of the projection D~_1 on both sides of the latter equality yields

If we substitute this in (2.6) we arrive at the other wanted relation.

0

In the final part of this section we discuss another Hilbert space which con-tains all ~ as mutually orthogonal subspaces. Let

n

with natural inner product

( 21T)

f

exp (-II x Ir /2) f (x) 9 dx, IRq

and with corresponding norm " • "

M2 (IRq)

2 13 Let ~ denote the algebraic direct sum of the Hq,s, i.e •

.;;;....;c.~~..;;..;;;;...;...;..;;.-"'-:....;;;..;~- R, n

each f E ~ is of the form f

=

I

n=O

f with R, ~ 0, f E: Hq, 1 ~ n ~ £.

n n n

Observe that Hq is the space of all harmonic polynomials in q variables. Let

J..fI

denote the closure of Hq in M2 (IRq) •

Our first aim is a characterization of the elements of ~. Lemma 2.14. Let 9 E

V

q and h E

V

q•

n m

Then we have the following equality

2 (n+m) /2.

r

(n +~ +q)

2.1Tq/2

9 (~) h (~) dO" 1 (f,;)

q-Pro.cE., We. have, 1 .

J

exp(-lIxlfI2) (Hqg) (x)

UfIh)

(x)dx (211') q/2 IRq OC> = I'

J

( r 2 ) n+m+q-l q/2 exp -

:2

r (2'11') .

o

I

g(~)h(~)dcrq_l (~)dr

sq-l 2 (n+m)/2 q/2 2.'11' <X>

I

exp -s s

()

(n+m+q-2) /2

f

o

_sq-1 2(n+m)/2.

_r

(n +~ +q) = ---~--~~-2.n-g/2

J

g(~)h(~)dcr 1(~) q-q-l S Corollary 2.15. Let n ~ m. <X> Then ~ ~ Hq in ~. Moreover ~fI = ~ Hq•

n m n=O n

Proof.

V

q 1

V

q in L

2(SQ-l).

- - n m

The second statement is trivial.

Corollary 2.16. Let n ~ O. Let {eq .

I

1 ~ j ~ dq} be an orthonormal basis

n,J n in

V

q. Then n is an orthonormal basis in

H

q• n Co.rollary 2. 17. Let n ~ 1.

The subspace

H

q has a reproducing kernel Kq with

n n

o

x .".. 0 .". y

x ""

0 or·y = 0

o

o

(2.7)

Proof. Let {eq .

I

1 ~ j ~ dq} be an orthonormal basis of ~. Fix x,y E IRq

n,] n n

with x

f:

0

f:

y.

Because of corollary 2.16 and property 1.8

In case y

=

0 or x q/2 dq 2.n

IIx If

I!

If _n_

pq(~ 2nr(n+!l) y O'q_l nUx!! 2 (Lemma 2.3)

o

we have Kq(x,y) n

o.

Of course the kernel Kci of

Hci

satisfies

y

• lIy I~ ;:

Because of property (1.5) we know that for each f E

Hq

n

X E

Lemma 2 1 . Let n ~ 0 and let x E IRq. Then

1 we have

(2.8) Because q ~ 2 we a. + 9: -2a. + q So we get Kq(x,x) n Further we have q K1 (x,x) and q KO (x,x) =

rei)

(2n +q -2)

r(n

+q -2) 2nr(n +

!)

r(n + t)

r(kI -

1) (2n + q - 2) (n - 2 + q - 1) (2n - 2 + q) (2 (n - 2) + q)

have for each (l ~ 0

1 _:5

_s....::..1:.

_. q :5

(~)n-1

IIx

If

n q n! :5 II x

It

(because dq 1 q) 1 :5 _{( q}

.9....:....l

₎ -1

Corollary 2.19. Let f € ~ and

x

€ IRq. Then

n n-1

I

f (x)

I

:<;; ( q -q 1) -2- II x Ifll f II

rnr

M 2(IR q )

Proof. From inequality (2.7).

(q - 1) II x Ifn q r(n + 1) •

Let

Qq

denote the orthogonal projection from ~ onto ~.

n n

Theorem 2.20. Let f € ~.

