Clifford Analysis (Research Notes in Mathematics, 76)

Research Notes in Mathematics '

F Brackx' R Delanghe &FSommen

Clifford analysis

Pitman Advanced Publishing Program BOSTON · LONDON · MELBOURNE

76

F Brackx, R Delanghe & F Sommen State University of Ghent, Belgium

Clifford analysis

Pitman Advanced Publishing Program BOSTON·LONDON · MELBOURNE

PITMAN BOOKS LIMITED 128 Long Acre, London WC2E 9AN PITMAN PUBLISHING INC 1020 Plain Street, Marshfield, Massachusetts 02050 Associated Companies Pitman Publishing Pty Ltd, Melbourne Pitman Publishing New Zealand Ltd, Wellington Copp Clark Pitman, Toronto First published 1982

© F Brackx, R Delanghe & F Sommen 1982 AMS Subject Classifications: (main) 30 A97, 46 F15, 42 A68 (subsidiary) 46 E20, 31 B65, 32 A30 Library of Congress Cataloging in Publication Data Brackx, F. Clifford analysis. (Research notes in mathematics; 76) Bibliography: p. Includes index. 1. Holomorphic functions. 2. Clifford algebras. 3. Distributions, Theory of (Functional analysis) 4. Harmonic functions. I. Delanghe, Richard. II. Sommen, F. III. Title. IV. Series. QA331.B77 1982 515.9'8 82-14993 ISBN 0-273-08535-2 British Library Cataloguing in Publication Data Brackx, F. Clifford analysi~>.-(Research notes in mathematics; 76) 1. Functions I. Title. II. Delanghe, R. III. Sommen, F. IV. Series 511.3'3 QA331 ISBN 0-273-08535-2 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording and/or otherwise without the prior written permission of the publishers. This book may not be lent, resold, hired out or otherwise disposed of by way of trade in any form of binding or cover other than that in which it is publish..~vJ 0

(A.~,v)o

2n [A.v~] 0

(A.,vii.)o. Next n 2n[~A.v] 0 = 2 [~v>..] 0

(~.Xv) 0

(vii.,X) 0

(vii.,A.)o

= (A.,vii.)o. 1.14

0

In accordance with 1.12 the function< given by

n

2

2n

[~A.]o

= 2n [A.~]o

L (- 1)n(A)(n(A)+1)/2

"A~A

A

is a non-singular trace on A satisfying >..,~

EA

antl representing the real linear functionals on A, i.e. for any fixed A. E A the function '>..:A ~R defined by

is a real linear functional on A, and conversely, for any real linear functional T on A there exists a unique >.. E A such that

In particular for >..

-_ 2n( - 1',n(A)(n(A)+1)/2 ~A· 11

In the special case where A = ~ we have n

2 JJo· Moreover

1.15 In a classical way a matrix representation of the Clifford algebra A may be obtained as follows. Order the basic elements eA, A E PN, of A in a certain way: n

{e(K) : K = 1,2, ••• ,2} and associate to each A E A the real (2n x 2n)-matrix 8(A), the entries of which are given by n

8(A)K,L = [Ae(K)](L)' K,L = 1, ••• ,2 , where [JJ](L) denotes the coefficient of the e(L)-component of JJ. This representation is an isomorphism between A and ~2n x2n. Notice that e(e 0 ) is the identity matrix I and that

If v(p) denotes a p-vector then the associated matrix e(v(p)) is symmetric when p = 0,3(mod 4), while the matrix e(v(p)) is skew-symmetric if p = 1,2 (mod 4). In this way it turns out that whatever p may be (0 :S p ~ n), e(v(p))

= (e(v(p)))T

T 2n X 2n where a denotes the transpose of a E ~ • Hence for any A E A

12

z. Modules over Clifford algebras: general properties z. 1 The aim of this section is to give some general information

concerning a class of topological modules over Clifford algebras and their duals. As will be seen, nice relationships between real linear and A-linear functionals on such modules may be established. They turn out to be of basic importance when function and distribution modules will be introduced in the following sections. In the sequel definitions and properties will be stated for left A-modules and their duals, the passage to the case of right A-modules being straightforward. In order to avoid any confusion, we use systematically the subscripts '(1)' or '(r)' to indicate that the module under consideration is a left or a right A-module. In the case of a bi-A-module, no subscript is used, although when it is e.g. considered as a left A-module, the subscript '(1)' is then added. 2.2 Let X(l) be a unitary left A-module, i.e. X(l)'+ is an abelian group and a law (A,f) + Af from Ax X into X(l) is defined such that for all A,~ E A andf,gEX(l)

(i)

(A+

~)f

Af +

~f

(A~)f

A(~f)

(iii)

A(f+g)

Af + Ag

( i v)

e0 f = f

(ii)

.

Notice that X(l) becomes a real vector space if m is identified with me 0

A0

c

=

A.

Moreover when speaking of a submodule E( l) of the unitary left A-module X(l) we mean that E(l) is a non empty subset of X(l) which becomes a unitary left A-module too when restricting the module operations of X(l) to E(l)• l) and Y( l) are unitary left A-modules then a function T:X( l) +Y( l) is said to be a left A-linear operator if for all f,g EX( l) and A E A

2.3

If X(

T(H+g)

H(f)

+

T(g).

The set of all left A-linear operators from X( l) into Y( l) is denoted by 13

Notice that, by defining for each T E L(X(l)'y(l)) and~ E A, (T~)(f) = T(f)~. f EX( l)'L(X( l)'Y( l)) is turned into a unitary right A-module. In the special situation where Y(l) coincides with A, the laiter being considered as a unitary left A-module over itself, L(X(l)'A) is called the algebraic dual of X(l) and it is denoted by X(~1g· Its elements are called left A-linear functionals on X(l) and for any T E x(~1g and f E X(l) we denote by the value ofT at f. By previous considerations X(~1g is a unitary right A-module. Finally observe that if X is a unitary bi-A-module then for each T E x(~1g and ~ E A we can introduce, next toT~ already defined, the left A-linear functional ~T by putting L(X(l)'Y(l)).



= . f EX.

Similarly, under the same assumption upon X, we have for each S E x*alg and ( r) ~ E A, the right A-linear functionals ~Sand S~ given by

= ~<S,f>, f E X,

<S~,f>

=

and S,~f>,

f E X.

2.4 Let X(l) be a unitary left A-module. Then a family P of functions p : X(l) ~R is said to be a proper system of semi-norms on X(l) if the following conditions are fulfilled: [P 1J

There exists a constant C0

~

1 such that for all p E P, A E A and

f,g E x(l) (i)

(ii)

p(f+g) s p(f) + p(g) p(Af) s C0 IA1 0 p(f), and p(Af)

IAip(f) if A € R.

[P 2J For any finite number p1 ,p 2 , ••. ,pk E P, there exist p such that for all f E X(l) sup j=1, .•• ,k

14

p.;(f) s C (f). ..J

p

€

P and C > 0

[P 3]

= 0 for all p E P then f = 0.

If p(f)

Notice that when considering X(l) as a real vector space, (X(l)'P) turns out to be a (real) locally convex Hausdorff space. Obviously a lot of definitions and properties from the general theory of locally convex spaces may be carried over without any restriction to the class of topological modules over a Clifford algebra which was just introduced. We shall restrict ourselves here to mention those notions which will be currently used throughout the book. For more details concerning the classical situation the reader is referred to e.g. [4]. 2.5 Let P and Q be two proper systems of seminorms on X(l)" Then Pis said to be weaker than Q (or Q is stronger than P), notation P < Q, if for each pEP there exist q E Q and C > 0 such that for all f E x( 1) p(f )

~

Cq ( f) .

Moreover P and Q are called equivalent, notation P ~ Q, if P


0 such that for all f Il 0

~

X(l) into Y(l) is denoted by of L(X(l)'y(l)). E X(~Jg is called bounded if E X(l)

Cp(f).

The submodule of X{~~g which is obtained in this way is denoted by x( 1) and it is ca 11 ed the dual of X( 1). Furthermore if (X(l)'P) is looked upon as~ real locally convex space then its real dual, i.e. the set of bounded (real) linear functionals on x(l)' is denoted by X(l)" 2.7 Now one may define proper systems of seminorms on L(X( l)'Y(l)) and X{ l) by considering special families F of bounded subsets of (X(l)'P). 15

Let B c X(l) be bounded, i.e. for each pEP there exists Cp sup p(f) fEB

~

0 such that

Cp;

then, for each q E Q, Pq,B p B(T) q,

>

L(X(l)'Y( l)) ~R, given by

= sup q(Tf), fEB

defines a seminorm on L(X(l)'Y( 1)). Now, if F is a family of bounded subsets of X( l) satisfying (i)

(ii)

u B = x(l) BEF if B1, B2 E F, then there exist C > 0 and BE F such that B1 u B2 c CB,

then PQ,F

= {Pq,B : q E Q, BE F} determines a proper system of seminorms on

L(X( l)'y( 1)).

In the special situation where Y = A we put for each B E F and T E pB(T)

x(1 )

= sup Il 0 • fEB

In this case we pay special attention to the families Fs and Fb consisting of either all finite or all bounded subsets of X. If X(l) is endowed with the corresponding proper system of seminorms we denote it respectively by X(l)s and XCl)b and we say it carries the weak or strong topology respectively. Of course, if X(l) is considered as a (real) locally convex space, then X(l)s and X(l)b stand for the classical dual spaces. 2.8 In what follows we shall establish some basic relationships between Alinear and real linear functionals on an A-module.

Proposition Let X(l) be a unitary left A-module and letT be a left A-linear functional on X. Then for each A E PN, T T is a real linear functional on eA X( l) satisfying (Te T,f) = (Te T, eAf) , A o Moreover for all f E X(l)' 16

f E X(l)"

= 2-n

I A

eA eA

conversely, letT be a real linear functional on X(l)" Then for each A E PN there exists a unique left A-linear functional TA on X(l) such that 'eTA= T. This functional TA is given by A

I

= 2-n

eB .

B

Finally T = 0 if and only if T T = 0 for some A E PN. eA Proof (i) If T E x*alg then clearly 'e Tis a real linear functional on X(l) for (1 ) A which = A A

= i A

n

2

I. 0 and pEP such that for all f E X(l)' A [ [ : ; C p(f). A

By Proposition 2.9 we have 2 [[ 0

- 2 = 2-2n ,L () B

: ; c2.2-2n

I

(p(eAeBf))2

B

: ; 2- 2n c 2 c~!eAI ~(p(f)) 2 Lie 8 ! ~ B

Hence, there exist c* Il 0

::;;

* )' whence T E X(l 2.10

>

0 and p E P such that for all f E X(l )'

c* p(f) c

Now we state a Hahn-Banach type theorem.

Theorem Let X(l) be a unitary left A-module provided with a proper system of seminorms P, let Y(l) be a submodule of X(l) and letT be a left A-linear functional on Y(l) such that for some C > 0 and p E P I[ 0 ::;;c p(g) for all g E Y(l)' Then there exists a left A-linear functional T* on X(l) such that (i) (ii)

T*IY(l)

= T;

for some c* > 0, Ilo::;; c*p(f),

f E X(l)'

Proof As for any g E Y(l) ll : ; [e 0 10 i[ 0

::;;

C[e 0 [ 0 p(g),

0

there ought to exist, in view of the classical Hahn-Banach theorem for real 20

locally convex spaces, a bounded real linear functional Ton X(l) such that TIY(l) = Te T and Il ~ Cle 0 10 p(f) for all f E x(l )" 0

By means of Proposition 2.9 and its proof, there exists T* E x(~~g such that Te T* =Ton X and Il 0 ~ C*p(f) for all f E X(l) with C* C0 Cie 0 I0 • 0

As T IY( 1) 1

eo

(T-T*)

Te 0 T we have again by virtue of Proposition 2.9 that on Y(l )'

o so

that T = T*IY(l)'

c

2.11 Corollary Let X(l) be a unitary left A-module provided with a proper system of seminorms P and let Y(l) be a submodule of X(l)' Then Y(l) is dense in X(l) if and only if for each T E x( 1) such that TIY(l) = 0 we have T = 0 on X( 1). 2.12 As will be seen in the next sections most of the A-modules of test functions which will be used currently, fit into the following framework. Suppose that X is a unitary bi-A-module and that Y is a real vector space such that, considered as rea 1 vector spaces, X = IT Y. AEPN Furthermore assume that each f E X may be written as f = I fAeA with fA E Y A

for all A E PN, the subsets A of N being ordered in a certain way. Finally suppose that X(l) andY are provided respectively with the systems of seminorms P and p* such that the locally convex topology on X induced by P is equivalent to the ::>roduct topology defined on X = IT Y. AEPN Then the following proposition characterizes completely the dual x( 1 ) of X( 1).

2.13 Proposition LetT be a bounded left A-linear functional on X(l)' Then for each BE PN there exists a unique bounded real linear functional r 8 on Y such that for any~= L ~AeA E X(l)' A

=

L

eAeB.

A,B Conversely if (T8 )BEPN is a sequence of bounded real linear functionals on Y then T : X(l) ~A defined by

21

= I

eAeB' ~=I ~AeA E X(l)'

A,B A sa bounded left A-linear functional on X(l)' roof

Given

any~=

I

eA~A E

X we have, using Proposition 2.8,

A

= A

=I eA A

2-n

~here

I eAeB A,B B

I eAeB A,B we have put for each B E PN, = 2-n , -I- E· Y. eB

We know from Proposition 2.9 that 'e T is a bounded real linear functional on X(l) and hence on IT Y and finally ~n Y itself. So r 8 is a bounded real AEPN linear functional on Y; its uniqueness follows from its construction. The converse is easily checked by computation. c 2.14 We close this section by considering inductive limits of topological A-modules. Let ((xi 1},Pj))jeN be a sequence of unitary left A-modules xt 1 ) each of which is provided with a proper system of seminorms Pj such that (i)

(ii)

for all j E IN,

xt

l) is a submodule of

xt;~;

the canonical injection Ij : X~)+ xt;~ is bounded.

Then on X(l) = _u xtl)' which is clearly a unitary left A-module, a filter.1eN ing set of seminorms Pind is determined in the following way. Let n = {p 1 ,p 2 , ••• ,pJ., ••• ) be a sequence of seminorms belonging to n P. jeN J and let y = (c 1 ,c 2 , ••• ,cj'''') be a sequence of positive real numbers. Furthe~ more let an arbitrary decomposition off E X(l) be written as f = ~ f., (J)

22

J

j

fj E X(l)" p

n,y

Then we put (f) =

f

I

inf = \f. L J

cJ.pJ.(fJ.).

(j)

( j)

One may easily check that P1- nd = {P

n,y

: n E 11

jEl-l

P., J

y

E 11 lR+ } jEl-l

is a filtering set of seminorms on X(l)" The topological left A-module (X(l)' ~ind) obtained in this way is called the inductive limit of the sequence ((X~l)'Pj))jSN and it is also denoted by x(l) = limjSNind xil)" * j Of course one has that T E X(l) if and only if Tj = TJX(l) E xj*(l) for each j E IN.

3.

The spaces C0 (K;A) and

c0 (n;A);

A-measures

3.1 Let K be a compact subset of lRr (r ~ 1). Then C0 (K;A) stands for the unitary bi-A-module of A-valued continuous functions on K. Of course C0 (K;A) =

11 C0 (K;lR) AEPN

and each f E C0 (K;A) is endowed with the norm llfll =sup Jf(x)J 0 , f E c(l)(K;A). xEK Clearly this norm is equivalent to the one defined in a classical way on C0 (K;A) = 11 C0 (K;lR). AEPN 3.2 Now ·;et r2 c lRr be open and let (~B)BEPN be a sequence of real measures inn (see e.g. [5]). Then for any semi-interval I contained in n,~(l) is defined by ~(I)

L

~B(I)eB

BEPN and the

function~

thus obtained is called an A-valueJ measure inn. 23

Notice that the support of supp

~

= u

~.

notation supp

~.

is given by

~8 •

BEPN If N(n;A) stands for the set of A-valued measures in n, then clearly M(n;A) is a unitary bi-A-module. An A-valued function f = L fAeA is said to A

be ~-integrable inn if for all A E PN and BE PN,fA E L 1 (n;~ 8 ) where L 1 (n;~ 8 ) denotes the classical space of R-valued ~ 8 -integrable functions. The set of all A-valued ~-integrable functions inn is denoted by L 1 (n;A;~). It clearly constitutes a unitary bi-A-module. In the particular case where ~ 0 is the Lebesgue measure in n and ~B is the zero measure in n for each B F ~. we write L1 (n;A). For any f E L 1 (n;A;~) we put

I f(x)d~ = L n

eAeB

A,B

I fA(x)d~B Q

and

I d~f(x) = L Q

B,A

e8eA

I fA(x)d~ 8 • n

Using a classical result and Proposi~ion 2.9 the dual C0 ( 1 )(K;A) of c( 1)(K;A) may be described in terms of c0 (K;R).

3.3

Theorem: (Riesz representation theorem) LetT E C 0 ~ 1 )(K;A). Then there exists a unique A-valued measure ~ in Rr with support contained in K such that for all f E c( 1 )(K;A)

= IK f(x)d~.

