Generalized functions and operators on the unit sphere

Generalized functions and operators on the unit sphere

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Graaf, de, J. (1984). Generalized functions and operators on the unit sphere. (Eindhoven University of Technology : Dept of Mathematics : memorandum; Vol. 8406). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1984

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EINDHOVEN UNIVERSITY OF TECHNOLOGY

Department of Mathematics and Com~uting Science

Memorandum 1984-06 January 1984

GENERALIZED FUNCTIONS AND OPERATORS ON THE UNIT SPHERE

by

J. de Graaf

Eindhoven University of Technology

Department of Mathematics and Com~uting Science

PO Box 513t Eindhoven the Netherlands

GENERALIZED FUNCTIONS AND OPERATORS ON THE UNIT SPHERE

by

*) J. de Graaf

Sl1~~ary Tw f 1" d f t" t h " t sphere "q-1 C lRq

__

~~=U~~. 0 spaces 0 genera Lze unc Lons on e unL "

are introduced. Both types of generalized functions can be identified with suitable classes of harmonic functions. Several natural classes of conti-nuous and conticonti-nuously extendible operators are discussed: Multipliers, differentiations, harmonic contractions/expansions and harmonic shifts. The latter two classes of operators are "parametrized" by the full affine

n

semi -group on lR .

AMS Classifications: 46F05, 46F10, 31B05, 20G05.

*)

Department of Mathematics and Computing Science, Eindhoven University of Technology.

1. INTRODUCTION AND NOTATIONS

In this paper I describe two natural theories of generalized functions on the unit sphere Qq-1 in ]Rq and some natural classes of linear operators acting on those generalized functions. The test functions in both theories are restrictions to Qq-1 of suitable classes of harmonic functions on open sets in ]Rq. Also the generalized functions in both theories are explicitly represented by classes of harmonic functions in ]Rq. The generalized func-tions appear to be "boundary values" of harmonic funcfunc-tions.

Both theories are much stronger (i.e. they contain more distributions> than Schwartz' theory.

The theory in this paper resembles hyper-function theory. There are fundamen-tal differences however: Harmonic functions in n variables satisfy one POE, whereas analytic functions in n variables satisfy the overdetermined system of Cauchy-Riemann equations. Further, the product of two harmonic functions is usually not harmonic, while the product of two analytic functions is al-ways analytic, etc.

The theories we introduce here are very special concrete cases of the general functional analytic constructions in [G

1], [G2], [G3], [E].

The classes of operators that we introduce are based on simple geometric considerations and on the properties of harmonic functions as derived in Section 2. For example a continuous linear operator is associated with each element of the full affine semi-group on ]Rq. In the Hilbert space L (Qq-l>

these operators are (strongly) unbounded in general. The precise "represen-tation properties" of these operators are not yet clear to me!

In the sequel I will stick to the following notations and conventions. For theory and proofs I refer to [SJ, [MJ.

q-l Q (ai R) x

=

r t;, ~

=

R n Bq iR) dw q w q

f

Harm(W)

sphere with centre a and radius R in IRq. , the unit sphere.

vectors in IRq.

, open ball with centre a and radius R in IRq. , the open unit ball.

, the usual Lebesgue measure in IRq.

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

n

.

, integrations take place over

n

q- 1 if not indicated otherwise.

harmonic functions i.e. functions ~ which

~

satisfy 2 +

dX 1

on W.

, the vector space of harmonic functions on the open

U

Harm(Bq(air», the vector space of functions which r>R

, the vector space of all harmonic functions on IRq. HarmW) U Harm(Bq{O;r», the vector space of functions which are

de-r>O

fined and harmonic on an open neighbourhood of ~. This neighbourhood may depend on the function. ("Harmonic germs" .)

HHP(q;n) _{, the vector space of harmonic homogeneous polynomials} of degree n in q variables.

N (q,n) = dim HHP (qin), we have, see [M], N (q,n) s K n q-2 , K is a constant.

q q

s

(c;), S f{

n - n,

n·lI, (.,.)

spherical harmoniCS, i.e. restrictions of elements in q-l

HHP(qin) to

n

.