Then f is a harmonic function on IRq which satisfies

I

f (x)

I

Proof. Pix x E IRq. For all g €

ftt·

we have

n-1

o

o

I

I

(Corollary 2. 19) ()O

s (

I

n=O / q

exp(~

q - 1 q

So the linear functional t : ~ ~ € defined by x t (g) = x is continuous. 00

I

[Q~g]

(x) I n=O

For all 9 E Hq we have t (g) = g(x). x

m

For m € IN put f

=

I

[Qqf].

m n=O n

Then f + f, m + "" in norm and f + ~ t (f), m + "", uniformly on compacta in

m m x x

IRq. Since the f 's are harmonic, the function f is harmonic. It is clear

m

that f satisfies the inequality (2.8).

0

Next we prove the converse of theorem 2.20. We need an auxiliary result.

Lemma 2.21. Let r > 1. Let f E Harm(Bq(O,r» and 9 E Harm(Bq). For each P,

1

s

p < r

Proof. We have

=

f(~)g(i;)da 1 (~) =

q-f(pE;;> [e-(log p)Ag](t:;)da 1(~) =

q-[e-(log

p)Af(p.)J(t:;)g(~)da

1(t:;)

Proof. From theorem 2.20 it follows that

if!

c M2 (mq) n Harm(mq). Let

q q

.a

q-l

f E M2 (m ) n Harm(m ) and assume that f .l M-. For all 11 E S a n d n ~ 0

We have 1 =

---=-...,.-(21T)q/2

J

exp(-lIx

If/2)f(X)K~(X,n)dX

=

m

q

o

From Lemma 2.21 it follows that

So for all 11 € sq-l and n ~ 0

a

[ "" 2 r n+q-l exp(-

2)

r n r a q-l

f

q-l S So fl q-1 .l L

2(Sq-l). Hence for each n E sq-l fen) = O. Since f is harmonic,

S

we conclude f

=

O.

$.E;tna~ks.

- From Theorem 2.20 it follows that for all x E IRq the linear !unctionals f ~ f(x)t f E ~ are continuous. So the space ~ has a reprOducirtgkernel

00 e n=O

H

q it follows from n~.~n<.~+·" that n

00

L

n=O

We have for f E

rfI

n

(lemma 2.14) •

So i t follows that Bq is a compact injective operator from

MJ.

into L

2(Sq-l). Next we define a unitary representation of O(IRq) in

MJ..

For all f E ~~ and R E O(IRq) the function f(RT.) (

~f1.

The following definition makes sense:

Definition 2.23. The representation rrq: O(IRq) + L(~~) is defined by

It is easy to see that

n

q is a unitary representation of O(IRq) in ~. Each finite dimensional subspace

rfI

is invariant under the operators rrq(R) ,

n R E O(IRq).

The representation R + rrq(R)

IrfI'

R E O(IRq),iS irreducible, because rrq(R) :=::

n

=

HqL

Bq and because the representation R +

L

I '

R E O(IRq) is irreducible.

R 00 R~

r1oreover

MJ.

=

$

rfI.

See theorem 1.6 (Gurevic). n

n=O n

Finally we give another description of the inner product (-,.)

restrict-ed to Hq. M2 (IRq)

Definition 2.24. Let p be a polynomial in q variables.

a

Let p(3) denote the differential operator p(--- , .•• ,

_aX

1

Lemma 2.25. Let f,g E

rfI.

Then (f,g) :=:: [f(3)9J (0) M 2(IR q ) d

ax) .

q

prodf. If f € ~ and 9 € ~ with n# m, then (f,g) q = 0 and

n

m

[f(a)g](O)

=

O. M_{2 (m )}

So we restrict ourselves to f,g €

Because of Lemma 2.14 we have

/fl.

n =

---=-0' q-1 f(t;)g(E;)dO' 1 (t;)

q-The homogeneous polynomial f is a linear combination of monomials of the form n II xk . with 1 S k.~ q. j=1 J _q-l J Lete - 1 on S • Put 2nr (n + 9.) 1

f

n a

=

II _{t;k. g} dO' 1 (t;)

r

(9.) a _q-l q-2 q-1 J S Then we have 2n

r

(n + ~) 1 n a

=

[ IT _{~ e,g]q} f(9.) tJ _j=1 2 q-l j

Because ~ is self-adjoint and because of Lemma 2.12 n

.