3.4 Now let C0 (n;A) denote the unitary bi-A-module of A-valued continuous functions in n. Obviously

c0 (n;A) = rr

C0 (n;R)

AEPN

and each f E C0 (n;A) can be written as f

c( 1)(n;A) K~ 24

= L fAeA, fA E C0 (n;R). The module A

is equipped with the following proper system of seminorms P. n be compact and put

Let

pK(f) = sup If (X ) I0. xEK

f E

c0 ( Q; A) •

Then P = {pK : K c n compact} induces the topology of uniform compact convergence on c( 1)(n;A). If (Ks : s = 1,2, ••• ) is the compact exhaustion of n determined by Ks = {x E Q: lxl :;; sand d(x,ll{~) ~ then Pis equivalent toP*= {pK

s

-}l,

s

1,2, ••• ,

: s = 1,2, ••• }.

Moreover P is also equivalent to the system of seminorms defined in a classical way on C0 (n;A) = rr C0 (n;JR). LetT E Ca*(l )(Q;A); then there exist AEPN s ~ 1 and C > 0 such that Il 0 < CpKs(f), f E

c( 1)(n;A).

As c( 1 )(n;A) is a submodule of c( 1)(Ks;A), by the Hahn-Banach theorem, T admits an extension T* to c( 1)(Ks;A) and hence, in view of Theorem 3.3, an A-valued measure ~ in Rr can be found with support contained in Ks such that for all f E c( 1)(Ks;A) =

Jnf(x)d~.

which implies that for all f E = 4.

c( 1)(n;A)

Jnf(x)d~.

The space V(n;A); A-distributions

4.1 Let K and n be respectively compact and open subsets of Rr. Then V(K;A) and V(n;A) denote respectively the unitary bi-A-modules of A-valued infinitely differentiable functions in Rr with support in K and of A-valued infinitely differentiable functions in Q having aompaat support contained in n. Obviously V(K;A) V(n;A) and for

each~

IT V(K;JR), AE.PN n V(n;JR) AEPN E V(K;A) (resp.

~

E V(n;A)),

~

L 4AeA with

~A

E V(K;R)

A

25

( res p.



A E V (rl ;lR) ) •

Notice also that if



=I AeA E V(Q;A), then its A

support is given by supp¢ =

u AEPN

supp A

4.2 Taken as a left A-module V(l )(K;A) may be provided with the proper system of seminorms PK = {pk : k E ~} with

aE]N r ,

r a1 = I aj, aa =_a__ a1 ax 1 j=1

Ia!

(J,

" r " ar axr

Clearly this system is equivalent to the one defined in a classical way on V(K;A) = II V(K;JR).· Next V(l)(rl;A) may be equipped with an inductive limit AEPN topology by putting

where (Ks)sElN is a sequence of compact subsets of ( i) (ii)

r2

such that

-

~s = Ks

KS

c

KS+1 00

( i i i ) II =

U

s=1

Ks •

4.3 Obviously this locally convex topology is equivalent to the one defined in a classical way on V(S"l;A) = II V(st;JR). Hence the dual V(l )(rl;A) of AEPN V(l )(rl;A) is completely characterized by Proposition 2.13. It consists of the so-called left A-distributions in rl. 4.4 If a= (a 1, ••• ,ar) is any multi-index in ~r then for any T E V(l )(rl;A), aaT is defined by ('daT,¢> = (-1)jaj

= > 29

while for S E v(r) (l!{;A) and T E V(r)(l!{' ;A), <SeT,¢> = <Sx,>, and this for all ¢belonging to respectively V(l)(lRr+r' ;A) and V(r)(lRr+r' ;A). The distributionS~ T is called the tensor product of SandT. Notice again that in general S 0 T 1 T 0 S. Now let S,T E v( 1 )(lRr;A), at least one of them having compact support. Then S*T E V(l)(lRr;A) is defined by <S*T,¢>

= <Sx

0 Ty,¢(x+y)>>

= >, and this for all ¢ E V(l )(lRr;A). Analogously, let S,T E V(r)(lRr;A), at least one of them having compact support; then S*T E V(r)(lRr;A) is given by <S*T,¢>

<Sx 0 Ty,¢(x+y)>

= <Sx, >, and this for all ¢ E V(r)(lRr;A). In both cases S*T is called the convolution of SandT. Notice that the convolution of A-distributions is not commutative and that, if e.g. S,T,U E v( 1 )(lRr;A), at least two of them having compact support, then the associativ~ law holds, while the distributive law remains valid in v*(l) . ,+,* As to the a-distribution, defined as usual by = ¢(0), ¢ E V(lRr;A) it may be considered as well as a left or as a right A-distribution. Moreover one may easily check that for all T E v( 1)(lRr;A) or T E V(r)(lRr;A) O* T = T*o = T. Furthermore, i f S,T E v( 1 )(lRr;A) are such that S*T is defined and P(D) is a hypercomplex differential operator with constant coefficients, then P(D)(S*T) = (P(D)S)*T.

* (lR r ;A) Analogously, for S,T E V(r) (S*T)(P(D)) = S*(T(P(D))).

30

Finally, for f =I fAeA and g A

I g8e 8 belonging to L1(:n{;A), define their· B

convolutions f*g and g*f by f*g(z)

J f(x)g(z-x)dx lR r

and g*f(z)

J r

g(x)f(z-x)dx

lR

g8 (x)fA(z-x)dx. I eBeAJ lRr B,A Then f*g, g*f E L 1 (1Rr;A) and in general f*g I g*f. A straightforward computation yields that if f and g are considered as left A-distributions, then their convolutions f*g and g*f, considered too as left A-distributions, coincide with the ones in distributional sense. Analogous results of course still hold when dealing with the case V(r)(lRr;A). 4.7 Taking account of Proposition 2.13 and the theory of [-valued distributions, the following fundamental theorems may be formulated. For the classical situation the reader is referred to [16]. 4.7.1 Theorem ("Principe du recollement par morceaux") Let {Sii:i E I) be a family of open subsets in lRr and let (Ti:i E I) be a family of distributions such that for each i E I, TiE v( 1 ){Sii;A) (resp. V(r)(Sii;A)). Furthermore assume that, if Si. n D. 1 ¢, then T. lSi. n Si. = T.ISi. n Si .• Then there exists 1

J

11

J

J1

J

a unique distribution T E V(l){Si;A) (resp. V(r){Si;A)) where~= i~I Sii' such that Tls-2.1 = T.1 for all i E I. 4.7.2 Theorem Let Si and V be open subsets of lRr such that V is compact and contained in r.. furthermore letT E v( 1 ){s-I;A). Then there exist a E ~rand an A-valued function f continuous in a neighbourhood of V such that T = 3af in V, i.e.

31

= (-1)iai J

aa~(x)f(x)dx, ~ E V(l)(V;A).

Rr 4.7.3 Theorem (1) Let n eRr be open and let V c n be open such that V c n is compact. Furthermore let (Ta)aE]0, 1] be an s-bounded family in v( 1 )(n;A) such that Ta s-converges to T0 if a~ 0+. Then there exist B E ~r and a family of A-valued functions (fa,fo :a E ]0,1]), all of them being continuous in a neighbourhood Wof V, such that fa converges uniformly to f 0 on Vand Ta = a8f a , To = a8f o in V. (2) If moreover (Ta>aE]0, 1] depends s-continuously upon a, then fa may be chosen in such a way that the function f given by f(a,x) = f a (x), f(O,x) = f 0 (x) is continuous in [0,1] x W. 5.

The space S(Rr,A); tempered A-distributions

5.1 Call S(Rr;A) the unitary bi-A-module of A-valued Papidly decPeasing . . Rr • Of course f unct-z-ons 1n S(Rr ;A) = IT

S(Rr ;R)

AEPN

and each~ E S(Rr;A) can be written as~

I AEPN

5.2 Considered as a left A-module S(l)(Rr;A) may be provided with the proper system of seminorms P = {pk : k E ~}where for each~ E S(l)(Rr;A)

k E

~.

a E

~

r

•

Its dual SCl)(Rr;A) consists of the so-called tempePed left A-distPibutions or left A-distPibutions of slow gPowth in Rr. 5.3 Using classical results (see [16]) and again Proposition 2.13 the following fundamental theorem may be proved. First recall that a continuous A-valued function g is said to be of slow gPowth inRr if there exist k E~ and C > 0 such that inRr, lg(x)l 0 :;; C(1+lxl 2 )k. Obviously, if g is continuous and of slow growth in Rr its associated A-distribution is tempered. 32

* r ;A). Then there exist a E ~ r and a continuous 5.4 Theorem LetT E S(l)(m A-valued function g of slow growth in mr such that T = aag in mr. 5.5 Now let n cmr be open. Then we define s(n;A) to be the set of functions ¢ E s(mr;A) such that supp ¢ c nand ¢together with all its derivatives aa¢ vanish on an, i.e.

Clearly s(n;A) is a submodule of S(mr;A) and hence s(l)(n;A) may be endowed with the topology induced by S(l)(mr;A). Its dual is denoted by S(l)(Q;A) and it is called the space of left A-distributions of slow growth inn. By the Hahn-Banach Theorem, each T E s*(l)(n;A) admits an extension " ' . * r ;A), TinE s(l)(n;A). * T E* s(l)(m r ;A) and obv1ously for every T E s(l)(m Notice t~at if n is bounded then S(l)(n;A) c V(l)(n;A) and that in this case T E S(l)(n;A) implies that T E V( 1 )(n;A). Moreover if n is bounded and an= an, then clearly s(l)(n;A) = V(l)(n;A), both algebraically and topologically, so that in this case s( 1 )(n;A) = v( 1 )(n;A). Furthermore an A-valued continuous function g is said to be of slow growth in n if it can be extended to a function h which is continuous and of slow growth in mr. Finally an A-valued function g is said to be of weak slow growth inn if there exist a constant C > 0 and k,k' E ~such that inn

I g ( x) I 0

::;:

c(1

+

I x 12 )

k

1

(

1 + ~a-n})

kI

•

Of course if g is of slow growth in n then it is of weak slow growth in n. Theorem Let Q cmr be open. (i) If T E s( 1)(n;A) then there exist a E ~rand g, continuous and of weak slow growth in n such that T = aag in n. (ii) I~ n is convex and f is continuous and of weak slow growth inn, then f E s(l)(r.;A). 5.6

* * r ;A) so that, (i) If T E s(l)(n;A) then T admits an extension "'T E S(l)(m in view of Theorem 5.4, a E ~r and an A-valued continuous function g, which '\, is of slow growth in mr, may be found such that T aag. Consequently '\, Tin = T = aagln. (ii) Without loss of generality we may assum~ that 0 En. Consider

~

33

the following diffeomorphism 8 between the open halfsphere Sr = {X E ffir+ 1 : r+1 + r (x 0 + 1 )2 + x1 + .•• + x~ = 1 and x0 > -1} and ffi= = {x E ffir+1 : x0 = 0} = ffi , determined by the projection from S~ onto ffir with centre (-1,0, .•. ,0). Then clearly the space S(~;A) is transformed into the space S(~' ;A) = V(~;A) where -1 r = {x E ffi r+1 : (x + 1) 2 + x2 +•.. +x 2 =1}. where~· = 8 ~. ~· being taken inS 1 o r Now define for each ¢EV(l)(~;A), ¢'(x') = ¢(0(x' )) and associate to f the distribution f' E D( 1 )(~' ;A) given by = . Then one may show that f' can be extended to an element of s( 1 )(~';A) and hence, using density arguments, that f can be extended to an element of s(n(~;A).

6.

o

The space

E(~;A);

A-distributions with compact support

6.1 Let~ be an open subset of ffir; then E(~;A) is the unitary bi-A-module of all A-valued C -functions in ~. Obviously 00

E(~;A)

IT AEPN

and for each ¢ E

E(~;JR)

E(~;A),

¢=

I

¢AeA with ¢A E

E(~;JR).

A

6.2 Considered as a left A-module, E(l )(~;A) may be provided with the proper system of seminorms P = {Ps,k s, k E W} where for each¢ E E(l )(~;A) sup

sup

icxi~k

XEKS

(Ks)seN being a compact exhaustion of~ and a E Wr. Its dual r( 1 )(~;A) consists of the so-called left A-dist~ibutions with compact

6.3

and

34

suppo~t

in

~.

Of course we have the inclusions

1.

Hilbert A-modules

To fix the ideas let H(r) be a unitary right A-module. Then a function (,) : H(r) x H(r) +A is said to be an inner product on H(r) if for all f,g,h E H(r) and A E A, 1.1

(i)

(f,g +h)

(f,g) + (f,h)

(ii)

(f,gA)

(f,g)A

(iii)

(f,g)

(g,f)

o

~

( i v) ( v)



0 and = 0 if and only if f = 0

.

0

0

Notice that from the definition itself it immediately follows that (O,f) = (f,O) = 0 and that (fA,g) = ~(f,g). MOI~eover, as for each A E A,= eo . is an inner product on H(r) the latter being considered 0

0

as real vector space.

Consequently, putting for each f E H(r)

llfll 2 = 0

we have that for all f,g E H(r) ll ~ llfllllgll and llf +gil~ llfll + llgll· 0

Hence 11·11 is a proper norm on H(r) turning it into a normed right A-module. 7.2 Definition Let H(r) be a unitary right A-module provided with an inner product (,). Then it is called a right Hilbert A-module if it is compiete for the norm topology derived from the inner product. 7.3 In a classical way the elements f,g E H(r) are called orthogo·wl if (f,g) = 0 while a subset {fi:i E I} of H(r) is said to be orthonormal if (fi'fj) = 0, i I j, and llfill 2 = 1, i E I. 7.4 Proposition (Bessel's inequality) Let (fi) be a finite or countable orthonormal subset of H(r)" Then for all f E H(r)

35

Proof M E lN

As for each i E lN, llfi(fi,f)ll 2

;;;

I

(fi,f)l~

we obtain that for any

M

o

l:

:sllf-

fi(fi,f)JI 2

i =1 M

= llfll 2

M

-l:

llfii 2

-L

l(fi'f)l~·

l:

l

~ !AI~

. 0



~

0 and = 0 if and

for all A E A and f E L 2 ,(r)(H;A;~), 0 such that B(x,R) c Then again by Proposition 9.2 and taking into account that E(y-x) is right monogenic in y E S,B, we get

S.

J

Cl(S,B)

E(y-x)do f(y) y

=

I

S'B

E(y-x).Df(y) dy.

( 9. 1 )

Observe that E(y-x).Df(y) is integrable on S since Of is continuous on S and

where R' > 0 is such that S c B(x,R'). of (9.1) tends to

fs

So if R ~ 0+ then the right hand side

E(y-x).Df(y)dy.

As to the left hand side of (9.1) it can be put into the form

Jas

E(y-x)do

Y

f(y) -

I

aB(x,R)

E(y-x)do f(y) Y

the second term of which can also be written as

53

I

e.e.eA (y.-x.)fA(y)dy. 1 ClB(x,R) J J J 1

j,i,A

Rm~ 1

L

I

eJ.eieA

B(x,R)

j, i ,A

(6 .. fA(y) + (y.-x.)Cl fA(y))dy lJ J J Y;

I

I

1 (m+1) f(y)dy + - 1Rm+l B(x,R) wm+1

1 G-x)Df(y)dy. Rm+ 1 B(x,R)

As for sufficiently small R f(y) = f(x) + e(R)

with

lim e(R) = 0 R+O+

and sup IDf(y) 1 0 yEB(x,R)

$

C

it is obtained that

I

lim R+O+

ClB(x,R)

E(y-x)do f(y) y

1-(m+1) vm+1 f(x) wm+1

= -

f(x)

where Vm+ 1 = ~+1 1 is the volume of the (m+1)-dimensional unit ball. taking limits for R + 0+ in (9.1) yields

Ias

E(y-x)do f(y) - f(x) Y

= I E(y-x).Df(y)dy. s

9.6 Corollary (Cauchy's Integral Formula) left monogenic in n then

Hence

c

Let S c n be as in 9.5.

Iff is

0

Ias

E(y-x)do f(y) Y

=~ L

f(x) for x E S

o

for x E sz-....s.

9.7 Proposition (Mean Value Theorem) then f(a)

=

I

B(a,R) Rm+ 1v m+1 for each R > 0 such that B(a,R) 54

f(u) du c

n.

Iff is left monogenic inn and a En

-

proof Take R > 0 such that B(a,R) on the ball B(a,R) gives

c

n. Applying Cauchy's Integral Formula

I

f(a)

CJB(a,R)

or f(a) As (u-a)Du

= m+1 one gets by Proposition 9.2

I

m+1

f(a)

B(a,R)

f(u) du.

c

9.8 Proposition (Cauchy's Integral Formula outside a ball). Let f be left monogenic in Rm+l,B(O,R) with lim f(x) =A· Then for each xSRm+l,B(O,R) f(x) = A -

J

aB(O,R')

E(y-x)do f(y) y

where R' is suitably chosen such that R < R' < lxl. Proof Take x E Rm+l,B(O,R) arbitrarily; choose R' such that R < R' < lx I and R" such that B(O,R') c B(x,R"). Applying Cauchy's Integral Formula on 0 B(x,R"),B(O,R') one gets f(x)

I

=

aB(x,R")

- as(6,R') r

E(y-x)do f(y). Y

(9.2)

But introducing spherical coordinates the first integral becomes

J and so

m

f(x+R"w)dS

s

w

lim J E(y-x)doy f(y) =A. R"-++oo aB(x ,_R") Hence taking 1imits for R" ->-+ oo in (9.2) it is obtained that f(x) =A -

r

.

l E(y-x)do f(y). aB(b,R') y

c

55

9.9 Theorem (Maximum Modulus Theorem). Let f be a left monogenic in the open and connected set n. If there exists a point a E n such that lf(x) 10

~

lf(a) 10 for all x

E

n

then f must be a constant function in rt. Proof

Put lf(a)l 0 =A and consider the subset nA rt>, = {x E n: lf(x)

of~

given by

= A}.