, the complex Hilbert space of square integrable functions q-l

n

.

q-l norm and inner product on L

2{n ) (f,g) (f(o) ,g(o» =

f

f(P

g(~)

dw

q II f II 2 (f ,f) .

The restriction of an arbitrary element in HHP(qin) to

n

q-1 is orthogonal to the restriction of an arbitrary element in HHP(q;m) to

n

q-1 if m

t

n. The mentioned restrictions of HHP(qin), n = 0,1,2, •.• , establish a complete

q-1 q-l

set in L2 W }. We do not introduce a special orthonormal basis in L2

en

) .

The restriction to

n

q-1 of any polynomial of degree m in q variables is a finite linear combination of restrictions of elements in HHP(qin) with

P

_n denotes the orthogonal projection of wri te ( P f) (t;)

=

s

(

t;) •

n - n, f

-From [M] we quote the estimate

with

lis II

~r

Is

(OI2dW}~

,

n lJ n - q

for any S € HHP(qin) .

n

2. SOME LEMMAS ON HARMONIC FUNCTIONS

q-1

(n ) onto HHP(q,n). Often we

q-1 Let f € L

2(n ). Decompose f in spherical harmonics f(E;)

( 1.1)

In the first lemma we give conditions on f such that i t can be extended to an harmonic function on Bq(OiR) for some R > 1. The extension is again de-noted by f.

LEMMA 2.1. (i) f € L

2(n q

- 1} can be extended to an element in Harm(Bq(O;R), R> 1, iff

00

I

< 00 for all r , 0 S r < R •

n=O

co

(ii) If f E Harm(Bq(OiR» then the sequence

L

rn Sn,f<P converges

uni-n=O formly to f on each ball Bq(QiR

Proof·

(i) ~) With (1.1) and r ~ Rl < R2 < R we estimate

(i)

~

(Rl)2n nq-2 1 2n 2 ~ 2 R +'2R2 ilSn,fll •

Wq 2

00

Both are terms of a converging sequence. Hence I r s f (E;> converges n=O n n,

-uniformly on each Bq(~;R1) and therefore belongs to Harm(Bq(~iR».

q q-l

=:» Suppose f E Harm(B (0 iR) ). For each r < R we have f (r E;) E L2 W ) • 00

Hence

n~o

/ n II Sn,f 112 (ii) See part (i).

< 00

If f and g belong to Harm(Bq(Q;R)} the product f· g is usually not harmonic. For this reason the following lemma is not a trivial result.

o

LE~ 2.2. Let f,g E Harm(Bq(OiR», R> 1. The restriction of the pointwise

product f • g to Qq-l can be extended to a harmonic function in Harm(Bq(O;R» • We will call this product the harmonic product of f and g.

Proof. Write

00

I

m=O

In case of absolute convergence we can write

00 f(~,> g(P =

I

,11,=0

s

(t;) m,g -(2.1 )

Let 1 < R1 < R. Uniform convergence of (2.1) on

n

q-1 follows from the estimate

Here C_fg is a constant which only depends on f and g. From the last inequa-lity it also follows that

II

Therefore the sequence (2.1) also converges in L

2-sense. Next we estimate the norm of the projection of f(~) • g(~) on the space of spherical har-monics of degree k. 00

I

m+n=t S (0) oS (0) n,f m,g

Note that the second sum in the above expression presents a homogeneous (not necessarily harmonic) polynomial of degree t. When restriced to

n

q-1 this polynomial can be regarded as the restriction of a harmonic polynomial of degree $

~

to

n

q-1• So the projection

P

yields zero.

I

II

I

S (0) oS (0)11 ~

n, f m,g

Q,=k m+n=Q,

where c

1 does not depend on k.

Hence, for all R

2, 1 < R2 < Rl < R

00

I

R~k

II

P

k (f 0 g) II 2 < 00 • k=O

Now apply Lemma 2. 1 • 0

LEMMA 2.3. Let f E Harm(Bq(O;R», R > 1. Let A ::IRq -+ Rq be a linear mapping.