2nr(n + 9.) n-1 a = _ _ _ --=;:..2_ a [IT

~

e,

D~+l (~

g) + Dq 1

(~

g) ] r(9.) q-l j=1 j n n- n q 2 n-l

Remembering that II ~.e ~ V~+1 and applying Lemma 2.12 again we ~et j=1 J n-l

f-

[IT

~.e,

q-l 1 J 2n-1r(n - 1 + 9.) n-l _ _ _ _ _ _ 2 _ _ 1_ [ IT °q_l j=1

Since

P

_k 9 € H~_l we can repeat this argument n-l times and finally we get n

1 n (l

=

-=---r

e , a q-l IT

P

k g] j=l j q 1 = -a q-l

f

q-l S n ( IT

~

g)

(~)da

1

(~)

aX.. q-j=l k J n [IT

-f-

g]

(0) j=l X_k j

Since f is a linear combination of monomials we get (f,g) q

=

[f(Cl)g](O) .

M2 (IR )

3. A space of functions on a real separable infinite dimensional Hilbert space In the first part of this section we construct a complex Hilbert space of functions of infinitely many real variables with the aid of the Hilbert spaces of harmonic functions, introduced in the previous section.

In the second part of this section a unitary representation of the full or-thogonal group on IRt2 in this Hilbert space will be introduced.

We use the following conventions and notations:

(3.1) IRt2 denotes the real Hilbert space of square summable sequences with inner product and norm denoted by ( ••• ) and II • II respectively.

For a,b,c,d E IRt2 we define

(a + ib. c + id) = (a.c) + (b.d) + i [ (b.c) - (a.d)] .

(3.2) (ej)j denotes the standard orthonormal basis of IRt₂; i.e. e_j

(3.3) IRtc denotes the subspace of IRt2 consisting of all finite sequences x i.e, only a finite number of entries of x are non zero.

(3.4) The space IRq is embedded in IRt2 identifying (x1' ... ,x

q) and _j=1

~

x,e. _{J J} E IRt2' Thus we get for example

IRq = IRt and so on . c

(3.5) O(IRt

2) denotes the group of all orthogonal operators (i.e. bijective isome-tries) from IRt2 onto IRt

2. (3.6)

o (

IR t ) == {V E 0 ( IR t

2)

I

V ( IR t )

=

IR t } •

c c c

(3.7) Eq denotes the orthogonal prOjection from IRt2 onto IRq. (3.8) The group O(IRq) is embedded in O(IRt

2) identifying R E O(IR q

) and the ortho-gonal operator RE + (I - E ) E O(IRt

2).

00 q q

Remark: u O(IRq) is a proper subset of O(IRt ). c q=l

Example: Define U E O{IRt 2) by

00

Then U

t

u q=l

n E IN •

O(IRq) and U E O(IRt ).

(3.9)

(3.10)

(3. 11)

(3.12)

We start with the definition of an inner product space in which all spaces ~ of the previous section can be embedded.

Definition 3.1. The linear space F(IRt₂) consists of all functions f: IRt2 ~ ~ with the following properties:

(a) f is continuous on IR!2' (b)

(c)

3 : f = foE

p>O p

I

e -II x If

121

f (x) 12dx < CD for p IS IN such IRP

In F(IR!2) an inner product is defined by

(f,g) F(IRt )

2

that f

The corresponding norm will be denoted by II. I'F (IR t ).

2

(Remark: CD

f

e-u2/2dU =

(2n)~)

.

_00

Definition 3.2. Let q € IN.

The mapping embq: ~ E F(IR!2) is defined by embqf = foE

q

foE • P

If no confusion arises, we identify f € ~ and embqf € F(IRt

2) and we denote embqf by f.