10

Of course nA I~ since a E nA. So let y E ~QA; this implies that lf(y)I 0 0;

and by differentiating twice

I

(ax.fA(x)) 2 +

A

Summing up over i

I

I

0, 1 , .•• ,m.

0,

1

= 0,1, •.• ,m yields

(ax.fA(x)) 2 + 1

i ,A

fA(x).a~~fA(x)

A

1

I

fA(x).~m+ 1 fA(x)

0

A

or

I

(ax.fA(x)) 2

. A 1•

= 0 for all x En.

1

This results finally into

a fA(a) = 0 inn for all

0,1 , ••• ,m and all

Xi

A E P{1, .•. ,n}, which means that f is constant inn.

c

9.10 Corollary Let n be a bounded open set in ~m+ 1 and suppose that f is continuous in n and left monogenic in n. Then sup lf(x)l 0 = sup lf(x)l 0 • xEn xEan

n

Proof As is compact and lf(.)i 0 is a continuous function on point a E n such that sup lf(x)l 0

=

n there

is a

lf(a)l 0 •

XEQ

If a E an then of course sup lf(x)l 0 = lf(a)l 0 xEan and the assertion is true. So assume that a E n. ponents

Decompose n into its com-

57

all of them being bounded, open and connected. components, say~ .• As

Then a belongs to one of the

J

if(x)

10

~

lf(a) 1 0 for all x

E

nj

it follows from Proposition 9.9 that f is a constant function in nj. sup lf(x)l 0

=

lf(a)l 0

=

sup

~

lf(x)l 0

XE()[2J.

XE n"

which yields the desired result.

sup

lf(x)l 0

~sup

XECl\-:

Hence

lf(x)l 0

XE\2

o

9.11 Theorem (Weierstrass) Let (fj)jeN be a sequence of left monogenic functions in n. If for each compact set K c g and each E > 0 there exists a natural number N(E,K) such that sup lf.(x)- f.(x)l xEK

1

J

0


0 was chosen arbitrarily it is obtained that IA 1 10

faI 10.3

h do f

= 0.

c

Iff e c0 (n;A) is such that for all I e I(g)

Proposition

faI

= 0 or

dof = 0

then for any I e I(n) 0

f(x)

=J

ai

E(y-x)do f(y), for all x e I. x

0

Proof Take I E I(n) and x E I. Choose c > 0 arbitrarily. uous at x there exists an n(c) > 0 such that

As f is contin-

lf(x+o)- f(x)l 0 < c if lol < n.

E(y-x)do f(y) Y

= (J

ai

Y

k

Jai

0

j=1

Y

=J

aJ

k

+

I J J=

By Stokes's Theorem

E(y-x)do [f(y)-f(x)] Y

k

E(y-x)doyf(y) -

. 1 aJ .

62

Y

u JJ. be a partition of I' 0.

11.2.5 Proposition Any left inner spherical monogenic Pk of order k may be written as

where the sum runs over all possible combinations (1 1, ••• ,1k) of k elements out of {1, ••• ,m} repetitions being allowed. Proof The proof is given by induction on k. trivial. Fork= 1 we have to show that

For k

0 the assertion is

m

m

By Euler's formula and taking into account that DP 1 = 0 we have at once m P1(x) = xo ax P1(x) + I x.1 aX. P1(x) 0 1 i =1 m m = xo[op1 - I eiax.P 1(x)] + I X; ax.P 1(x) 1 1 i =1 i =1 m

Now assume the stated formula to be true for k; again by Euler's formula m

(k+1)Pk+ 1(x) = x0 ax Pk+ 1(x) + 0

I i=1

xi ax.Pk+ 1(x) 1

and as DPk+ 1 = 0, 69

m

L zi

(k+1) Pk+ 1(x) =

ax.Pk+l(x).

i =1

1

so by the hypothesis made m

1

pk+1(x) = l..11 ••• 1 k-"0 X

••• X

51

k=O (1 1, ••• ,1k)

5t

"

f(O}J)

••• 0 X

11

1k

converges uniformly on the compact subsets of A' to the null function for any combination (s 1, ••• ,st) E {1, ••• ,m} t • As eo if (s1, ••• ,st) = (11, ••• ,lk)

a

••• a

xst

><s 1

v1

1 (O)

{

1• • • k

0

otherwise

it follows that >..

=Cl

s1 ••• st

xs

••• Cl

1

xs

f(O) t

for all (s 1, ••• ,st) E {1, ••• ,m}t. This implies that the given series (11.5) c coincides with the Taylor series of f in A' and hence in A too. 11.3.7

Proposition

If the series

00

L

L

V1

k=O (1 1, ••• , 1k)

1 (x) >..1 1 1"""k 1"""k,

considered as a multiple power series, converges normally on the compact subsets of the open set A which contains the origin, then it represents a left monogenic function f in A. Moreover the given series is precisely the multiple Taylor series about the origin of f in A. Proof Clearly the sum of the given series is an A-valued analytic function in A and, termwise differentiation being allowed, we get immediately that Df = 0 in A. Now as f is left monogenic in A, there exists an open neighbourhood of the origin A' c A in which f may be developed into its multiple Taylor series 00

f(x> =

L

L

v1

k=O (1 1, ••• , 1 k)

1 (x)

1• • • k

a x1

••• a 1

x1

f(O). k

Hence the multiple power series 00

L

L

k=O (1 1, ••• ,1k) 75

converges normally in A' to the null function. On the same line of argument as in the proof of Theorem 11.3.6 it follows that "11"" .lk = ax

11

••• ax

1k

f(O)

for all (1 1 , ••• ,1k) E {1, ••• ,m}k, whence the given series coincides with the multiple Taylor series about 0 of f in A' and hence in A too. c 11.3.8 Remark Analogous results are valid for the local behaviour of a monogenic function f near a point a I 0. It is obtained that if f is left monogenic in an open set n then for each point a E n there exists a neighbourhood Aa of a, contained in n, such that in Aa f may be developed into a unique normally convergent Taylor series of left inner spherical monogenics 00

f(x) =

I k=O

P f(a) (x)

k

where

m 1 P f(a)(x) =rr k

I 11 =0

m

I k=0

(x(a 1 ) ••• (x 1 -a 1 )ax ••• ax 1 f(a) 1 1 k k 11 k

V~a) 1 ( x). ax ••• a f (a) 1""" k 11 x1 k

with (a) k z (a) 1 ••• z 1 , (1 1 , ••• , 1k) E {1,2, ••• ,m}

1

k

and

11.3.9 Theorem Let n be an open connected set and let f be left monogenic in n. Then the following are equivalent statements: ( i)

(ii)

76

=0

in n; there exists a point a En such that f

for all (1 1, ••• ,1 k) E {1, ••• ,m} k and k E lN. ~

Clearly (i) implies (ii). A= {y E n:ax

••• a 1 x1 1

So let

f (y )

= 0,

V (1

k

1 , ••• ,1 k ) E { 1 , ••• , m} , k E lN }•

k

From the hypothesis of (ii) we know that A I ~. We will show that A is both open and closed in Q; by the connectedness of Q it will follow that A must be n and so f = 0 in n. To see that A is closed in n let y E An n and let (y.) .SN be a sequence in A such that y =lim y .• As each a ••• a f(x) J J j-><x> J x1 x1 1

k

is continuous it follows that f(y) = lim a ••• a f(yJ.) = o, x1 x1 . 1k J->oo 1 k for all k E lN and all (1 1, ••• ,1k) E {1, ••• ,m} k • Soy E A is closed in Q. To see that A is open in n, let y E A and let R > 0 be such that B(y,R) c n. Then in lx-yl < R ax

11

f(x)

••• ax

=

~ k=O

(

L (l 1 , •••

,1k)

0

and, consequently, B(y,R) proof. c 11.4

c

A.

Thus A is open in n and this completes the

Expansions of the Cauchy kernel

11.4.1 In order to solve the problem on the radius of convergence for the Taylor series posed in 11.3.5, the Cauchy kernel E(y-x)

y-x --"--m-+. .1. . , ly-xl

YFx

is developed into a series of inner spherical monogenics in x, depending UROn y, which for each R > 0 converges normally in x E B(O,R) and in y E Rm+ 1,B(O,R). 11.4.2 Starting point is a known series expansion for the potential (m (see [15])

>

1)

77

00

( k-tm-2\

1 Iy-x 1m-1

L \

k

k=O

!x lk

) !Yim+k-1

Lk,m+1()

(11.6)

or 00

1 Iy-x 1m-1

L k=O

k 1 i.:lL<xv} k! ' y !Yim-1

(11.7)

which converges, together with all possible series of derivatives, normally in lxl < R < IYI for each R > 0. Each of the terms in (11.6) is a spherical harmonic in x and moreover is harmonic in y E Rm+ 1 since ~ 1 is the fundaJy I o mental solution of the Laplacian ~m+ 1 and the corresponding terms in (11.6) and (11.7) may be identified. The action of the operator Ox on (11.6) and (11.7) leads to the following expansions of the Cauchy kernel in lx! < R < !YI for each R > 0: 00

E(y-x) =

{1)

w1

1••• 1k

(y)

for (1 1, ••• ,1k) E {1,2, ••• ,m} k , k E ~ 0 • 11.4.3 Clearly Kk,m+ 1,y(x), k E ~. is a left and right inner spherical monogenic of order k in x, and it is easily seen that there exists a constant c > 0 such that m

2n/2

1Ko,m+1,y(x)lo

=~

1Kk,m+1,y(x)lo

~ Cm(1+k2)1xlk, k E ~o·

(11.12)

, (1 1, ••• ,1k) E {1,2, ••• ,m} k , k E Jll, are left and Next all functions w1 1 • • • 1k right monogenic in R~+1 with 1i m W1

X-+oo

1 (X) = 0

1• • • k

and are homogeneous of order -(m+k). Hence identifying the homogeneous parts of (11.6) and (11.7) it is obtained that also the functions 1 (k+m-1'\ 1 wm+ 1 k+1 } IYim+k

K (x) k,m+1,y v1

(11, ••• ,1k)

1••• 1k

(x) .w

11 ••• 1k

(y)

are left and right monogenic in y E R~+ 1 with limit value zero at homogeneous of order -(m+k).

oo,

and are

11.4.4 Let us have a closer look at the structure of the functions w1 1 ••• 1'k defined for all (1 1, ••• ,1k) E {1, ••• ,m}k and k EIN. By definition we have i n Rm+ 1( m > 1) 0

79

(-1)k+1 wm+1 H (x) -[ 11···1k - D lxlm+Zk-1 1

=

m-1

o[ ax

••• ax 11

1 lxlm-1 1k

]

]

where H1 is a real-valued homogeneous polynomial of degree k. 1 • • •1 k

w11 ••• 1k only

takes its values in sp{e 0 ,e 1, ••• ,em}. H (x) 11 ••• 1k g (x) = 11••• 1k lxlm+Zk-1

Clearly

Next the function

1 1· t · homogeneous of order - ( m+ k - 1 ) an d as W · monogen1c · 1n · ,...,m+ 1s ~0 11 • • • 1k 1s follows that g1 1 is harmonic in ~m0 + 1 • Now

1... k

f),

g

m+1 11 ••• 1k

X E

(x)

~m+1 0

,

which means that H1 1 is a spherical harmonic of order k. 1••• k we have that H1 ···1 (x)

w

11 ••• 1k

'x)

= DCI.

. , x) =

-~ r· .1 k

o[ lx 1lm+2-1k

Summarizing

]

where g1 is a real-valued harmonic function in Rm0 +1 homogeneous of 1 •••1k order -(k+m-1) 1 while H1 1 is a real-valued spherical harmonic of order , ••• k k. Furthermore 1 (x) = ·m1Zk+ 1 [ lx I2 DH 1 1 (x)-(m+2k-1 )xH 1 1 (x)] 1"""k lxl 1"""k 1"""k (1, •••• ,,k) Hk+1 = --._.---.-lxlm+Zk+1

w1

(11·····1 k)

where Hk+ 1 is a harmonic homogeneous polynomial of degree (k+1) taking values in sp {e 0 ,e 1, ••• ,em}' which is not monogenic since 80

and

11.4.5

Finally notice the following orthogonality relations:

f

W (x) da W (x) = 0 11 ••• lk x s1 ••• st

as(6,R)

for all k, t E J.l, (1 1, .•• ,lk) E {1, ••• ,m} k , (s 1, ••. ,st) E {1, ••• ,m} t and R > 0. Indeed, by Proposition 9.2 this integral is independent of the radius R and so it equals

J

1i m

R~oo aB(O,R)

W

1 1··· 1 k

( x) da

J R-(k+m) W

lim

R~oo

Sm

W

s1 ••• st

X

11••• lk

( x)

(w) Rm da R-(t+m)W (w) w s 1 ••• st

11.5 The radius of convergence of the Taylor series 0

11.5.1 Let f be left monogenic in B(O,R). We saw in 11.3 that there exists an open neighbourhood of the origin A in which f may be developed into its Taylor series (see (11.2)) 00

f(x) =

L

Pkf(x)

k=O where for each k E J.l (see 11.3))

is a left inner spherical monogenic. This Taylor series and all possible derived series converge normally in A.

81

11.5.2 By Cauchy's Integral Formula (Corollary 9.6) we have for any O 0 only depends upon the dimension m and where R' may be chosen arbitrarily in ]O,R[. 11.5.3 Remarks (i) 82

The expansion (11.2) together with (11.15) constitutes a natural

generalization of the Taylor expansion

where d k 1 [(zdu) u].g(u)du J ;iB( ,R')

for a holomorphic function g of one complex variable.

~ K (y) is left inner spherical Iu I +1 1,m+ 1,u monogenic of order 1 in y. Substituting this function for f in formula (11.14) yields the orthogonality relation (ii)

We saw in 11.4.2 that

Kl,m+1,u(x) Iu lm+1

0

_

kl -

1

w1 m+

(k+m-1\ k+1 ) in~

(iii) In 11.3 we also saw that Taylor series (see (11.4' ))

J aB( ,R)

f may be expanded into its multiple

00

L

f(x) =

L

f ( o) 1 ( x ) ax ••• ax 1"""k 11 lk

V1

k=O (1 1, ••• ,lk)

which, considered as a multiple power series, converges normally in it is well known (see [11]) that '"' ( )_k , _-_1 L. k.'

Iy-x lm-1

<x v

• y

>k

1

lyl m- 1 •

~-

Now

(11.7')

k=O this time being considered as a multiple power series in the real variables x0 ,x 1, ••• ,xm• converges normally for lxl < (/Z-1)IYI· This leads to m m ( -1 ) k E(y) (11.9') E(y-x) = L ---r! L x1 ••• x1 ay ••• a L 1 k 11 y\ k=O 11 =0 1 k=O 00

00

L

L

k=O (11, ••• ,1K)

v

(y) 1 (x) w1 11""" k 1""" lk

(11.10')

83

which, considered as a multiple_power series in the real variables x0 ,x 1, •• ,x~ converges normally for Jxl < (/2-1)JyJ. So by Cauchy's Integral Formula we arrive at 00

f

f(x)

wl ••• 1 (y)day f(y) (11.13') 1 k

as(b,R')

which, still considered as a multiple power series, converges normally in B(0,(/2-1)R). From the uniqueness of the multiple Taylor series expansion (Proposition 11.3.7) it follows that (11.4') and (11.13') coincide in A and 0 hence in B(0,(~-1)R) too. This means that the original multiple Taylor series (11.4') converges normally in B(O,(iZ-1)R). This domain of convergence is indeed smaller than the domain of convergence of the corresponding Taylor series (11.2) considered as a series of left inner spherical monogenics. 0

1~

Laurent series and pointwise singularities

12.1

Outer spherical monogenics

Let the function g:aB(O,R) function~ven in 1Rm+1.-...aB(O,R) by 12.1.1

Lemma

f(x)

=

f

~A

be continuous and consider the

E(y-x)day g(y), Jx I I R.

as(b,R) Then (i) (ii)

f is left monogenic in §(O,R); f is left monogenic in 1Rm+ 1-...8(0,R) with lim f(x) x~

= 0 and admits

in 1Rm+ 1-...8(0,R) the following normally converging series expansion: 00

f(x)

= - L Qkg(x) k=O

where wl

with 84

1 (x) 1"""1(

J.ll

1 • 1"""k

k E I'J,

I

V1

aB(O,R)

1 (y)doy g(y).