Suppose II All = Rl < R.

f ( q-l

Deineg~) ;f(A~) EL

2W ). g can be extended to a harmonic function in

Harm(Bq(O;.E...» •

- Rl

00

Proofo Again write f(P =

I

S f(t;). Consider S f(A x). This is a

homo-n=O n, - n ,

-geneous polynomial of degree n. With (1.1) i t follows

ISn,f(A P

I

~ IIAUn

(N(~,n»)~

_{lis n,} f"

.

q Hence, lis _n, f(Ao)II ~ _Rl n (w N (q,n) ) ~ II S f II

.

q n, Since 00 (P k g)

(9

=

Pk

I

Sn,f(AP n=k

we have :s; (w Ntq,n»!:l q 00 Let Rl < R2 < R. Let 1 R2 :s; L < -R ' ₁ then L2k II P 112 kg :s;

{I

(L;l)n N(q,n) + 2 n=O 2 R2

Since this is true for all L < - < R we conclude Rl Rl

for all L < R Rl

3. A METRIZABLE SPACE OF GENERALIZED FUNCTIONS

q-l

A theory of generalized functions on Q is a Gel'fand triple

q-l q-l

Here S W ) is the test space of smooth functions. The space T (Q ) can be regarded as the continuous dual of S(Qq-l). Moreover, S(Qq-l) is embed-ded in T(Qq-l) via L

2(Qq-l). In this section we take for the elements of S(Qq-l) restrictions of functions which belong to

Harm(Bq(~;l».

So each

f E S(Qq-l) can be extended to a function f E Harm(Bq(O;R» for some R > 1

q-l q

DEFINITION

3.1. A sequence (f ) c S(Qq-1) is said to converge iff n

(f ) c Harm(Bq(O;R», for some R > 1, and (f ) converges uniformly on

n - n

q .

B (0 ,R). This is equivalent to saying that (f (R~» n -for some R > 1.

converges in L (Qq-1)

2

For T Wq- 1) we take Harm(Bq(O;

1».

It "contains" (possibly diverging) series

00

\' \' r2n II S F II 2

of spherical harmonics L Sn f(~) with the property that L

n=O ' n=O n,

< 00

for all r, 0 < r < 1.

DEFINITION

3.2. A sequence (F ) c T{Qq-l) is said to converge iff (F

(r~»

n n

-converges in L (Qq-l) for each 0 < r < 1. 2

REMARK 3.3. S(Qq-l) is a space of type S B and T(Qq-l) is a space of type Y, T with Y Y,B q-l 1 1 2 ~ L2 (Q ) and B -2 (q - 1) I +

{"2

(q - 1) I - ilLB} • q-1 Here ilLB denotes the Laplace-Beltrami operator on the unit sphere Q and I _{denotes the identity operator. See [G}

1

J,

[G2

J,

[G3

J.

All general consi-derations of these papers apply here. SeQ q-1 ) and T(Q q-l ) are complete nu-clear topological vector spaces. T(Qq-l) is Frechet (i.e. metrizable). S(Qq-l) is an inductive limit of Hilbert spaces. A few general functional analytic results are presented here in an ad hoe manner.

DEFINITION

3.4. Let f E S(Qq-1), F E T(Qq-1). The pairing <f,F> is defined

by

< f I F> = ( f (R

~)

, F (R-1

~)

) (3.1 )

The inner product makes sense for R > 1 sufficiently small. The result does not depend on the choice of R. This can easily be seen by decomposing f and F in spherical harmonics.

It is a trivial exercise to prove that the mappings f ~ <f,F> and F ~ <ffF> are sequentially continuous. Moreover, all continuous linear functionals can be represented in the way of (3.1):

THEOREM

3.5. For each continuous linear functional

~

€ S'(Qq-1) there

exists

F~

E T(Qq-1) such that for all f E S(Qq-l} one has

~(f)

=

<f,F> •

Proof. Let

~

E L

2(QQ-1). Denote the solution of the Dirichlet problem on Bq (O;l) with

~

as a boundary condition again by

~.