So ~ will be considered as a proper subspace of F(IRt

2). With these conven-tions we have the following relaconven-tions:

(f,g) F(IR! ) 2 (f,g) q M2 (IR ) and

tfI

c Hq+P c F(IRR. 2) n n for q,p E IN for q,p € IN II f I'FCIR! ) = 2 f,g € ~

Lemma 3.3. Let m,n,p,q E IN with m

F

n. Then HP 1. Hq in F {IR 9..

2} •

n m

Proof. Let r max{p,q}.

By (3.10) HP c Hr and ~ c Hr ,

n n m m

Since Hr 1. Hr in ~f, by (3.9) we conclude HP .1 ~ in F(IR9..

2).

n m n m

In the sequel the following subspaces of F(IR9..

2) will be of importance. Let

H

denote the subspace

n

Let

H

denote the subspace

~ ~.

q=l 00 00 u q=l

o

We recall that ~ Span ( u

n=O ~) t

n i.e. the space of all harmonic polynomials in q variables.

Observe that

H

is the direct sum of the

H

'5, so

H

is the space of all har-n

monic polynomials of an arbitrary number of variables. For the spaces of de-finition 3.4 we construct a Hilbert completion with respect to II. If(IRt )'

2

To this end some auxiliary results are needed,

Lemma 3 ,Let f E

H

and let x E IRt2' Then

I

f (x)

I

~ _{exp (-2-)}

IIx

Ir

_II

_{f If (IR9. )}

2

Proof, Since f E H I there exists apE IN such that f € ~ c ~f! for all q ? p,

From theorem 2.20 it follows

I

f (x)

I

So

I

f (x)

I

If

(E x)

I

p II

xli

2 exp (-2-)11 fll F(IRt ) • 2

o

(a) Then (fk(x»k is a convergent sequence in ~ for all x € IR~2.

(b) Moreover the function f: IR~2 ~ ~ defined by

f (x)

=

lim fk (x) ,

k-"

satisfies the following conditions:

( -I) _... If( ) I _{x ~ exp} (II _{- 2 -}x If)l' _~m Ilf IL _{k 't-(IR ~ ) ,}

and k+<x> 2

(ii) f is continuous on IR~2.

Proof. (a): Fix x E IR~2.

By lemma 3.5

So (fk(x»k is a convergent sequence. (b) (i): Observe that

If(x) I = lim Ifk(x) I ~

k-..

IIx If ,

:s; exp (-2-) hm

II

fk IhIR 11.) •

k+<x> 2

(ii): Fix x € IR11.2 and e: > O.

Let k > 0 such that for all R, ~ k

2 II f _{k -} f IL R, 'f (IR R, ) :s; exp -( (II x

II

+ 2 1) ) ~ b •

2

Let 1 > 0 > 0 be such that Ifk(x) - fk(y) I <:

f

for all y ( IRt2 with IIx-yll<:o.

Then for all t <:: k and all y wi th II x - y

II

<: 0

:::;

~

ex

(II

x If _

01

x

II

+ 1) 2) + .£ +

~

(hl_

<II x II + 1) 2, <:

~

So for all y €

mt

2 with II x - y II < 0

If(x) - fey)

I

< e: •

o

Defini tion 3.7. Let M denote the Hilbert completion with respect to II • II F (:IR R,2)

of

H

which consists of all functions on mt

2 which are pointwise limits of Cauchy sequences in

H.

The corresponding inner product and norm in

M

will again be denoted by ( • ) fA and II • 1

_M,

Let

M

denote the closure of

H

in

M.

n n

Each element of M is the pointwise limit of a sequence of harmonic functions each of an arbitrary but finite number of variables.

Since

H

L

H ,

n ~ m, and

H =

n m u

nEIN

H

n it follows that M n L M , m n ~ m and that

M =: q) M. n=O n

The space ~~ is a closed subspace of

M

because of (3.9). Since

M

consists

n

of homogeneous functions, we have M n ~

=

~.

n n

(3.13) Let pq,

P

and pq denote the orthogonal projections from

M

onto ~,

M

and

n n n n

W

respectively.