1" •• k

Differentiation under the integral sign being allowed we have at once that f is left monogenic as well in B(O,R) as in Rm+ 1,§(0,R). Also it is easily checked that lim f(x) = 0. Reverting to the expansion (11.11) of the Cauchy kerne 1

~

E(x-y) =

I( k=O

~

w1

(1 1 , •••

,lk)

which converges normally in in Rm+ 1,§(0,R)

IYI

00

(

~

f(x) = -

\

k=O

1 (x)

1" •• k

< p


R.

I

Kk,m+1,x(y)doy f(y)

(12.5)

as (o, R' )

In view of (11.12) they also satisfy the Cauchy estimate (12.6)

where the constant C > 0 only depends on m, and R' may be chosen arbitrarily in ] R,+oo[. (iii) We saw in 11.4.3 that the function _1_ (1 +m-1) 1 K (u) wm+ 1 1 +1 IYim+l l,m+1,y is left monogenic in y E R~+ 1 with limit value zero at"" and moreover equals

I (s 1 , ••• ,s 1 )

W

~

1

••• s 1

(y).V s ••• s (u). 1 1

Substituting this function for f(y) in the formulae (12.4) and (12.5) gives rise to

I

I

r

wt

t (x) j vt .• •t (y)doyWs s (y)Vs s (u; k 1 · · · -~ r· · 1 (t 1 , ••• ,tk)(s 1 , ••• ,s 1 ) 1• • • k aB(O,R) 1

I

_ 1 (k+m-1)(1+m-1\ - wr-1 k+1 1 +1 ) m+ aB(O,R) In view of the orthogonality relations (12.3) the left hand side reduces to

87

eSc

=

) (

t1, ••• ,tk , s1, ••• ,sl

)

}:

(t1, .•. ,tk)

wt1 ••• tk (x) vt, ••• tk (u)

0( t 1• • • • • t k ) ' ( s 1' • • • ' s 1 ) ( k+m-1 ) 1 \k1 wm+ 1 IxI m+k +

k,m+ 1,x (u).

K

Consequently the following orthogonality relation holds:

= ____ 1 (k+m-1\ w m+ 1

J

k+1

[ aB( ,R)

12.1.5 In view of the considerations made above it is quite natural to introduce the following definition. Definition A function Qk which is homogeneous of order -(m+k) and left (right) monogenic in R~+ 1 is called a left (right) outer spherical monogenic of order k. Observe that any outer spherical monogenic has limit value zero at oo. As an example we mention the functions w1 1 , (1 1, ••• ,lk) E {1, ••• ,m}k, 1• • • k

all of which are left and right outer spherical monogenics of order k. If f is left monogenic in Rm+ 1,B(O,R) then the function Qkf is a left outer spherical monogenic of order k. The right A-module of left outer spherical monok genies of order k is denoted by Q(r)" 12.1.6 Proposition A basis for the right A-module Q~r) of the left outer spherical monogenics of order k is given by {Wl

l

1• • • k

and dim Qk

: (1 1, ••• ,lk)

E

k {1,2, ••• ,m}}

= 2nM(m,k).

k Proof Let Qk be an arbitrary element of Q(r)" have in Rm+ 1 that

Then by Theorem 12.1.3 we

0

Qk(x)

=

~ S=O

(

L (1 1 , ••• ,ls)

and hence, putting x = rw, 88

Wl1··· ls (x)

~11 ••• ls)

from which it apparently follows that the series on the right hand side only contains the term s = k and so

I (1 1 ' .•• '1 k) To prove the right A-linear independence in Rm+ 1 of the set {W 0

k

11 ••• lk

(1 1 , •.• ,lk) E {1,2, ••• ,m} }, let

I (11, ... ,lk)

W (x) ~ 11 ••• lk 11 ••• lk

0 for all x E Rm0 +1•

Take an arbitrary combination (s 1 , ••• ,sk) E {1,2, .•• ,m}k. (1

I

1 )

(

1•···• k

J aB(O,R)

Then

Vs ••• sk (x)do x W1 ••• 1k (x)\} ~ 1 •• .1k 1 1 1

0

for any R > 0, which in view of the orthogonality relations (12.3) turns into ~

s 1••• sk

=

o.

As the sequence (s 1 , ••. ,sk) was arbitrarily chosen it follows that all constants ~l 1 are zero. o 1• • • k 12.1.7 Let Qk be a left outer spherical monogenic of order k; then by 11.4.4 Qk can be expressed as

or Q k

D[

Hk

l

\x\m+2k-1J

or still

Where g is an A-valued harmonic homogeneous function of order -(m+k-1) in 89

Rm+ 1, Hk is an A-valued harmonic homogeneous polynomial of degree k and Hk+l is an A-valued harmonic homogeneous polynomial of degree (k+1) satisfying x.Hk+ 1(x) (m+2k+1) - - - JxJ2

D Hk+ 1(x) Moreover as

and

the left outer spherical monogenics of order k are eigenfunctions of the in R~+l with respective eigenvalues (k+m) and (-k). operators rand

r

12.1.8 The restriction to the unit sphere Sm of a (left) outer spherical monogenic Qk(x), notation Qk(w), is called a (left) surface outer spherical monogenic. From the considerations made in 12.1.8 it follows that

which means that Qk(w) is an A-valued surface spherical harmonic of order (k+1) and hence an eigenfunction of the Laplace-Beltrami operator ~~+ 1 with eigenvalue -(k+1)(k+m), i.e.

12.2 Laurent series 12.7..1 Here the behaviour of a function which is monogenic lar domain is investigated. The aim is to obtain a Laurent where the inner and outer spherical monogenics will play an as the positive and negative powers of the complex variable classical case.

in an open annuseries expansion analogous role z do in the 0

12.2.2 Theorem Let f be left monogenic in the annular domain G = B(O,R 2 B(O,R 1 ) (0 < R1 < R2 ). Then in G the function f may be expanded into a unique Laurent series

90

00

00

k=O

k=O

)'

where both series converge normally in B(O,R 2 ), respectively Rm+ 1,B(O,R 1 ). The left inner spherical monogenics are given by

with

J

W11 ••• lk(y)do/(y), (l 1 , ••• ,lk) E {1, ••• ,m}k

aB(O,R) while the left outer spherical monogenics are given by

with lJ

l = ( V1 ••• l (y)doy f(y), (1 1 , ••• ,lk) E {1, ••• ,m}k 11··· k aB(b,R) 1 k

the radius R being arbitrarily chosen in ]R 1 ,R 2 [. Proof Take x E G arbitrarily; then there exist R~ and R~ (R~ < R~) such that G' = B(O,R~),B(O,R') c G and x E G'. By Cauchy's Integral Formula

--0

-

f(x) =

J

aG'

E(y-x)do

Y

f(y)

or

with

I

aB(O,R~)

and f2(x) = -

I aB(O,R')

Conclusion by Lemma 12.1.1, Theorem 11.3.6 and Proposition 12.1.7.

o

12.2.3 Remark Assume that f is left monogenic in the annular domain G = §(O,R 2 ),B(O,R 1 ) (0 < R1 < R2 ). Then we just sa~ that f may be expanded 91

into a Laurent series of left inner and outer spherical monogenics which cono m+1 verge normally in respectively B(O.R 2 ) and m 'B(O.R 1 ). Just as in the case of the Taylor series expansion (see 11.5.3) it is possible to expand f into a so-called multiple Laurent series, where the homogeneous terms are not bracketed together. and where the domain of convergence is severely altered. In addition the annular domain has to satisfy a supplementary condition, which says that there exists an open subset 0 G' = B(O,R~ )'B(O.RU with R1 < R~ < R~ < R2 • such that ( /2"+1 )R~ < ( /Z-1 )R~. The expansion of the potential

'I --rr(-1 , k 0 such that f E M(r)(B(a,R),{a};A). 12.3.2 Let a E 1Rm+ 1 be an isolated left singular point of the function f; then in an annular domain centered at a f may be developed into a Laurent series

k=O

k=O

Definitions (i)

The isolated left singular point a off is called a left pole of order p if the second series in the Laurent expansion of f about a breaks off from k ~ p. This breaking off second series is called the singular part of f at a. (ii) The isolated left singular point a off is called a left essential singular point if the second series does not break off. 12.3.3 Definition A function f is called (left) meromorphic in an open set Q if there exists a subset S c Q such that (i) (ii) (iii)

S has no accumulation point in Q; f is (left) monogenic in ~S; f has a (left) pole at each point of S.

Notice that condition (i) implies that no compact subset of Q can contain an infinity of points of S; this means that S is at most countable. So if f is left meromorphic in 1Rm+ 1 there are three possibilities accordingly as the set S of left poles of f is empty, finite or countable.

94

12.3.4 Consider a function f which is left meromorphic in Rm+ 1 If the set of left poles is empty then the function f is called left entire. Theorem

If f is left entire then f admits the expansion

I (

f(x)

k=O

v1

L

(1 1 , ••• , 1 k)

1 (x).a

1""" k

~

f(o>)

••• a

~

1

k

converging normally in the whole space Rm+ 1 • Proof The Taylor series expansion off about the origin is valid in the -m+1 whole spaceR • c 12.3.5 Consider a function f which is left meromorphic in Rm+ 1 and suppose that the set of the left poles off is the finite set {a 1, ••• ,aj}. If pi denotes the order of the left pole ai and Gi(x) its singular part then j

L

f -

Gi

i =1

So f admits in Rm+ 1 the following expansion:

is left entire.

I ((1

f ( x)

k=O

L

vl 1 ••• 1 k ( x) • "1 1 ••• 1

1 , ••• ,lk)

J

. p.-1 J

1

(

I I .

+

1

(a.)

I

=1 k=0 (1 1 , ••• , 1 k)

(.)

)

w.1 1 1 Cx>.~l1 "" 1···k

1 •

1"""k

12.3.6 For meromorphic functions with a countable set of poles we prove Mittag-Leffler's Theorem. Theorem Let {a.}.ElN be a sequence of distinct points in Rm+ 1 without an 1 1 accumulation point in Rm+ 1 , such that with each point ai there corresponds a natural number pi and a function p.-1 G.(x)

=

1L· (

1

L

\

k=O

(1 1 , ••• ,lk)

w(ai) 11 ••• lk

(x)~(i)

)

11···lk.

Then there is at least one left meromorphic funct:on f in Rm+ 1 the left poles 95

of which are exactly the points a;• off at a; is G;(x).

E ~. and such that the singular part

Proof Order the points a; in such a way that their respective Euclidean norms form a non-decreasing sequence

o~

Ia 1 I ~ Ia2 I ~ • • •

For each i E ~a suitable polynomial Z; will be constructed such that the series 00

is the desired left meromorphic function. If possibly a 1 should be the origin then take z1 = 0. For each a. the singular part G; is left monogenic in 1 Rm+1 '{a;}· Take a and a' to be fixed and such that 0

R' or a'alajl

>

R which means

that B(O,R) Put

c

Bj.

Now define two functions f1 and f 2 in the following way.

t

f 1(x) =

L (Gi(x)-Zi(x));

i=1 0 clearly this function f1 is left meromorphic in B(O,R) with left poles at a 1 , ••• ,at' (t' ~ t) and corresponding singular parts Gi(i = 1, •.• ,t'). Next put 00

L

(Gi(x)-Zi(x)),

i=t+1 where it is known that for each i

>

t

sup !Gi(x)-Zi(x)\ 0 XEB(O,R) 00

00

L (Gi-Zi) will converge normally on 2 i=t+1 i=t+1 B(O,R); so by 0 the Weierstrass Theorem (Theorem 9.11) f 2 (x) will be left monogenic in B(O,R). Hence the function As

L

~converges the series

00

fl + f2 =

L

(Gi-Zi)

i =1 0

is left meromorphic in B(O,R) with left poles at a 1 , ••• ,at, and corresponding singular parts Gi(i = 1, ••• ,t'). Now let R->-+oo to obtain the desired function. c 12.3.7 Remark The structure of the most general left meromorphic function f in Rm+ 1 with prescribed poles and singular parts is

where h is left entire. 12.3.8 Definition Let f be left monogenic in the open set n except for the left singular point a E n. This point a is said to be a removable left sin9ularity of f if there exists a left monogenic function h in n such that h(x) = f(x) for all x E ~{a}. To determine whether or not a left singular point is removable the following criterion is available. 97

12.3.9 Theorem Let n be an open set containing the origin and let f be left monogenic in ~{0}. The isolated left singularity at the origin is removable if and only if there exist R > 0 and M > 0 such that for all xE{y:O

1

-u- or Iu IIY I > 1

1 Du lulm-1 • 1-m

(1)

k,m+1,y

>

( 13. 1)

1, or still

I

s(y,u) =

k=O ( 1 1, .•• , 1 k) On the contrary ifM> IYI or luiiYI -s(y,u) =

I k=O




f(.Y)day E(y-x)

aB(O,R-d 1

and thus in I u I

>

R'

I

f{.Y)doy E(y-~)

I aB(O,R-E:)

f{.Y)doy s(y,u)

sf(u) =



0 is suitably chosen.

13.4 Let us give two examples of spherically transformed functions. 13.4.1

For the constant function f(x) = e0 we have

13.4.2 Let Pk be a left inner spherical monogenic of order k; then sPk(u) = (Pk =

(*)

(i5 k(u)

5

lul~-1)1-~ 1

lulm+2k-1

),5u

-m

5 = Pk(u).( - ( ) = Pk u.

1 ) ---1 u lulm+Zk-1 -m

luI

um+2k+1

m+2k-1 m-1

which is clearly a right outer spherical monogenic of order k. 13.5 Making use of the expansion (13.2) for s(y,u) it is obtained that sf(u) =

I

L

(

I

f{Y)daywl

k=0(1 1, ••• ,lk) ClB(O,R-E:) 1

(v

1 (y) \ 1 1 (.!) 1""".k 1 1··· ku _

°u)

Iu Im-1 1-m • 104

But as

I

I

r\

(l1' ... ,lk) aB(O,R-d

L L (l 1, •.•

a

I

((-1)m

(1 1, ••• ,lk)

B f'(y)aaywl, ••• l (y)\vl (,) , u k ) , ••• lk iulm-1 1-m

f(,Y)day-wl

aB(O,R-E)

(\

J

f{Y)day(-1)kw 1

,lk) aB(O,R-E)

(~) ~-1

1 CY))vl 1 1··· k 1··· k u lui

10-um

B (y)\v (1) 1 u 1••• 1 k ) 11••• 1k u Iu Im-1 1-m

this expansion for sf(u) in lui >*reduces to sf(u) =

L

sPkf(u)

k=O which is obviously the Laurent series expansion of sf(u) in mm+l,B(O,*) 13.6 Assume that g is left monogenic in ffim+l,B(O,R) with lim g(x)

= o.

X-><x>

Then g(x) is right monogenic in mm+l,B(O,R) with lim g(x) X-><x> function

= 0, and the

°u

1 1 sg(u) = g(-) 1 1-m ii !ulm-

is right monogenic in B(O,*). It is called the spherical transform of g. To put it in an integral form we proceed as follows. By Cauchy's Integral Formula outside a ball

9(x)

-I

g{.Y)doy E(y-x)

aB(O,R+E) 1 and thus in Iu I < R sg(u)

J

g(Y)day E(y-~)

aB(O,R+E)

I

1 !ulm-1

Bu 1-m

g(y)day s(y,u)

aB(O,R+E) 105

where E > 0 is suitably chosen. 13.7 Let us give two examples of spherical transforms of left monogenic functions outside a ball.

;:~ e0 •

13.7.1

For g(x) = W0 (x) = E(x) we get sg(u) = - 1-

13.7.2

Let Qk be a left outer spherical monogenic of order k; then

~+1

B

sQk(u) = (Qk(i)

lul~-1)1-~

_ - ( )-I m+2k-1 m+2k+1 - 0k u u u 1 1-m which is clearly a right inner spherical monogenic of order k. 13.8 Making use of the expansion (13.4) for s(y,u) it is obtained that sg(u)

i

=

r

( J

9(Y)dcry\ ••• 1, (y)(w], ... J,

k=O(l 1, ••• ,lk) aB(O,R+E)

(~) 1•1~-1 ,n:'.}

But as 1 1 °u sQkg(u) = Qkg(-_) 1 -1u lulm-m

( \

J

(1 1, ••• ,1 k) aB(O,R+t:)

(\

l:

J

(1 1, ••• , lk) ClB(O,R+E)

-

-

g(y)aayvl

\(1 1 °u 1 ;w..1 1 -=-> 1 -11··· k - 1··· k u lulm-m

k (y)\(_ 1)kw 1 1 °u 9G>doy-,1 ••• 1k

z, ••• z, A. . 1 k 11 ••• 1k'

then

;;

sup L I IV 1 1 (x)l 0 IA. 1 ••• >-. 1 10 EC 1• • • k 1 k x m+ 1 k =M ( 1 1 , ••• , 1 k ) N m m

;;

I xECm+1 k=M

m

N

L ••• L lk1

L •• • L

IT k=M

Jr

1 =1 1

lk=1

pl ... pl IAl

1 =1 1 k

lz, 1 10 • •• 1z, k 10 1>-, 1 ••• 1k I0

k

I0

••• 1

1

k

which tends to zero if inf(M,N) .... + oo. By Weierstrass's Theorem (Theorem 9.11) there ought to exist a function f* E M(r)(Cm+ 1(o,2-m/ 2p);A) such that f*{x) = lim SN(x) N-+oo

109

uniformly on each compact subset of C 1(o,2-m/ 2p). + -m/2+ m+ that f*(O+x) = f(x) in cm(0,2 p). c

Moreover it is clear

n

14.3 Definition If cRm is open then an open neighbourhood n of Din Rm+ 1 is said to be x 0 -nor'1Ti.al i f for each x E ~2 the 1 ine segment {x+te 0 } n \~ + is connected and contains just one point in n. + m 14.4 Proposition Let~ cR be open and let f be an A-valued analytic function in Then there exist a maximal x0 -normal open neighbourhood Q of 1 in Rm+ and a unique f* E M(r)(st;A) such that f*(O+x) = f(x).

n.

n

n;

Proof Let y be an arbitrary point in then there exists p* such that f(x) may be developed into its Taylor series in Cm(y,p*). So by the above Theorem 14.2 there exists a function f~ E M(r)(Cm+ 1(y,p);A) such that f;(o+x) = f(x) in Cm+ 1(y,p) n Rm

= Cm(y,p), with p = 2-m/ 2p*. Moreover if = f; in cm+ 1(y,p) n cm+ 1(i,p'). Indeed, o in cm(y,p) n cm(z,p') and as the function

cm+ 1(y,p) n cm+ 1(z,p') 1 ¢then f; (f;- fi)(O,x)

= f(x)-f(x)

f;- fi is left monogenic in Cm+ 1(y,p) n Cm+ 1(y,p') it follows that f~-ff vanishes in this connected set. So it is possible to gather the extensions to one left monogenic extension of f in the x0 -normal neighbourhood +u+ cm+ 1cYl yHl

of

+

s-2.