For each r, 0 < r < 1, lji(r

0

belongs to SWq-1). Let

~

E

s'

Wq-1) be given. The functional

q-1

~ ~ R, (lji (r'», r fixed, is continuous on L2 W ). Hence, by Riesz' theorem

Q-l there exists gr € L

2(n ) such that ~(lji(r'}) = (lji,gr)' Replacing lji by .),g ) = (1jJ,g (r .}) = (ljJ,g ). Define

r r 1 r

1r

FR, by F~(r~) = gr(~)' It is harmonic and reproduces i in the desired way.

0

q-l

Now we come to some natural classes of operators which map

sen

)

continu-ously into itself. Most of these operators use the harmonic extension of the test functions for their definition.

3.A. Multipliers

Let h E

s(n

Q-1) be fixed. Consider the mapping f

~ ~

f Lemma 2.2 we see that h • f E S Wq- 1) •

3.B. Differentiation operators

Let ~ € m.q• The operator f H- (a· 'il) f is defined as follows. First extend

af af

f to a harmonic function, then calculate a

1 -,,-oX1 + .•• + a -,,-q oX and restrict

q

this to

~q-l.

Instead of the constants we can also use multipliers, thus

getting differential operators with variable coefficients. An interesting

subclass of this type is obtained in the following way: Take a matrix A € m.qxq. The operator f H- (x,A 'il) f maps S Wq-1) into itself. If A I

then (x,A'il) _{an . If A is antisymmetric,}

a

AT _{-A the vector fields} (X,A \/)

are tangent to

~q-1,

they are linear combinations of the moment of momentum

operators in quantum mechanics. They are the infinitesimal generators of

rotation operators in L2(~q-l}.

3.C. Harmonic contractions

Take a matrix A € m.qxq with II A II ~ 1. Define (L f) (t;)

=

f (A ~). In this

A -

-definition the harmonic extension of f is used. From Lemma 2.3 we obtain

that

LA

maps

S(~q-l)

into itself. If A is orthogonal the harmonic extension

of f is not needed because then II A ~ II == 1. Notice that LAB

1:

LA 0 LB in

general!

THEOREM 3.6.

The operators mentioned in 3.A, 3.B and 3.C map SW q-l )

con-tinuously into itself.

The proof can be given by ad hoc arguments or by applying [G 3

J.

Finally we come to the question whether the operators 3.A, 3.B and 3.C q-l

can be extended to operators from the distribution space T(Q ) into i t

-q-l q-l q-l *

self. If a mapping

L :S(Q

) ~ S(Q ) has a L

2(Q )-adjoint

L

which

q-l

maps S(Q ) continuously into itself, then L can be extended to

I :

T(Qq-l)

~

T(Qq-l) by <f,LF>

=

<L*f,F> which is a continuous linear q-l

functional on S(Q ). This easily proves the extendibility of the multi-pliers.

The extendability of differential operators with constant coefficients follows because they map Harm(Bq(~;l)) into itself. The general differen-tial operators are extendable because they are compositions of differendifferen-tial operators with constant coefficients and multipliers.

*

The extendability of LA with A orthogonal follows from LA

II All < 1 then 00

I

RnSn,f(A~)1I

:::;cllfll n=O LT. If A

i f R II A II < 1. This implies the extendabili ty. Cf. [G

2

J.

If II A II < 1 the

q-l q-l

operator LA is even smoothing, i.e. i t maps T(Q ) into S(Q ).

We will not discuss the extendability of L for the general case II All :::; 1 A

4. A SPACE OF GENERALIZED FUNCTIONS WITH A METRIZABLE TESTSPACE

In this section we consider a different Gel'fand triple

The test space E(Qq-l) consists of restrictions to Qq-l of functions in

HarmC~q).

We will (somewhat loosely) identify ECQq-l) and

Harm(~q).

DEFINITION

4.1. A sequence (fn) c E(Qq-l) is said to converge iff (fn)

converges uniformly on each ball Bq(O;R) for all R > O. This is equivalent to saying that (f (R 1;;» converges in L

m

q-1) for all R > O.

n - 2

For U

m

q- 1) we take Harm(O). It "contains" (possibly diverging) series of

00 00

spherical harmonics

I

Sn,F(~)

with the property that

L

r2nlls FII2 < 00

n=O n=O n,

for r sufficiently small.