Lemma 3.8. Let n € IN.

(a) pq +

P ,

q + 00, strongly .

n n

(b) pq + I, q + 00, strongly .

Proof. (a): Take f E

M.

Let e: > O. We choose g €

H

such that IIf - gil <

t

Take q E IN with g E Hq. Then for p ~ q

IlPPf - P f II ~ IIPP(f -g)II

M + IIPPg - P gllM + lip {g - f)IIM

n n

M

n n n n

<

Lemma 3.9. Let x € IR1

2. (a) For all f € M

I

IIx If

If(x) ~ exp (-2-l

II

filM (b) For all f € M

n

Proof. (al: This is a direct consequence of theorem 3.6(b) (i).

(b): Fix f €

M .

n

Then (pqf) is a Cauchy sequence with limit f. From corollary 2.19 it follows

n q I f (x)

I

=

lim I [pqf] (x) I == lim

I

[Pqf] (E xl

I

$ n n q q--- q -liE xlln

s...:.J..

n -1 q II pllfll <_ ~ lim sup ( q )

M

q-+«>

Irl!

n

Because of the previous lemma

M

f n ~ 0, and M have reproducing kernels. n

o

The computation of these reproducing kernels requires the following results.

Lemma 3.10. Let f €

M

and let y € IRt

2. Then

Proof. Follows from property (1.6).

Lemma 3.11. Let y €

nu

2• 2n

Then

(K~(YIY»q

is a convergent sequence in IR with limit

II~I~

Proof. Let n ~ 1. From the proof of lemma 2.18 it follows

liE

ylln (2n +q -2) (n -2 +q -1) ... (q -1) ---.,;q=..,.-_ (2n -2 +q) (2(n -2) +q) ••• (q) n! _ II ,,112n

So lim Kq(y,y) _{n -} ~. _n! The case n = 0 is trivial. q

-o

Defini Hon 3. 12. The functions Kn' n :2: 0 and K from

:m 1/,2

x

:m 1/,2 into mare

defined by

n

K ( _{n x,y -}) _ (x.y) _n!

K(x,y) = exp(x.y)

Theorem 3.13. Let y Em1/,2 and let n :2: O. Then we have (a) (b) (c) (d) n Kn ( 0 , y) E M and

II

K (., y)

II

M =

!Lx!L .

n n

.frlT

For each f E M n

II ¥II

2 K ( • , y) E M and II K ( 0 , y) II M = exp (-2-) • For each f E M fey) = (f,K(o,y»M •

Proof (a): For q,p E IN we have

=

Kq+P(y y)

n '

From Lemma 3.11 it follows that (Kq(o y» is a Cauchy sequence. Because of n ' q

Lemma 3.9 lim Kq(o,y) exists pointwise and is an element of M •

q~ n n

Let x E m2

2, Since

So with Lemma 3.11 and Lemma 1.11

lim

K~(X'Y) = ~Ixllnllylln (II~II

q-l'CO

y n

1fYII)

n

11

K ( _{n ., y}

)11

_M=

lim IIKq_{nO, Y} {

)11 -

_{M - - -}

lIyll

q-l'CO

Irl!

(b): Fix f EM. Then

n fey)

₌

lim q~ lim q~ [pIf] (y) = n (f , K~ ( 0 , y) )

M

(f,Kn(o,y»M (c); From (a) it follows that

00

exp(O.y) EM.

2 So K(-,y) E

M

and IIK(·,y)II

M

=

exp(II~1I

). E Y q liE yll) q 1 n = - , (x,y) n.

(d) Since point evaluation in M is continuous and Af

=

K is the reproducing kernel of M.

$

n=O

M

it follows that

n

o

Due to the fact that the elements of

M

can be approximated by harmonic func-tions in finitely many variables, certain properties of harmonic funcfunc-tions carryover to the functions of M. The following two theorems are results of this type. The first theorem is a generalization of the classical Mean-value theorem for harmonic functions.