Next, if Q1 and Q2 are two x -normal neighbourhoods of Q and f~ , f~ 0 ''1 >12 are the corresponding left monogenic extensions of f, then Q1 u Q2 is again an x0 -normal neighbourhood of~- As f5 1 ID 1 n nz and f5 2 IS11 n nz are left monogenic extensions of f(x) in n 1 n nz, it follows that

which means that both extensions may be gathered to one left monogenic extension in n1 u rlz. Hence the union of all x0 -normal neighbourhoods of Q in which there exists a left monogenic extension of f is the maximal x0 -normal neighbourhood of Q with a left monogenic extension. Clearly both this neighbourhood and the extension are unique. c 110

14.5 Definition Let S1 clRm be open and let f be an A-valued analytic func. + tion 1n Q. The maximal left monogenic extension f* of f, as constructed in proposition 14.4, is called the left Cauchy-Kowalew3ki exten3ion (C-K extension) of f. 14.6 Corollary Let f be an A-valued analytic function in the whole of IRm. Then its (left) C-K extension is a (left) entire function. 14.7 Remark Iff and g are A-valued analytic functions in IRm and A E A then the left C-K extension of f + gA is given by (f+gA)* = f* + g*A· 14.8 Theorem Let f be an A-valued analytic function in~ clRm. function f* given by 2k+1 oo k [ X k ] f*(x) I (-1) 5 (Zk+ 1) 1 ~mf(x) k=O

Then the

is left monogenic in a neighbourhood Q of S1 in 1Rm+ 1 and satisfies f*(O+x) f(x) in S1. If moreover f is analytic in the whole of IRm then f* is left entire.

n;

Proof Let K be a compact subset of CK and AK' depending upon K, such that

then there exist positive constants

sup ~~~f(x)l 0 ~ cK(2k)!A~ XEK

111

In this way it is easily seen that the given series and all derived series will converge normally in

- , _1_[=() IrK IrK

0

Kx]-

U

OGo

Kc::;Q

This means that f* E E(r)(n;A).

Clearly f*(O+x) 2k+1 k [ xo k J (-1) ~+ 1 (Zk+,J! ~f(x)

00

of*(x) = ~ k=O 00

( -1)k

~

2k-1 xo (2k-1)! ll~f(x) +

k=1

f(x) and in n

00

l:

( -1)k

2k+1 xo 1 (2k+1)! ll~+ f(x)

k=O

o. In the particular case where f is analytic in the whole of Rm then for any compact set K c: Rm and any >. > 0 there exists a cK,>. > 0 such that la!Akf(x) I

sup

X€K

XIll

~

CK ,(2k+IBI)!>.k. 'A

So the cited series and all derived series will converge normally in Rm+ 1• c 14.9 Remark It follows from the above theorem that if f is a real-valued analytic function inn c:Rm, then its C-K extension takes values in sp{e0 ,e 1, ••• ,em} (see also 11.2.7(ii) and Theorem 11.3.4.). 14.10 Proposition Let f be an A-valued analytic function inn c:Rm and let f* E M(r)(Q;A) be its left C-K extension. Then for any multi-index s E ~m. a!f*(x) = [asf(x)J* in n.

X

X

-+

Proof The function a~f is analytic in n and possesses a left C-K extension ~* in a certain x~-normal neighbourhood Q' of Q. On the other hand the

fu~ction

a!f* is left monogenic in Q and its restriction to n is easily shown X

-+

to be precisely a~f. In view of the uniqueness of the C-K extension, Q c: Q' X and the statement follows. c 14.11 112

Now assume f and g to be left entire functions; then their restrictions

fiRm and giRm are A-valued analytic functions in Rm.

so the product

remains analytic in Rm. It therefore possesses a left entire C-K extension · ex t ens1on · · denote d by in Rm+ 1• Th 1s 1s

and called the Cauchy-Kowalewski product (C-K product) of the left entire functions f and g. Of course an analogous definition holds for right entire functions; the corresponding notation is GR. 14.12 (i) (ii) (iii) ( i v)

14. 13

Remark The C-K product shows the following properties. The C-K product is associative. If fiRm. giRm = giRm. fiRm then fG Lg 1elf = fQL1 =f. m+1 M(r)(R ;A) ,+,GL is a real algebra.

gGLf.

Examp 1es

14.13.1 As the C-K extension of a Clifford number A is A itself, it is clear that AQL~ = AGR~ = A~, YA,~ E A. 14.13.2 The C-K exter.sion of X; (i = 1, ••• ,m) is z; = xieo- x0 e;; the C-K extension of x.x. is ~(z.z. + z.z.) if I j or z~ if i = j; this means that 1

zielzj

J

c.:

1

1

J

J

1

Zi@RZj = zr(zizj + Z/i)

1

Vij(x) if i f j

and z 1.Glz 1. = z.GRz. .. (x) if i = j. 1 1 = z~1 = 2! v11 As the left- and right C-K product of Z; and zj coincide, the subscripts L and R may be dropped. Moreover the C-K product of z; and zj is commutativ~ (see 14.12(ii)) and so we arrive at n1 nm n 1 ! ••• n ! v1 z 1 G •.• G zm 1 ( x) m 1" •• k where n; stands for the number of times that i appears in (1 1, ••• ,lk). 113

14.13.3 Further we have

where ni and ni are the number of times that i appears in (1 1, ••• ,lk) and (s 1, ••• ,st) respectively. 14.13.4

Iff and g are left entire functions with Taylor expansions

f(x) = ~L k=O

(

\' L

(1 1, ••• , 1k)

vl

1 (x)ax

1• • • k

1

••• ax 1

f(o))

1 k

and

vs ••• st (x) axs ••• axs g(o)) 1

g(x)

1

t

then their left C-K product is expanded in the following manner: m , oo f IT (ni+ni\lv fGLg(x) = L L L l. ni JJ 11 ••• lks 1••• st(x) k,t=O (1 1, ••• ,lk)(s 1, ••• ,st) 1=1

a

xl

••• a

xl

1

f(O).a k

xs

1

••• a xs

g(O) t

14.4 Remark Assume that f is analytic in the whole of Rm; then it admits in Rm a Taylor expansion which can be written in the following forms: m m 1 f(x) = I TI I k=O 11=1 ....

00

00

k=O (1 1, ••• , 1 k)

1 A ••• x1 k 11••• 1k l nlxl n1···· m· 1

00

k=O n1+••• +nrn=k

n1 •I • • • nm·I

n1 n m,* x1 • • • xm /\n

n

1• • • m

where ni stands for the number of times that i appears in the combination (1 1, ••• ,1k) and the meaning of the An* n is obvious. We already know from 1• • • m

Corollary 14.6 that the left C-K extension f* of f is a left entire function. Its Taylor expansion about the origin holds in the whole of Rm+ 1 and reads 114

f*(x)

m

1

00

I

I

TI

k=O

m

1 1=1

v1 (x) ~ I 1 •• •1 k k=O (1 1 , ••. ,1 k)

k=O (1 1 , ••• ,lk)

)I

11 •••1 k

n1! ... nm!

~ k=O

14.15 Proposition (Leibniz's Rule) then for any multi-index RE ~m

1~( feL g) X

I ( ! ) a~feL

=

++

o~

B

a

X

Iff and g are left entire functions

a!-C:g. X

Proof In view of Proposition 14.10 and Remark 14.7 we have consecutively, denoting the C-K extension by *,

a~

(fGLg) =

+

r!) ...,af"'"'L

I

c;c:s \a 15. 15.1

l • m

~

p(x,a)

k

m

I

zl,"""z\ ali ••• alk

lk=1

k!

vl

1 (x) al ••• al •

1... k

1

k

127

But V.l

-i,-

1 ••• 1 ( x) = rtT zl Q • • • 1· • • k 1· nm · 1

zl

9

k

where ni stands for the number of times that i appears in (1 1 , ••• ,lk), and so, using 14.13.2, p(x,a)k =

I (l,. •..• lk)

n

k! I

n

zl

I

1• • • • m•

Q

•••

1

Q zl

k

a 1 ••• a 1 1 k

m

I

=

l,. •••• lk=1

zl

(!) •••

1

® zl

k

al ••• al • 1 k

This results finally into -+ k -+ k® p(x,a) =( m I z1 a 1)kQ = p(x,a) • 1=1

So we may conclude that the k-th power of the function p(x,a) is a left and right inner spherical monogenic of order k. 15.7.4 So we are led to consider series of the form

Y p(x,a)k "k

( 15 .1)

k=O which, if the coefficients >.k are chosen to be real, will represent left and right monogenic functions in their respective regions of convergence. As far as the investigation of the convergence of (15.1) is concerned we shall consider (15.1) as well as a series of spherical monogenics and as a multiple power series in the real variables x0 ,x 1, ••• ,xm. Proposition expansion

If the function f of one complex variable with Taylor series

k=O is holomorphic in the open disc {~ E [: and right inner spherical monogenics 128

1~1
l~ xE(C)

-+ -+ 2 sup [<x,a> xE(C)

+

x~I;I2Jk/2

00

I k=O the latter numerical series being convergent due to the holomorphy of f in

kl

< p.

( i i)

As

p(x,a-+)k

-+ -+ (<x,a>e 0

-

x0 -+a) k 129

the series (15.2), now considered as a multiple power series in the real variables x0 ,x 1, ••• ,xm, reads: k 1_ k0 k0 k1 k k k k 1••• ama_,. 0 • 'L. lc k I 'L. ( 1) m k0 ! ••• km! xo x 1 • · •xm a 1 m k=O k0 +••• +km=k oo

But as

a..2

_,. _,.

(153) •

_,. 2

(ala)=- lal, we have

(-1)slal 2s a

if

k

2s

if

k = 2s+1

k 0

which implies that a

is in sp{eo,e1, ••• ,em}; moreover

So investigating the absolute convergence of (15.3) leads to the series k k1 k k1 k k lx 0 1 °lx 1 1 ••• 1xm1 mla 1 1 ···laml mlal 0 •

00

2 k=O

(15.4)

It is clear that if x belongs to the region (B) then

or

L

k1 k

I

•

k

I

IX 0 I

k0

k

k1

k1

k

Ia I 0 IX 1 I Ia 1 I ... Ixm I mIam I

ko+••• +k m=k o· • • • m· which means that (15.4) is convergent in (B).

k

k

m< p

c

15.7.5 Remark The regions (A) and (B) introduced in the above proposition are optimal. Geometrically spoken they can be interpreted as a cylinder and a pyramid respectively.

130

15.7.6 The function ~

e r;; --

1

TI

L

r;;

k

k=O is holomorphic in the whole of [.

So the function

00

L ~

(z1a1

+ ••• +

k=O is left and right entire.

zmam)k It is denoted by

exp(x ,a). For a= (1,1, ••• ,1) E Rm the function exp(x), already introduced in 15.2 is obtained. Notice that

L kT1

00

exp(x,a) =

( <x,a>e -+ -+

0 -

x0 -+)k a

k=O

15.7.7 The expansion 00

I

1-r;;

holds in

r;;

k

k=O

lsi


e0

-

x0 ~a

~ ~ )2 ~ 2 (1- <x,a> + x02 1a1

which obviously is defined in

However the convergence of

L

00

~->-

(<x,a>e 0

-

~k

x0 a)

k=O considered as a multiple power series in the real variables x0 ,x 1, ••• ,xm only holds in the smaller region

15.8 Now we construct a generalization of the function ~u/~:, (u,d

E

a:

x 0: 0 •

15.8. 1 Consider the following functions appearing in the integral expressions of the inner and outer spherical monogenics constituting the Laurent series: qk(u,y)

(-1)k

k

-

= ~ --~Y~(~m+~1 m

L 1 k=0

u 1 • ··~ dyl •.. 'dyl E(y) 1 k 1 k

V (u) W (y) 11"""lk 11"""1k

I (11''""'lk) _ (k+m-1) 1 k+1 IYim+k -

w

11"""lk

(y) v

11···lk

( u)

K (u) k,m+1,y •

we know that those functions, which take their values in sp{e 0 ,e 1, ••• ,em}' 132

are left and right inner spherical monogenics in the variable u, while they are also left and right outer spherical monogenics in the variable y. Also recall that qk(u,y) is a direct generalization of the classical expression (-1)

k

K!

(u d )k 1 ~ z;;

where u and z;; are complex variables. for the Clifford norm of qk: Jqo(u,y) lo

By (11.12) we have at once an estimate

2n/2

=

IYim

and jqk(u,y)Jo

C (1+k2) (k+m-1) - m k+1 5

k

lui~

?

'

k E ~o·

15.8.2 So we are led to consider series of the form 00

L >..kqk(u,y) k=O which, if the coefficients >..k are chosen to be real, represent left and right monogenic functions in both variables u and y separately in a certain region, and which moreover take their values in sp{e0 ,e 1 , ••• ,em}. Proposition expansion

If the

f~nction

f of one complex variable with Taylor series

00

f ( z;;)

I

k=O is holomorphic in the open disc {z;; E C

Jz;;J < R}, then the series

00

L

JckJ qk(u,y)

k=O converges normally in the region {(u,y) E Rm+ 1

x

R~+ 1 : JuJ

< R

Jyj}

to a left and right monogenic function in u and y separately.

133

Proof Obviously

I

lckl lqk(u,y) 10

k=O -:;

em

00

1

lckl vm(k) l~lk' y

I

IYim k=O

where vm(k) is a polynomial of degree m in k. Hence the series converges normally in {(u,y) E 1Rm+ 1 x 1R~+ 1 : ~~~ < R}. c Now take the entire function

15.8.3

00

1

TI

k z:;.

;

then

is holomorphic in u E [ and z:;. E [ 0 separately. By the above Proposition 15.8.2 this is immediately generalized to the function 00

H(u,y) =

L

1

TI qk(u,y)

k=O which is thus left and right monogenic in u E 1Rm+ 1 and y E 1R~+ 1 separately with lim H(u,y) = 0, and takes its values in sp{e 0 ,e 1 , ••• ,em}. y->«>

15.8.4 A straightforward computation shows that H(u,y) satisfies the equations lul 0,1, ••• ,m (15.5) d H(u,y) H(r TuT,y)dr 0, 0 Ui while

J

-

H(O,y) = qo(O,y) = IY~m+1 • Conversely the function H(u,y) i~ completely determined by those equations and that initial value. Indeed it can be proved that Proposition Let f(u,y) be an A-valued analytic function in (u,y)ERm+l xlRm+l 0 134

which is left monogenic in u and right monogenic in y, and satisfying the equations 0

and the initial condition

-

y

f(O,y) Then f(u,y)

IY lm+l •

= H(u,y).

15.8.5 Remark Denoting by r u the spherical Cauchy-Riemann operator acting on the variable u, we have, as H(u,y) is left and right monogenic in u, that

and H(u,y)(ar

"'* r r u )w

1 +-

0,

whence r uH = Hr"'*u = - r

arH.

So for the function G(u,y)

r Iu I

1

= TUT Jo

u

H(r TUT,y)dr

appearing in the equations (15.5), we have

I lui 0

=-

(-r a H)dr r

H + G

or (1 - ru)G(u,y)

H(u,y)

and analogously G(u,y)(1 - "'* ru) = H(u,y). It then follows that the function G(u,y) is left and right monogenic in 135

u E Rm* 1; notice also that the equations (15.5) yield the left and right monogenicity of G(u,y) with respect to the variable y. 15.8.6 Now we give an estimate for the function H(u,y). Proposition Let 1u1 > IYI and let s E ~. . Rm+1 x Rm+1 such that 1n 0

Then there exists a constant Cs 1

~ Cs(1 + 1~1 )s+m - - e y IYim

>

I _Yu I

Proof As

and the left inner spherical monogenics are eigenfunctions of the spherical Cauchy-Riemann operator, we have

and hence, using the Cauchy estimate (11.17) 1(1- r )sH(u y)l u ' ~ C'

~

C'

< =

C ~ (1+k2)(1+k)s (k+m-1) lulk k~O k! \ k+1 ~

( +k)s+m ~k L 1

oo

k=O

k!