DEFINITION

4.2. A sequence (F ) c U(Qq-l) is said to converge iff

n

(F ) c Harm(Bq(O;r», for some r > 0, and (F ) converges uniformly on

n - n

Bq(O;r). This is equivalent to saying that (F (r E;) converges in

n -L

2(Qq-l) for r > 0 sufficiently small.

REMARK

4.3. E(Qq-l) is a space of type T(Y,B) and U(Qq-l) is a space of

type cr(Y,B}. See [EJ. For Y and B see Remark 3.3. All general (topological)

q-l q-l

considerations of [E] apply here. In particular E

en

)

and U

en

)

are

com-plete nuclear topological vector spaces. E(Qq-l) is a Frechet space. U(Qq-l)

is an inductive limit of Hilbert spaces. Some of the results in [E] are

DEFINITION

4.4. Let f E

E(~q-1)

I F E

U(~q-l).

The pairing <f,F>U is de-fined by

< f I F> U = ( f (R 1;) F (R -1 1;) )

The inner product makes sense for R > 0 sufficiently large and does not depend on the choice of R.

It is a simple exercise to prove that the mappings f ~ <f,F>U and

(4.1)

F ~ <f,F>U are sequentially continuous. Without proof we mention, cf. [E],

THEOREM

4.5. For each continuous linear functional tEE'

en

q-1) there

q-l q-l

exists F~ E U(~

)

such that for all f E E(~

)

one has t(f)

=

<f,Ft>U'

Now we come to some natural classes of operators which map

E(~q-l)

conti-nuously into itself. Most of these operators use the harmonic extension of

the testfunctions to the whole of

m

q for their definition.

4.A. MUltipliers

Let h E

E(~q-1)

be fixed. With Lemma 2.2 we see that the mapping f

~ ~f

= h • f acts from E Wq-1) into itself.

4.B. Differentiation operators

Just like in 3.B we can introduce the operators (a· V), (x,A V), etc. The comments in 3.B also apply here.

4.C. Harmonic contractions and expansions

Take any matrix A E ]Rqxq. Define (L f) (I;) = f(A!;) with the aid of the

A

-harmonic extension of f. The comments in 3.C also apply here.

4.D. Harmonic translations

Let w E ]Rq. Define (T f)(l:) = f(l:+w). T clearly maps Ecnq-1) into itself.

w - w

THEOREM 4.6.

The operators mentioned in 4.A, 4.B, 4.C and 4.D map E(O q-l )

continuously into itself.

The proof can be given by ad hoc arguments or with the aid of [EJ.

Finally a few words on the extendibility problem. The operators 4.A and 4.B are extendible to operators from U(Oq-l) into itself. The proof runs along similar lines as in the cases 3.A and 3.B. The extendibility of the operators 4.C and 4.D is an open problem.

With each element [A;~] of the affine (semi) group on ]Rq we can assoc~dte the operator L[A;W] by

(L[A;W] f) (~)

In general we have

L oL /=L •

[BiZ] [AiW] [BA;Bw+z]

-

-

-

-As yet I do not know in which way the operators L "represent" the [AiW]

REFERENCES

[E] Eijndhoven, S.J.L. van, A theory of generalized functions based on one-parameter groups of unbounded self-adjoint operators. T.H.-Report 81-WSK-03, Eindhoven University of Technology.

[G1] Graaf, J. de, A theory of generalized functions based on holomorphic semi-groups.

Part A: Introduction and Survey. To appear in Proc. KNAW.

[G2] Idem. Part B: Analyticity spaces, trajectory spaces and their pairing. To appear in Proc. KNAW.

[G3] Idem. Part C: Linear mappings, tensor products and Kernel theorems. To appear in Proc. KNAW.

[M] Muller, C., Spherical Harmonics. Springer Lecture Notes in Mathematics, Vol. 17, Springer Verlag/ Berlin etc. 1966.

[S] Seidel, J.J., Spherical Harmonics and Combinatorics. Preprint,