Theorem 3.14. Let f E M, let x E _IR12 and let r > O. Then

f (x) lim 1

f

f (.;) do 1 (.;>

=

.

q~ C1 r q- q-q-1 q-l S (E x,r) q

(3.14) Proof. Put

a.

q 1

f

q-l S (E x,r) q f{~)do 1(~) •

q-The sequence (Pqf)q is a Cauchy sequence in M with limit f. Since [Pqf] (x)

=

[pqf](E xl and since pqf is harmonic we have

q

Ia.

-[pqf](x)

I

= 1

I

(f (~) - [Pqf](t;) )do 1 (~)

I

q q-l q-o _q-1 r _q-l S (E x,r) q

J

2

s 1 _{exp(tilL)lif - pqfll,.fOq_l u;;)}

q-l 2 o _q-1 r _q-1 S (E x,r) q So lim

a.

- [pqf] (x)

_o

or q-+«> q lim Ci lim [Pqf] (x) f (x)

.

q-+<x> q q-+«>

The second theorem is a weak version of the min-max principle of harmonic functions.

Theorem 3.15. Let D be a bounded open subset of IR~2 with boundary

r

such that D is weakly closed in IRi

2. As for harmonic functions, we have V D

I

f (x)

I

s

XE sup

ncr

If

(n)

I .

Proof. Let r > 0 be such that D c {x ( IR~2

I

II xii < r}. Then we have

2

V - If(x) I ~ eX_{P (2)lIfIl}

M

XED

Since

r

=

D\D, d

=

sup If(n) I < 00.

nEr

Now suppose the assertion (3.14) were not true.

Then there exists y E D such that if(y)

I

> d. Let p > 0 be such that

:S;

:'

If(y) I = d + P. Since D is open there exists z E E D for sufficiently large q

q such that

(a) _{If(z) - fey) I <}

t

_p

(b)

From (b) i t follows

1 < - p

3 for all XED .

We obtain

IcFlf](z) I = If(y) - fey) + fez) - fez) + CPqf](z)

I

~

?: If(y) I - If(y) - fez) I - If(z) - [Pqf](z) I >

1 1

> d + p -

3

p -

3

p =

Further we have

sup ICpqf](n)

1'5

sup <If(n)

1+

ICpqf](n) - fen)

1<

nEr nEr

1 < d + } p . So we are in the following position:

There exists a function, viz pqf, on IRq which is harmonic and satisfies ICPqf](z) I > d +

t

p and sup ICpqf](n) I < d +

t

p •

nEr Since D is open in IRt

2, observe that EqD is open in IRq. We have z E E D.

q

We are ready if we can prove that the boundary of E D is contained in E

r.

q q

Now Eq is a compact operator and D is weakly compact in IRt

2, whence Eq(D) is compact in IRq

Therefore E (0) is closed in IRq and E (D) c E (0). Since

q q q

~\E (D) C {E (O}}\E (D) C E (D\D)

q q q q q

the boundary

r

of E (D) is a subset of E

(r).

q q q

E

(n

(3.15) This is contradictory to

I

[Pqf](z)

I

> d +

~

p > sup nEE (f) q

I

[Pqf] (n)

I

2: sup

I

[Pqf] (n)

I .

nEfq

o

Next we present two orthonormal bases in

M.

One of them arises from an expli-cit construction, the other is given expliexpli-citly. This construction is by in-duction and is based on a generating principle due to Muller. This principle is formulated in the following lemma.

Lemma 3.16. Let f denote the restriction to sq-l of a harmonic homogeneous polynomial p of degree m in q variables.

Let n 2: m. Let F: sq + ~ be defined by F(/1 -

t2~

+ te 1)

=

Aq+l(t)f(~)

q+ n,m

q-l

~ E S , t E [-l,lJ •

(For the definition of Aq+1 see (1.20).) n,m

Then F is equal to a restriction to sq of a unique harmonic homogeneous polynomial of degree n in q+l variables.

Proof. See [Mu], definition 4 and lemma 15.

o

Before we proceed we mention that the numbers dq

n dim

(H

q

) satisfy the

rela-n tion n

L

m=O as a consequence of corollary 2.7. Definition 3.17. Let e_O,l

=

1.