IYim+

m+s ( oo ) 1 d Iu Im+s L k\ 1~1 k IYim dlulm+s \ k=O 1

( m+s

:;; C' --m \ L cm:s) IY I j=O J :;; C _1_ ( 1 + 1-Yu I) m+s s IYim

136

(m+s)! 1-yulm+s-j) eiYI (m+s-j)!

J~l.

o

0

Notes to Chapter 2 In his book [Di] Dinghas writes: "Die Grundlegung der Funktionentheorie einer komplexer Ver~nderlichen als einer zusammenh~ngenden und selbstst~ndigen Disziplin is im wesentlichen das Werk von Cauchy, Riemann und Weierstrass". In this sentence three distinct approaches to holomorphic function theory of one complex variable are implicitly mentioned, namely the Weierstrass approach based on power series, the Cauchy approach based on complex differentiability and the Riemann approach based on the so-called Cauchy-Riemann equations. As is well known these three approaches are equivalent. Generalizations of the theory of holomorphic functions of a complex variable have ever since been developed along one of these three lines. In [L] P. Lounesto has beautifully analyzed how Clifford algebra can be used to generalize the Cauchy-Riemann operator. Let us briefly recall the three possibilities he mentions. Taken= 2, consider the Clifford algebra C(v 2 , 0 ) and introduce the operator e 1 + e 2 a~ • If one considers mappings f from c1 into c1, i.e. f: xe 1 + ye 2 ~ ue 1 + ve 2 satisfying the condition (e 1 a~+ e 2 a~) (ue 1 + ve 2 ) = 0 (i), then conformal maps of c1 are obtained unless the derivative of the map is zero. On the other hand, if mappings f from c1 into c0 ~ C~ are considered, i.e. f: xe 1+ye 2 ~ ue 0+ve 12 • satisfying the condition (e 1 ax+ e 2 a~)(ue 0 + ve 12 ) = 0 (ii), then the classical Cauchy-Riemann equations are obtained. Finally if spinor valued mappings f on c1 are considered. i.e. f:xe 1 + ye 2 ~ uf 1 + vf 2 , where f 1 = ~1 (e 0 + e 1) and f 2 = ~1 (e 2 - e 12 )

:x

:x

satisfying the condition (e 1 + e 2 a~)(uf 1 + vf 2 ) = 0 (iii), then it comes out that (iii) is equivalent to (i). Clifford algebras constructed over an m-dimensional real orthogonal vector space (m > 2) thus seem to be very appropriate to define differential operators in higher dimension which generalize the Cauchy-Riemann operator. Moreover these operators satisfy nice symmetry conditions. Note that if we put V = R6,m and if Y = R~,m + R6,m (= R + V) is the orthogonal space provided with the quadratic form X= xo +X~ (xlx) =X~ - 2, then the generalized Cauchy-Riemann operator D introduced in Section 8 is invariant under the group Spoin(Y) = Spoin(O,m) where Spoin (O,m) ~Spin (m+1,0). In this context also the paper [SW] by Stein-Weiss should be mentioned. in which systems of partial differential equations with constant coefficients are considered generalizing the classical

x

137

Cauchy-Riemann equations and being invariant under irreducible representations of the rotation group SO(n). Observe that Fueter, when building up a theory of regular functions of a quaternionic variable, already generalized the Cauchy-Riemann operator by introducing the operator D = a~+ i a~+ j a~+ k a~ (see [F]). As was demonstrated by Sudbery in [S] the definition proposed by Fueter of regular functions of a quaternionic variable, namely as null-solutions of the equation Df = 0, was the only plausible one to get a class of functions which would generalize the class of holomorphic functions of a complex variable in an appropriate way. Indeed, if a function of a quaternionic variable is called regular in an open subset n of R4 if it can be represented by a quaternionic power series about each point of n, then all ~-valued analytic functions in n are obtained, clearly a too large class when compared with the situation in the plane. On the other hand, if a function f:n +~is said to be regular in n if it admits a quaternionic derivative at each point of n, then only constant and linear functions remain, clearly a too small class of functions when again compared with the situation in the plane. Analogous statements hold in the Clifford algebra setting so that again only the Riemann approach is left. This situation clearly differs from the one in holomorphic function theory of several complex variables since there Hartogs's Theorem ensures the equivalence of the Weierstrass and Riemann approaches. Fueter and his students developed quaternionic analysis up to the fifties; a complete bibliography of their work may be found in [Hae 1], while a survey of the basic results of this theory is given by Deavours in [Dea]. More recently Sudbery (loc. cit.) and GHrsey-Tze in [GT] again payed attention to quaternionic analysis. Fueter's students Bosshard and Nef started a hypercomplex function theory in the framework of Clifford algebras in respectively [B] and [N], but only in the middle of the sixties this approach was taken up again independently by Iftimie in [I], Hestenes in [Hes], Delanghe in [Del 1] and [D~ 2] and Hile in [Hi]. More or less recently similar investigations were started by Goldschmidt in [G], Lounesto in [L] and Ryan in [Ry 1]. The latter author uses in [Ry 2] r.omplex Clifford algebras to study nullsolutions of a complex extension of the Cauchy-Riemann equations introduced in Section 8. Needless to say that Cauchy's Integral Formula plays a central role in function theorY· 138

It was shown by Habetha in [Hab] that, if one wishes to generalize cl ass·;cal function theory by considering algebra-valued functions in such a way that a 'simple' Cauchy Formula still holds then one has to restrict to the algebra of complex numbers, the algebra of quaternions or a Clifford algebra. Observe that Cauchy's Integral Formula (Theorem 5.9) is valid for open bounded domains in Rm+ 1 having sufficiently smooth boundary while in the theory of several complex variables it is normally given for special domains, namely polydiscs. Attempts to overcome this problem led to the integral formulae of BochnerMartinelli and Bergmann-Weil, whereby either the holomorphy of the kernel was lost or the applicability was again restricted to special domains (see [BT 1]. Only some ten years ago, to be more precise in 1970, Henkin and Ramirez succeeded in constructing an integral representation formula having a holomorphic kernel and this for functions which are holomorphic on a pseudoconvex domain (see [Hen] and [Ra]). Hence integral formulae for the solutions to the a-equation on such domains could be obtained (see e.g. [C]). Finally note that, due to the non commutativity of the Clifford algebra, the pointwise product of monogenic functions is not anymore monogenic. The product established in Section 14, though not being easy to handle, enables us to construct some kernel functions which are of basic importance in transform analysis (see Chapter 5). In the case of regular functions of a quaternionic variable such a product was already introduced by Haefeli in [Ha]. Bibliography [BJ

P. Bosshard, Die Cliffordschen Zahlen, ihre Algebra und ihre Funktionentheorie (Thesis, Universitat ZUrich, 1940). [BT 1] H. Behnke and P. Thullen, Theorie der Funktionen mehrerer komplexen Veranderlichen, Zweite Auflage (Springer Verlag, Berlin, 1970). [C] P. Charpentier, Formules explicites pour les solutions minimales de l'~quat;on au= f dans la boule et dans le polydisque de [n, Ann. Inst. Fourier 30 (1980), 121-154. [Dea] C.A. Deavours, The 4uaternion calculus, Amer. Math. Monthly 80 (1973) 995-1008. [Del 1]R. Delanghe, On regular-analytic functions with values in a Clifford algebra, Math. Ann. 185 (1970) 91-111. [Del 2] , On the singularities of functions with values in a Clifford algebra, Math. Ann. 196 (1972) 293-319. 139

[Di] A. Dinghas, Einf6hrung in die Cauchy-Wierstrass'sche Funktionentheorie (Bibliographisches Institut, Mannheim, 1968). R. Fueter, Die Funktionentheorie der Differentialgleichungen ~u = 0 [F] und ~~u = 0 mit vier reellen Variablen, Comment. Math. Helv.7 (1934) 307-330. [G] B. Goldschmidt, Verallgemeinerte analytische Vektoren in mn (Thesis, Universit!t Halle, 1980). [GT] F. G6rsey and H.C. Tze, Complex and quaternionic analyticity in chiral and gauge theories I, Annals of Physias 128 (1980) 29-130. [Hab] K. Habetha, Eine Bemerkung zur Funktionentheorie in Algebren, In: Function Theoretic Methods for Partial Differential Equations, Darmstadt 1976, Lecture Notes in Mathematics 561 (Springer-Verlag, Berlin, 1976) 502-509. [Hae 1]H. Haefeli, Hyperkomplexe Differentiale, Comment. Math. Helv. 20 (1947) 382-420. [Hae 2] , I funzionali lineari delle funzioni analitiche di una variabile quaternionale, Rend. Aaaad. Naz. dei XL (4) 2 (1952) 65-110. [Hen] G. Henkin, Integral representation of functions in strictly pseudoconvex domains and applications to the a-problem, Math. Sb. 82 (1970) 300-308, Math. USSR Sb. 11 l1970) 273-281. [Hes] D. Hestenes, Multivector functions, J. Math. Anal. Appl. 24 (1968) 467-473. G. Hile, Hypercomplex function theory applied to partial differential [Hi] equations (Thesis, Indiana University, 1972). [I] V. Iftimie, Fonctions hypercomplexes, Bull. Soa. Sai. Math. R.S. Roumanie 9 (1965) 279-332. P. Lounesto, Spinor valued regular functions in hypercomplex analysis [L] (Thesis, Helsinki University of Technology, 1979). [N] W. Nef, Die Funktionentheorie der partiellen Differentialgleichungen zweiter Ordnung (Hyperkomplexe Funktionentheorie), Bull. Soa. Fribourgeoise Sa. Nat. 37 (1944) 348-375. E. Ramirez De Arellano, Ein divisionsproblem und Randintegraldarstel[Ra] lungen in der komplexen Analysis, Math. Ann. 184 (1970) 172-187. [Ry 1] J. Ryan, Clifford analysis with generalized elliptic and quasielliptic functions, to appear in Appliaable Analysis.

140

[RY 2] J. Ryan, Complexified Clifford Analysis, preprint. [S] A. Sudbery, Quaternionic analysis, Math. Proc. Camb. Phil. Soc. 85 (1979) 199-225. [SW] E.M. Stein and G. Weiss, Generalization of the Cauchy-Riemann equations and representations of the rotation group, Amer. J. Math. 90 (1968) 163-196.

141

3 Spaces of monogenic functions

In this chapter we study the space M(r)(Q;A) of null solutions of the generalized Cauchy-Riemann operator D (§16) and characterize its dual and bidual (§§22 and 23). These characterizations rely heavily upon Runge type theorems (§18) \'lhich are obtained by using properties of the so-called regular solutions at infinity of D (§17). Just as in holomorphic function theory, the Runge approximation theorem plays a basic role in solving the equation Of = g where g belongs to some given class of functions or distributions (319) which in its turn enables us to introduce the notion of a primitive of a monogenic function (§20). In our case also a Mittag-Leffler type theorem may be derived from it (§21). Finally in Section 24 Hilbert modules with reproducing kernel are introduced which consist of square integrable monogenic functions and which generalize the classical HL 2 - and H2 -spaces. Further notice that only properties of spaces of left monogenic functions are proved, the passage to the:right monogenic case being straightforward. 16.

The space M(r)(Q;A)

16.1 Let Q be an open subset of Rm+ 1 ; then we called M(r)(Q;A) the unitary right A-module consisting of all left monogenic functions in Q and analogously M(l )(Q;A) the unitary left A-module the elements of which are right monogenic functions in n. Furthermore let Harm(Q;A) denote the unitary bi-A-module of all A-valued harmonic functions in n and let Harm(~;R) stand for the space of real valued harmonic functions in~. Then, considered as real vector spaces, Harm(n;A)

=

u

Harm(\l;R).

AEPN

Obviously M(r)~n;A) is a submodule of Harm(r)(n;A), which in its turn is a submodule of C(r)(n;A). Hence both of these spaces may be endowed with the c(r)(n;A)-topology defined in Section 3. 142

If Harm('.,;lR) also is provided with the topology of uniform compact convergence then clearly the c(r) (c;-1;A) - and the product topology on Harm(r) (q;A) are equivalent. As Harm(:J;JR) is a real Schwartz space, the same is true for Harm(:;;A) and M(r)(~2;A), Finally it is well known that Harm(;2;JR) is a Frechet space, whence Harm(r)(J;A) becomes a right Frechet A-module. In view of the Weierstrass Theorem stated in 9.11 we immediately get 16.2 Theorem M{r)(~;A) is closed in Harm(r)(~;A) and so it is a right Frechet A-module. Notice that from the above results it follows that M(r)(a;A) is a real Frichet-Schwartz space which in its turn implies 16.3 Theorem (Montel) Let B be a subset of t·1(r)('.?;A). if and only if B is closed and bounded.

Then B is compact

16.4 We close this section by considering the case where ~ = Rm+ 1• r4(r)(Rm+ 1 ;A) is called the ;;p:1UC of Zeft r'ntir>c funr~tion;; in 1Rm+ 1• By Theorem 12.3.5 any f E M(r)(Rm+ 1;A) admits the expansion 00

f( X)

=

where for each k E IN, ), 1 1 = _a_ 1••• k ax 1 in the whole spaceRm+1 • If Q= {V 11

E A, converging normally ... ~ f ax 1 x- 0 1

_

k k 1 : k E IN, (1 1 , ••• ,lk)E{1, ... ,m} .},

1' .. k

then by P(r) the right A-span of Q is meant; it is called the

spac!e of Zeft

inncn• Dpher>icaZ mcnogenicr{,

Clearly P~~~· the right A-module of left inner spherical monogenics of order k (see 11.2.2.), is a submodule of P(r)' We thus have 16,5 Theorem Each left entire function may be approximated by left inner spherical monogenics uniformly on every compact subset K of Rm+ 1, i.e. P(r) is dense in t4(r)(Rm+ 1;A). 17.

The space

M(r)(~;A)

17.1 In this section we shall deal with a submodule of M(r)(g;A), namely M(r)(n;A) consisting of those elements in M(r)(n;A) which are regular at infinity with respect to the funda~ental solution E of D. Let E denote the fundamental solution of D (see 8.9) given by 143

E(x) = - 1-

~+1

X

lx lm+l

, X

f: 0.

Recall that DE = ED = 8 * (Rm+1 ;A) as for E E V(r) * (lRm+1 ;A). F1nally . as well for E E V(l) let us recall that M(r)(n;A) may be identified with a submodule of 1 )(n;A).

v(

17.2 Definitions Let n be an open subset of 1Rm+ 1 such that 1Rm+ 1,n = K is compact and let f E M(r)(n;A) (resp. f E M(l )(n;A)). Then * (lRm+1 ;A ) ( resp. T E V(r) * (lRm+1 ;A ) ) 1s . called a left (resp. ( 1.) T E V(l) right) asymptotic extension off if there exists a compact set K* c1Rm+ 1 such that K c K* and Tis an extension of fj(~m+l\K*). (ii) f is called regular at infinity with respect toE if and only if any left (resp. right) asymptotic extension T of f satisfies the condition E•DT = T (resp. (TD)*E = T). 17.3 Definitions * (lRm+ 1;A ) ( resp.~ T E V(r) * (lRm+ 1;A ) ) for wh1ch . DT ( resp. TO) has ( 1. ) T E V(l) compact support is said to be regular at infinity with respect to E if E.DT=T (resp. (TD)•E = T). (ii) T E V(l)(G;A) (resp. T E VCr)(n;A)) is said to be a solution of Din n if DT = 0 (resp. TD = 0) in n. 17.4 Remarks 17.4.1

Take f E M(r)(n;A) c VCl)(i"l;A). 0 =

I

J

<j>(u)dcr/(u)

a(supp)

Then for any

+

I

I

)(n;A) ,

<j>(x).Df(x)dx

supp (<j>D) (x). f(x )dx,

supp which implies that = -D> = 0. 144

E V(l

(<j>D)(x).f(x)dx

supp

=



Hence the D-derivative off, both as a function and as a left A-distribution, vanishes in D. This indicates that the Definition 17.3 (ii) has been suitably chosen. DD = ~m+ 1 e 0 , it follows immediately from the general theory 17.4.2 Since DO on elliptic operators (see e.g. [10]) and the Proposition 2.14 that any solution in Vtl)(D;A) (resp. v(r)(D;A)) of DT = 0 (resp. TD = 0) is a E(D;A)function. Hence any distributional solution in ~ of DT = 0 (resp. TD = O) is a left (resp. right) monogenic function in D. The proofs of the properties listed in the following proposition are left to the reader. 17.5

Proposition

(i) Each T E E{l)(Rm+ 1;A) (resp. E{r)(Rm+ 1 ;A)) is a solution of Din Rm+ 1,supp T which is regular at infinity with respect to E. (ii) E is regular at infinity with respect to E. (iii) Let T1 and T2 be solutions of Din the complement of some compact set K cRm+ 1 and coincide there. Then, if T1 is regular at infinity with respect to E, so is T2 • . ) LetKcRm+ 1 becompact,letT EV(l)R * ( m+ 1 ;A)(resp.V(r/R * m+ 1 ;A)) (1v 1 be a solution of D in Rm+ 1,K which is regular at infinity with respect to E m+1 * m+1 * and let T2 E E(l )(R ;A) (resp. E(r)(R ;A)). Then T1 * T2 (resp. T2 * T1 ) is a solution of D which is regular at infinity with respect to E. In part. * m+1 ;A) (resp. V(r)(R * m+1 ;A)) of the formS= E*T (resp. 1cular each S E V(l)(R * (Rm+1 ;A) (resp. E(r) * (Rm+1 ;A) is a solution of D which is T*E) where T E E(l) regular at infinity with respect to E.