Let {e 1,e 2} be an orthonormal basis for H2, n 2: 1.

n, n, n

Let n,i E IN with i > 2.

There exist unique q E IN such that

Because of the relation (3.15) there exist unique m,j E IN u {oJ with

o

$ m < nand 1 $ j $ dq such that m

(3. 16)

n

i

=

L

k=m+l

The function e 0: IRt2 + ~ is defined by

n,~ E _q+ lx ~

a

e 0 (x) n,~ te 1 +

11 -

t2~,

t E [-1,1],

~

E sq-1 q+

a

So from {e_O,l} U {en,i

I

n E IN, {eO 1} U {e 0 n E IN, 1 ~ i ~

, n,~

E _q+ tX

1 ~ i ~ dq} we obtain the set n

dq+1 } and so on.

n

O.

{ }

H

2.

Theorem 3.18. For each n E IN let e 1,e 2 be an orthonormal basis for n

, n, n,

Let e 0' n,i E IN, i ~ 3 be constructed as in definition 3.17.

n,~

Then we have

(a) For all n,q E IN with normal basis of ~. n q ~ 2 the set {e 0 n,~ I 1 ~ i ~ d q } is an ortho-n

(b) For all n E IN the set {e 0 i E IN} is an orthonormal basis of

n,~

M .

_n

(c) The set {eo, 1 } U {e

n, i n,i E IN} is an orthonormal basis of

M.

Proof. (a): The proof is by induction. First we note that

mal basis of

H2.

for all n E IN the set {e 0

I

1

~

i

~

d

2

} is an

orthonor-n,~ n

n

Next we assume that for all n E IN the set {e 0

I

1

n,~ is an

orthonor-mal basis of ~.

n

Starting from this assumption we show that {e 0

I

1

~

i

~

d

q

+1} is an

ortho-n,~ n

normal basis of Hq+1 for all n E IN.

n

Fix n E IN. Let il E IN with

d~

< il

~ d~+l.

As we have seen there exist unique m

1,jl such that e 0 is given by formula

n'~l

(3.16) in which we put i i

1, m = mt and j = jl.

Because of lemma 3.16 e 0 E Hq+l.

Because of lemma 1.12(b) for all x E IRq+1\{O}

IITt(n +

~2)

L 1

""2 n q+ x

( + 1 ) II xII A (-II -II .e_q+₁₎

r(n + _q ____ ) n,m x

2

So also the e . with 1 ~ i ~ dq are given by formula (3.16) in which we put

nt~ n

m == nand j == i.

k . . l ' i dq+1

Ta e ~1,12 E IN with ~ 1

1, 2 ~ n .

For j = 1,2 take the unique mn,jn E IN U {a} such that e . is given by

.. .. . . n,l£,

formula (3.16) in which we put i == i£" m == m£, and J == J

t . We note that 1 ~ jt ~ dq , t 1,2. mR. Put a, (e . , e . )

M"

n'~l n'~2 Lemma 2.14 says + 1

f

e i (ll)e . (n)dO" (n) n , 1 n'~2 q

Take variables t E [-l,lJ and

~

E sq-l such that n

and dO" (n) '" (1 - t2) 2 dO" 1

(~)dt

q q-2n

--,----,.--:"- r

(n + 2n (q+1) /2 -1 1

f

_q-l

f

S

Because of the induction assumption we get

Finally, Lemma 1.12 and Lemma 2.14 yield

a

=

{ I

dq+1} ,~+1

Thus we have proved e . 1 ~ i s i s an orthonormal basis of

n-n,~ n n for arbitrary n. (b): Fix n € IN. Because u q~2 sis of M • n

tf.

is dense in M , the set {e . i E IN} is an orthonormal

ba-n n n,~

(c): Since M (i

n=O

M this statement follows trivially from (b).

n

The second basis consists of homogeneous polynomials which are not harmonic in general.

Definition 3.19. Let MI denote the set of multi indices

00 Let Is I

=

_I

j=O Sj' s!

=

s 1 ! s2 ! •.. and

a

s,t IT <5 1 s.,t. j= J J Put MI == {s E MI

n

I

lsi

=

n} for n ~ O. For each s € MI we define q>: s IRR,2 ~ It by lP (x)

=

s II j=l s. J (x.e . ) J

Put ~ = {q>

I

s € MI} and ~

=

{q>

I

s € MI }.

s n s n

Theorem 3.20. Let n E IN. Then we have (a) Vs EMI : q>s c Mn·

n

(b) The set ~ is an orthonormal basis of M •

n n

(c) The set 4> is an orthonormal basis of

M.

q

Proof. (a): We first construct auxiliary harmonic functions gs,k with s € MI and k,q E lN large enough.

n Fix k E IN.

For 1 s j s k, q ~ k we define the operators D~,k E L(IRR,2) by

f

m 0 q o

::+1

1 s j - 1 D. k e . mk m ~ J, J+ j S m ~ q - 1

o

(3.17) (3.18) (3.19) (3.20) (3.21) The q Dj,keh operators _{0, k} q J, q 0, k maps J, Dq _{, k '} e

=

), J 0, have Span k 'i' q 2 L

II

D4 kxll = j=l J, E Dq+P

=

Dq q j,k j,k h ,. j + Ink, 0 :5m:5q- 1

.

the following properties

{e, Ink

I

o

:5 m :5 q - 1} isometrically onto IRq

.

J+

II

xii 2 for all x E IRkq .

k

For all SEMI and k,q E IN with

n

g~,k:

IRJl.₂ -+ (£ by

I

SJ' - n and q ~ k we define the function j=l k gq k(x) = s, q q IT K (0, kx,e,) 1 S j ) ' J

Because of (3.17) the j-th factor is a harmonic homogeneous polynomial of degree s, in the variables x x x So gq is a harmonic

ho-J j ' j+k'···' j+(q-l)k· s,k

mogeneous polynomial of degree n in the variables x

1' ••• ,xqk.

k k

Claim: For S,t E MI , k E IN with

I

s,

I

t,

=

n, p ~ 1 and q ~ k we

n j"'l ) j=1 J have Proof: Put (l Then we have (s!t!)

~

( l

=

--~--~--~ (21T) (q+p) k/2 Because of (3.20) q q+p K t _{j ) '} (0, kx,e,) _J

=

k q II K (e"e,)6 t 1 tj J ) Sj' j

II

xii 2

f

IR (q+p) /k 2 k q+p q+p q q II K (0, kx,e,)Kt (0, kx,e,)dx. , 1 s, J, J , ) , J J = ) J e

Application of the transformation of variables x -+ (Y1'.'.'Yk) with

y,

=

Dq+Px yields because of (3.19)

k

- ( I

j=l 2 lIy.1I )

/2

J k e 2 lIy.1I _ _ J _ q+p q

IT K {y.,e.)K (y"e.)dY1, .. dYk

1 s, J J t. J J j= J J k = (s!t!)

~

IT j=l [ 1 ~+p

I

e 2 q+p q K (y.,e.)K t (y.,e.)dy.J s, J J . J J J s! IRq+p k q IT K (e.,e.)o t . 1 s. J J s " . J= J ] J

This proves the claim.

k

Let <P € ¢> • Let k E IN with I s , = n.

s n J

j=l

We will prove that gq ~ <P , q ~ ~, in s,k s

From claim (3.21) i t follows that for q

J J

M •

n ~ k, P ~

a

== s! k IT j=l q+p K (e"e.) Sj J J - s! k IT j=l q K (e.,e,) Sj ) J

Because of Lemma 3.11 the sequence k

IT

j=l (gq k is a Cauchy sequence in

M •

Let

s, q n

strongly to Dj,k € L(IR!2) defined by

{ e, D, ke. mk = e J J, J+ m e m+₁ m

=

0 1 ~ m < j m ~ j

Kq (e"e,» is convergent. Hence s. ) J q

J q

1 ~ j ~ k. The operator Dj,k tends

=

a

h ~ j + mk, m ~

a .