1~.6 Proposition Let (Tj)jEJN be a sequence in v(1 )(Rm+ 1 ;A) (resp.

V(r)(Rm+ 1 ;A)) consisting of solutions of D outside some compact set K, all of them being regular at infinity with respect to E. Then if (T.) .EJN conm+1 * m+1 J J . * ;A) (resp. V(r)b(IR ;A)), Tis also a solution of verges toT 1n V(l)b(R D in Rm+ 1,K which is regular at infinity with respect to E. ~

Consider for instance the case of left A-distributions. Use the con. V(l * )b (Rm+ 1 ;A ) to show that DT = 0 1n . Rm+ 1'K. of the operator D 1n Apply furthermore Proposition 17.5 and the fact that the convolution is separately continuous in Vb(Rm+ 1 ;JR) (see [16]) to prove that E*DTJ..... E*DT in v*(l)b(Rm+1 ;A). o t 1nu1ty · ·

145

17.7 Definition Let r2 be an open subset of 1Rm+ 1 such that K = 1Rm+ 1,n is compact; then we call M(r)(r~;A) (resp. M(l)(n;A)) the set of left (right) monogenic functions in n which are regular at infinity with respect to E. It should be noted that M(r)(r~;A) (resp. M(l )(n;A)) is a submodule of M(r)(n;A) (resp. M(l )(n;A)). We have even more, namely 17.8 Theorem M(r)(r~;A) (resp. M(l )(Q;A)) is closed in M(r)(n;A) (resp. M(l )(n;A)) and hence a right (resp. left) Fr,chet A-module.

Proof Let for instance (fj)jSN be a sequence in M(r)(n;A) which converges to f in M(r)(Q;A). 0 Let furthermore H be a fixed compact subset in 1Rm+ 1 such that 1Rm+ 1,n = K c H and let l/i E E(IRm+ 1 ;IR) with 0 ::: l/i 0 we have for each ;-;mm+1 E*Df(x)

f

E(x-t) Df(t)dt

Rm+1

J0R J m w(war s

+ ~a )f(x+rw)dr dw w

149

1 =---

um+1

N0\'1

I( sm

I

f(x+Rw)-f(x) )d,J.)- - 1- IR dr rf(x+rw)dw. wm+1 0 r sm

it is easy to check that r I

ff(X+rw)dw

Jsm

=0

while lim R++oo

J

(f(x+RuJ)-f(x))dw

Sm

=

-wm+ 1f(x).

Hence E*Df(x)

= f(x) for all x E Rm+ 1

which means that f is regular at infinity with respect to E. 18.

c

Runge type theorems

18.1 In the present section several Runge type theorems will be proved, namely: (i) Let K be a compact suEset of Rm+ 1 the complement of which is connected and let f be monogenic in an open neighbourhood of K. Then f may be uniformly approximated on K by monogenic polynomials. (ii) Let K be a compact subset of Rm+ 1 and letS be a countable subset of Rm+ 1,K having one point in each bounded component of Rm+ 1,K. Then each f which is monogenic in an open neighbourhood of K can be uniformly approximated on K by 'rational functions' having their poles in S. (iii) Let K and n be respectively compact and open subsets of Rm+ 1 with K c n such that n contains no relatively compact component of ~K. Furthermore let f be mo~ogenic in an open neighbourhood w of K with w c n. Then f can be approximated uniformly on K by elements of M(r)(n;A). ( i v) Let n be an ope:1 subset of Rm+ 1 and 1et ex. be a subset of Rm+ 1 ~ having one point in each bounded component of Rm+ 1 ~. Then a monogenic function f in Q may be approximated uniformly on each compact subset K of n by 'rational functions' having their poles in a. (v) Let K be a compact subset of Rm+1 and let ex. be a subset of K having one point in each component of K. Then a monogenic function- in Rm+ 1,K which 150

is regular at infinity with respect to E may be approximated uniformly on each compact subset H of Rm+l,K by 'rational functions' having their poles in a. When r·eferring in the sequel to 'The Runge Approximation Theorem', we in fact mean version (iii). For convenience we again only treat of left monogenic functions. 18.A The case M(r)(K;A) 18.2 Let K be a compact subset of Rm+ 1• Then we define M(r)(K;A) to be the set of those functions f for which there exists an open neighbourhood wf of K such that f E M(r)(wf;A). It is clear from this definition that the considered neighbourhood depends on the function chosen and that for the classical addition and scalar multiplication M(r)(K;A) becomes a unitary right A-module. In what follows, when speaking of subsets of M(r)(K;A) which are 'uniformly dense' in M(r)(K;A) then this density property should be understood in the sense of the norm induced by c(r)(K;A) on M(r)(K;A). 18.3 A central role is played by Lemma Let K be a compact subset of Rm+1 and let ~ be an A-valued measure in Rm+ 1 having its support contained in K. Then ;d~(x)f(x) = 0 for all f E M(r)(K;A) if and only if ~*E = 0 in Rm+ 1,K. Proof As to the necessary condition, let a E Rm+ 1,K and put f(x) Then f E M(r)(K;A) so that ~*E(a)

=

Jd~(x)

E(a-x)

E(a-x).

0,

or \I*E vanishes in Rm+ 1,K. Conversely let f E M(r)(K;A). Then there exists an open neighbourhood w of K such that f E M(r)(w;A). If w1 is an open subset satisfying K c w1 c w and cp E V(u.;;IR) with cp(x) = 1 in w1 , then fcp E V(w;A) so that, considering ~ is a right A-distribution, we have on the one hand that (p,f¢>

= Jd~(x)fcp(x) = Jd~(x)f(x)

While on the other hand

151

0.

Consequently

;d~(x)f(x)

= 0.

c

18.4 Theorem (Runge (i)). Let K be a compact subset of Rm+ 1 the complement of which is connected and let f E M(r~(K;A). Then f can be approximated uniformly on K by functions in M(r)(R + 1 ;A), i.e. M(r)(Rm+ 1 ;A) is uniformly dense in M(r)(K;A).

Proof Let B(O,R 1 ) = B1 be an open ball in Rm+ 1 such that K c B1 • Then obviously M(r) (B 1;A) is a submodule of M(r) (K;A). ~le now claim that every function f which is left monogenic in an open neighbourhood of K may be approximated uniformly on K by elements of M(r)(B 1;A), i.e. M(r)(B 1 ;A) is uniformly dense in M(r)(K;A). By means of the Hahn-Banach and Riesz representation theorems, it clearly suffices to show that each A-valued measure ~ in Rm+ 1 supported on K which annihilates M(r)(B 1;A) is also zero on M(r)(K;A). Let~ be such a measure. Then for each hE M(r)(B 1 ;A) we have by assumption that fd~(x)h(x) = 0 so that, in virtue of Lemma 18.3. ~*E = 0 in Rm+ 1,B 1• Since ~*E is an analytic A-valued function in Rm+ 1,K and Rm+ 1,K is connected, ~*E = 0 in Rm+ 1,K, which implies again by Lemma 18.3, that /d~(x)f(x) = 0 for all f EM ~(K;A). Next we prove that M(r)(R~r ;A) is uniformly dense in M(r)(K;A). 0 To this end consider a sequence of closed balls ]j = Bj(O;Rj) such that_K c B1, Rj < Rj+ 1 , j = 1 ,2, ••• , and Rj t"" and put for convenience K = B0 • By the preceding result, for each j E ~. M(r)(Bj+ 1 ;A) is uniformly dense in M(r)(Bj;A). Now take f = f 0 E M(r)(K;A). Then again by the.first step of the proof, given£> 0, a sequence (f.) "eN may be found such that J J fj E M(r)(Bj;A) and

152

sup jf.(x)-f. 1(x)j 0 ;,-fo:.. J J+ 2J+• xEB. J

Fix j E ~and consider the sequence (f.J+ k)ook=0

c

M( r )(B.;A). J

Then for s


- • Now t a ke a i E r,'"i n (Rm+ 1--.....:~r) c r, i ..: and put S = {ai : i : r.u, ].J*E = 0 in each wi, whence ].J*E = 0 in w. This proves the necessary condition. As to the sufficient condition, choose f E: M(r)(IRm+ 1-....K;A) arbitrarily and let K be a suitable compact neighbourhood of K which is still contained in n w. Then, in view of Cauchy's representation formula outside a compact set (see Corollary 17.15) for each x E: 1Rm+ 1-....K n

f(x)

r

=- j

Hence

aK

E(t-x)dotf(t). n

J

!d).J(x)f(x)

supp].J

- r

JaK

d].J(X) JaK E(t-x)dotf(t) n [Jd].J(X)E(t-x}]dotf(t)

n

-J

).J*E(t)dotf(t)

aK

0.

158

n

[J

18.20 Theorem (Runge (v)) Let K be a compact subset of Rm+ 1 and let L be a subset of K having one point in each component of K. Then R(r)(L) is . m+1 dense 1n M(r)(R '-K;A).

Proof Obviously R(r)(L) is a submodule of M(r)(Rm+ 1,K;A) since each of its elements is a right A-linear combination of functions having the form lE(x-ai)

ax 1

1

••• ax 1

k

Using the Hahn-Banach theorem, it clearly suffices to prove that if T E c(;)ORm+ 1,K;A) annihilates R(r)(L) then it vanishes on M(r)(Rm+ 1,K;A) too. To this end, let again (Kj)j~ 1 be the compact exhaustion of n = Rm+ 1,K considered so far and let T E co(r)(Rm+ 1,K;A) be such that it annihilates R(r)(L). Furthermore assume that T is bounded by PK. and choose j E ~ large enough such that both K and {x E Rm+ 1 : d(x,K) ~ 1} ~re contained in m+1 J {x E R : !xi <j}. Then an A-valued measure~ supported on K. may be associated with T such that for all f E c(r)(Rm+ 1,K;A) (see Seciion 3) =

Jd~(x)f(x).

Now let Rm+ 1,K. = nj u n1 u n{ u ••• be the decomposition of Rm+ 1,K. into J . J . 0 its components, n~ heing the unbounded one. Then K c w where w = U n~ i;;1 1 is a bounded open neighbourhood of K such that each of its components meets K. Hence, for each~ = 1,2, ••. , n1 n L ~~so that, taking a1 E n1 n z, i = 1,2, .•• , S ={a~ : i = 1,2, ••• } c L. As ~*E is an A-valued analytic function in Rm+ 1,supp ~. for each i ~ 1 there exists an open neighbourhood ~ . of aJ1: in which ~*E admits a Taylor development with coefficients a~

1

(a~) 1

0. 159

j so th a t , a.1 . Hence ~*E = 0 in w = u ~~.

Consequen tl y

in ~J1:.

~* E

-

• = 0 1n

for all f E M(r){IR 19.

m+1

~

. >1

'K;A).

1~

1

j c 1Rm+ 1'supp~ be1ng · connecte d ,

~i

~* E

=0

In view of Lemma 18.19 f d~{x)f{x) = 0

c

The equation Df = g

19.1 In this section we study the equation Df = g where g is an element of a given class of functions or distributions, namely E{r)(~;A), 1 ){n;A) and * { m+ 1 ) m+ 1 . S(l) lR ;A , n clR be1ng open. Of course, as DD = DD = ~+ 1 e 0 we may immediately obtain the existence of a solution f belonging to the considered class {see e.g. [20]}. Nevertheless in each of the cases we shall give a direct construction whereby we want to draw special attention to the pure function theoretic method developed in the case of s(1 )(1Rm+ 1 ;A). We also wish to point out that a same procedure may be followed to study the equation DJf =gin nJ clRM-J open. Hereby J = {i 1 , ••• ,ij}, 0 ~ j < m, is a proper subset of {0,1 , ••• ,m} = M such that J F {1 , ••• ,m}. Furthermore L { ) M-J and xJ iE~ xiei is then identified with the element xi iE~ of lR

v(

A

DJ stands for DJ = Di.

L

e. 1 In such a way D may be

while if 0

~

iE~

--0--. If J {i}, i EM, then DJ is denoted by axi written as D Moreover if 0 E J, then

J, then

D}iJ = DJDJ = e 0 t.~,

~ being the Laplacian in IR~. ~

In the special case where J = {0}, we put

= t.m·

Finally frequent use will be made of the following sequences of subsets of n. For j G 1 we put wj = {X E n: I>: I

1


..... } J

~

j and d(x ,1Rm+ 1,n)

> .... } • = J

and as usual, Kj = {X E :1: lxJ

160

1

19.A.

The case E(r)(n;A)

19.2 Theorem Let g E E(r)(n;A). Of = g.

Then there exists f e: E(r)(n;A) such that

Proof Put G1 = w2 , Gj = wj+ 1'Wj_ 1 , j ~ 2. Then (Gj)j~ 1 is a locally finite open covering of n. Let (~.) .> 1 be a partition of unity subordinate to J J= (G.).> 1 with~- E V(G.~). Then g = E ~-gin E( )(n;A). Now call for J J= J J j=l J r each j, gj = E*~jg. Then gj e: E(r)(lRm+ ;A), Dgj = ~jg and hence, if j ~ 3, Dg.J = 0 in w-J- 1 or g.J e: M( r )(K.J- 2 ;A). In virtue of Theorem 18.9 there exists h. E M(r)(n;A) such that sup l(g.-h.){x)j 0 ~ J...J. Consequently the series J xe:K. J J 2 J-2 00

L

gl + gz +

j~3

g.-h. J

(19.1)

J

converges in c(r)(n;A) to an eleme~t f. We claim that fEE(r)(n;A). Indeed, consider an arbitrary closed ball B(a,r) contained inn. Then fork sufficiently large, B(a r) c ~k and so g1 + g2 + ~ (gj-hj) E E(r)(B(a,r);A), j=3

while, using the Weierstrass theorem (Theorem 9.11) 00

L (gj-hj) E M(r)(B(a,r);A). j=k+1 Hence f E E(r)(B(a,r);A). M~reover, for each k 0

L

Df(x) =

j~1

19.B.

The case

Dgj(x) = ~ jg(x) = g(x). j=1

~

1 and x e: wk

c

v( 1 ,(n;A)

* 19.3 Theorem LetT e: V(l)(n;A). DS = T.

Then there exists

* )(n;A) such that s e: V(l

Proof Put again G1 = w2 , G.= w. 1 ~. 1 , j ~ 2, and consider a partition of ----J J+ Junity (~j)J~ 1 subordinate to the locally finite open covering (Gj)j.:;: 1 of n, with ~j e: V(Gj ;JR). Then in v(n (n;A) 00

T=

L

~/·

j=1

161

* )Ui;A), sj E V(l * )(Q;A) Call for each j? 1, Sj = E*

3 be the

1

Hl = 51 + 52 +

L

(Sj-hj),

j=3

* we have that H1 E V(l

)(r~;A)

and 1

p

L <sk-hk,<J>>

= <S 1+s 2 ,q,> + As for each k

f;

*

L

+

k=1

<sk-hk,<J>>.

k=p+1

p + 1, I<Sk-hk,¢'>1 0 ~~ sup lct>(x) 10 , the sequence (H 1>1;;: 3

converges in v( 1 ),s(n;A).

Call

2

XHI

s = 51 + 52 + L (Sj-hj}. j~3

Then OS= ct>1T +

cj> 2

T+

L

00

O(Sj-hj)

j~3

L ct>} = r.

[]

j=1

19.C. · The case s *0 ) (lRm+1 ;A) 19.4 Theorem Let g E c(r)(1Rm+ 1 ;A) be of slow growth in 1Rm+ 1• Then there exists f E c(r)(1Rm+ 1;A), being of slow growth in 1Rm+ 1 too, such that Of= g in 1Rm+ 1• Moreover, if g has the growth factor 1, then f can be chosen in such a way that it has the grmtth factor 21 + 4. Proof Let g E c(r)(1Rm+ 1 ;A) be of slow growth in 1Rm+ 1 with ig(x)i 0 ~c*(1+1xl for some 1 E ~ and c* > 0. 162

f

Consider the sequence of open balls (Bk)k, 1, all of them being centred at the origin and having respective radius k~ Take a partition of unity (¢j)j~ 1 in V(Rm+ 1 ;R) subordinate to the locally finite open covering (Gj)j~ 1 m+1 o R oo of R , where G1 = B2 , Gj = tij+ 1'Bj_ 1, j ~ 2. Then g = I ¢jg in j=1 o m+1 . for a 11 j :;c 1 , the sequence of c(r)(R ;A). Not1ce that, as 0 ~ ¢j :: constants (C.) .. 1 with J J>

cJ. =

lgJ. (X) I0'

sup m+1 xElR

gJ.

¢J· g'

satisfies estimates of the type

Put for each j f . (X) J

Then Dfj

~

1,

= E*g . (X ) J

= gj and sup xElRm+1

Now, as for j ~ 3, Dfj

Js.J+ 1

= 0 in Bj_ 1• fj admits a Taylor development about

0

co

f.(x) J

=

I

s=O

P f.(x) s J

with

where C' depends upon the dimension m and R is chosen arbitrarily in ]O,j-1[ (see 11.5.2). As for any E > 0 there exists C > 0 such that for all s E ~ E

C'(S+m+ 1)(1+S 2 S+l

)

~ CE (1+E)S,

we get that 163

IPs f.(x)l J 0

C ((1+dlx1)s (1+J.)l+1

5

-

E:

J-1

Call for each k ~ 3, 1+2 hk(x) = 2 P5 fk(x). s=O m+1 Then hk E M(r) (~ ;A) and for all x E ~m+l for which (1+e:)lxl have that co

) 1 +1

S=l+3


I0

;';

c

sup sup l3o.ijJ(x) lo [ai<M xEK

c

sup sup [3a.f (X) I0. [a[:;;M xEK

But, as f is an A-valued harmonic function in S1, a suitable compact neighbourhood K of K, with K c w , may be found such that for some constant ll ll

0 [[ 0

"::

C* sup XEK

jf(x)io· ll

171

As = -

we finally obtain that I lo:;; c* sup lf(x) lo · xEK 11

or Tu E M{r)(n;A).

We thus have proved

22.5 1 Propositio~ Let ¢ E ~j where j E ~ is fixed. Then for any u E M(l) (~m+ 'Kj;A) the right A-linear functional Tu defined on M(r)(n;A) by = Ju(t)D(f¢)(t)dt

(22.2)

belongs to M{r)(n;A). 22.6 Proposition If F. E M(l)(~m+ 1 ,K.;A) is the indicatrix of Fantappie of * J m+1 J T E M(r)(n;A) then for all s E ~ , the function aSF. is the indicatrix of J Fantappie of the functional a8T defined by = (-1)1SI, vf E M(r)(n;A) Proof It is clear that also a8Fj E M(l)(~m* 1 ,Kj;A); then by the foregoing Proposition 22.5 it is the indicatrix of Fantappie of the functional T1 E M{r)(n;A) given by = jaSFj(t) D(f¢)(t) dt, v f E M(r)(n;A) where ¢ E V(n~) is such that ¢(x) = 1 in some open neighbourhood w¢ of Kj contained in n. So

By Leibniz's rule a8(f¢)

=

I y:;;B

172

(~)a 8 -Yt.aY¢

=

-~. L (~)aB-Yf.aYq,.

a13f.q,

Ys:S

yilO

Now for

y

I 0



(-1)1BI!Fj(t).D[(asf).q,](t)dt l;;; 2n 12 sup Il fEB gEB* which means that e- 1 is continuous. 22.14

Obviously (i) and (ii)

imply (iii).

o

Using the above proposition we finally get

Theorem J is a topological isomorphism between M(r)b(n;A) and "'M(l )(Rm+1 ' n;A)ind" Proof From the considerations preceding Proposition 22.9 we already know that J is an algebraic isomorphism. To prove its continuity, it suffices to show that its restriction to each E~ ll !I . is continuous. To this end, * J, • J take T E Ej arbitrarily and consider J T = [Fj]j E M~l)" Then for any q E Pind 0

=c
0, 88r denote the boundary of the open (m+1)dimensional sphere 8r centred at the origin with radius r and consider the unitary right A-module L2(r)(88r;A) of A-valued square integrable functions on 88r, where Lebesgue measure is taken on 88r. 24.13 Definition Let R > 0; then ML 2 (r)(88R;A) consists of those elements in M(r)(8R;A) such that lim r-+R


0, there ought to exist N(s) E~ such that llfj-fki!R ~ s if j,k ~ N(d. By the Corollary 24.15 (fi)i~ is a Cauchy sequence in M(r)(BR;A) so that, by the completeness of this space, there exists f E M(r)(BR;A) t~ which (fi)ieN converges. Now take 0 < r < R. Then, as for each~ E H (aBR) (see [22])

Jasr

lfj-fkl~dS ~

llfj-fkiiR

~

sz

so that, by passing to the limit for j JaBr

lf-fkl~dS ~

s 2 for all k r

Again using the property that ( J we find that aBr lim r~R

~

~ oo ,

N(s). !f-fki~dS)O-

=

0

f(t:K + x) E E(r)(K;A). 0

Find g 1 E E(r)(K;A) such that D0 g 1 (x) = -f(cK + x) inK (see also Theorem 19.2). Call f 1 = h 1 + g 1 • Then f 1 E M(r)(l( x]O,cK[ ;A). Proceeding in the same way, assume call

204

tha~

h 1 _ 1 , g 1 _ 1 and f 1 _ 1 have already been constructed,

let g1(x) be such that D0 g1 (x) = -f 1_1(EK + x) inn and put f 1 = h1 + g1 • Then 1n view of the growth condition satisfied by f, there ought to exist 10 E ~ such that f 1 is continuous in K x [O,EK[. Hence 0

0

1im

fl

x .....a+

(X+

X

0

0

= fl (X

)

°

+ 0)

0 0

exists in v(l)s(K;A).

But, as in K x ]O,EK[,

f

f 1 (x+o>. 0

* belongs to V( 1) (K;A). 0

c

26.3 Remark It should be noticed that the monogenicity is not necessary the weakest condition upon f in order to get some of the implications stated in Theorem 26.2. To this end we refer to the results proved in the previous section. 26.4 In what follows a relationship will be established between the existence of boundary values and the solution of the equation OS = T. Theorem * Assume that f E M( r )(n+ ;A) admits a distributional boundary value -'--'--....::._ a+f in V(l)s(~;A). Then there exists a unique distributional extension f* off in v( 1 )(n;A) such that of*

= a(x 0 )

~

a+f(x). .....

Moreover the support of f* is contained in n+ u n. ~

Let K c ~ be a compact interval. Following the proof of Theorem 26.2 there exists 1 E ~. where 1 may be chosen to be even, say 10 = 2s, and 0 0 0 f 1 E C(r)(K x [O,+oo[) n n;A) n M(r)((K x ]O,+oo[) n n;A) such that 0

0

0

1

a of

x 0 10

= ( -!J. ) s m

205

*

0

exists in V(l)(K~A). Next extend f 1

o

n n where

to yK in (K x lR)

0

* * 0 *" Then fK E v(l)(K xlR) n n;A),fki(K Moreover, if 10 K x [-o.o] c

n

= 2s,

f;.

= (-~)syK

X

. *0 ]O,:cx{) n n= f, Whlle fKi(K ]-oo,Q[)nn:o. in (K x JR) n n.

and let ¢ E V(K x ]-6,0[ ;lR).

* ~ = ( -1) 0 such that

Then, using Cauchy's Theorem

( ( D( - t.m ) s

}yK)dx

Kx[O, o]

I

1 +1

(-1) o

do((-1\n)s<J>)yK

( Kx[O, o])

-n

1


Representation of distributions by monogenic functions

v(

In this section we show how distributions in 1 ) (~;A), s( 1 ~ U1;A) and E( 1 )(~m;A) may be represented by monogenic functions in ~\~and R +l~m respectively, where ~and ~ satisfy the conditions of the previous section. 27.1

A. 27.2

The cases v( 1 )(~;A) and s( 1 )(~;A) Let us recall that if

fe:

M(,r )(~\~;A) has the boundary value

BVf = lim f(x+x ) - lim f(x-x ), xo~+ o xo~+ o then the canonical extension f* of f satisfies the equation

A converse of this result will now be formulated. 27.3 Theorem ;et T(x) e: v( 1 )(~;A) (resp. s( 1 )(~;A)) be given and let F be a solution in V(l)(~;A) (resp. s( 1 )(~;A)) of the equation

Then F is the canonicai extension of f = FI~\~"* V(l)s(~;A) (rcsp. S(l)s(~;A)).

210

e:

"* M(r) ( ~\~;A) and BVf = T in

Proof We only work out the proof for v(1 )s(~;A), the other case being treated analogously. First observe that as DF = 0 in ~\Q,f = FI~\Q indeed belongs to M(r)(~\Q;A) (see 17.4.2.). Moreover, as f has a distributional extension F to~. by means of Theorem 26.2 the boundary values lim f(x±x ) ...0+

X +

0

0

>

= a-f exist in-+v(1 )s(S'l;A). Let f* be the canonical extension off; then F-f* = 0 in ~\~ and so supp(F-f*) c n. Now let K c ~ be a compact interval and put ~K = (KY~) n ~. Then in ~K'F-f* may be written as 0

n

F-f* =

Y.

o(h)(x 0 ) e eh(x)

k=O

*

0

where eh c V(l)(K;A). Consequently, as Of* = o(x 0 ) e BVf in

~.

we have that in

~K

o(x0 ) e (T-BVf) = D(F-f*)

L

(o(h+l)(x0 ) e eh(x)+o(h)(x 0 ) e D0 eh(x))

O~h~n 0

from which it follows that en (x) Hence F = f* in ~K and as moreover o(x0 ) e (T-BVf)

o(x 0 ) e

0 in K.

o0 e0 (x)

0 0

we find that T = BVf in K. Using the 'principe du recollement', we get the desired result.

Q

* -+ * -+ 27.4 Theorem (Representation Theorem) LetT E v(l)(~;A) (resp. s(l)(~;A)). Then there exists f c M(r)(Q\n;A) such that BVf = T. Furthermore if g ~ M(r)(~\~;A) is such that BVg = T, then f-g is extendible to a function h

c M(r)(~;A).

~Put f = Flm~ where F is a solution in v(l)(~;A) (resp. s( 1 )(~;A)) of the equation OS= o(x) e T(x). Then Theorem 27.3 implies that BVf = T. 0 .... As to the second part, let g c M(r)(~\~;A) be such that BVg = T. Then, denoting by g* c V(l)(~;A) (resp. S( 1 )(~;A)) the canonical extension of g we get that D(F-g*) = 0 in ~ so that h=F-g*extends monogenically f-g to ~. 0

211

B.

The case E(l) (lRm;A)

27.5 The representation of E{l)(lRm;A) may be given in a much more direct way than in the cases V( 1l(n;A) and S(l)(0,;A) and this by using the so-called Cauchy tmnsform of T E E( 1) (lRm;A). A

27.6 Definition given by

LetT E E{ 1 )(lRm;A).

Then the Cauchy transform T of T is

+-

T(x)

~+1

0 such that B(y,E) clRm+ 1,supp T. Furthermore let a E V(lRm;lR) be such that a = 1 in supp T and a = 0 in B(y,E) n lRm. Then in B(y,E) the function f given by 1

-+

U+X

f(x) = - - wm+1 u lu+xl is continuous and its value does not depend upon 212

Moreover f coincides

" o m " with T in B(y,£)~ and hence T indeed admits a continuous extension to Rm+l,supp T.

Clearly lim T(x)

0 and we claim that in

X-+oo

svf = T = lim X

0

-+0+

v( 1 )(Rm;A)

(T(x+x ) - T(x-x )). 0

To this end observe that V(l)(Rm+l;A) given by

-

x

lxlm+l

0

T has the distributional extension Text in

*(o(x 0 )

@

T(x)).

Furthermore, as

using Theorem 27.4 we get that

1im

=

X -+0+ 0

whence for any >

4> E

V( 1) (lRm ;A)

= lim X -+0+ 0

T

Finally, as Tadmits the distributional extension t to v*( )(Rm+l ;A). it " ex 1 follows from Theorem 26.2 that T satisfies the stated growth conditions. c 27.8 Now we formulate a characterization for the Cauchy transform of distributions in E(1 )(Rm;A). Theorem Let K cRm be compact and let f E M(r) (Rm+1 'K;A) be such that lim f(x) = 0 and that for some k E ~ X-+oo

sup l(d(x,K))k f(x)i 0 < + oo. d(x,K) 0 such that B(P,r) c B(O,R 1 ),B(O,R 2 ). Then, using Lemma 28.2, h = f 1 + f 2 in B(P,r) where Df 1 = 0 and Df 2 = 0. Put 0 y = Dh in B(P,r). Then o

1 -

m

o

m

y!B(P,r) n S = w(a r +wa w)h!B(P,r) n S = 0. r o

o

m

Consequently, as Dy = 0 in B(P,r) and as moreover y!B(P,r) n S = 0, we get that y = 0 in B(P,r). Since P E Sm has been taken arbitrarily we finally obtain that Dh = y = 0 in B(O,Rl),B(O,R2 ). c 28.5 By means of the foregoing theorem each f E a(r)(Sm;A) admits a unique (left) monogenic extension f* to a suitable open neighbourhood of Sm in Rm+ 1; it is called the Cauchy-Ko~aZe~ski extension of f. Of course; if g E M(r)(B(0,1+E)'B(1-E);A), 0 < E < 1, then g!Sm E a(r)(Sm;A) and the Cauchy-Kowalewski extension g* of g!Sm coincides with g. Now call

m o _ M(r)(S ;A)= ind lim M(r)(B(0,1+E),B(0,1-E);A); 1>£>0 then the previous considerations yield that a(r)(Sm;A) and M(r)(Sm;A) are isomorphic right A-modules. But, as each of the spaces M(r)(B(0,1+E)'B(0,1-E);A) may be provided with its natural topology (see 16.1), the inductive limit topology can be defined on M(r)(Sm;A) (see 2.15) and hence be carried over to ~(r)(Sm;A). Analogously a(l)(sm;A) is isomorphic to M(l)(Sm;A) =lim ind M(l)(B(0,1+E)'B(0,1-E);A) 1>£>0 216

from which it inherits a locally convex topology. *(m *(m . 28.6 The duals a(r) S ;A) and a(l) S ;A) are respect1vely called the spaces of right and left analytic functionals in Sm. If r\r)(lRm+ 1-.....sm;A) -+O denotes the right A-module of all left monogenic functions in lRm+ 1-.....sm which vanish at infinity, then in what follows it will be sketched how this space represents a( 1)(Sm;A). The details of the proof are omitted since they may be reconstructed by using arguments similar to those developed in Section 22. LetT E a( 1 )(Sm;A) and consider its Cauchy transform

1

A

T(x) = - - '

( 28. 1)

IX-wl

then f is left monogenic in lRm+ 1-.....sm and lim T(x)

0.

Furthermore

X-->oo A

T(x)

Xo (-PkT)(x) B( 0,1)

where for H clRm+ 1 , XH stands for the characteristic function of H. PkT and QkT are given by -PkT(x) = _1_ wm+ 1

Hereby

(k~m11) + w , ,w

and

(28.2)

0 and

00

Pkf*(x) +

I

Qkf*{x)

k=O

is the Laurent expansion off*, then by the Cauchy estimates (see (11.17) and 218

(12.6)) there ought to exist C , 0 such that for all k

E .N

Hence the sequence (Pkf,Qkf)k8N with Pkf = Pkf*ISm and Qkf = Qkf*ISm satisfies the stated estimates. Conversely, let (Pk,Qk)k8N be a sequence of surface spherical monogenics fulfilling estimates of the prescribed type. Put for each k EN, Pk(x) X ) X • lXI k Pk (TXT and Qk ( x ) = 1x1 - ( k+m) Qk(TXf). Then 1t may be shown that 00

f*(x) =

00

I

pk(x) +

k=O

I

Qk(x)

k=O

is left monogenic in an annular neighbourhood of Sm.

I k=O iS analytiC ir. Sm.

"'

I

Pk(w) +

Hence

Qk(w)

k=O D

28.8 Now let Sk be an A-valued surface spherical harmonic of order k. clearly Sk E a(r)(Sm;A) and so, by Theorem 28.7, \(w) =

I

pl \(w) +

1=0

I

QlSk((ll) •

Then

(28.3)

1 =0

But from the sections 11.2.1 and 12.1.9 it follows that

and

Consequently, as ~+ 1 sk in (28.3) and for which Theorem

-k(k+m-1)Sk, all terms P1 Sk and Q1_1sk appearing f k vanish. Hence we obtain

Let Sk be an A-valued surface spherical harmonic of order k.

Then

i.e. each surface spherical harmonic of order k may be decomposed in a unique 219

way into the sum of a surface inner spherical monogenic of order k and a surface outer spherical monogenic of order (k-1). Notice too that we have (see also (11.14) and (12.5)): Pk\ () w -- -1- (k+m-1)J k+1 m Kk m+1 Wm+1 S '

I

o(ll

c,))do

I

W

Sk(w 1 )

and

these expressions give rise to the Cauchy estimates (see also (11.17) and (12.6)):

~~~m

IPk\(w) lo;;:

c(k~~1 1 )( 1 +k 2 ) ~~m

15kL,))

lo

and

(28.4)

Observe furthermore that the decomposition Sk = PkSk + Qk_ 1sk is consistent with the dimension 2nN(m+1,k) of the space of A-valued surface spherical harmonics of order k. In fact, by the Cauchy-Kowalewski theorem for analytic functions in Rm we obtain that the dimension of the A-module of (left) inner surface spherical monogenics of order k equals 2nM(m;k), whereas the dimension of the A-module of (left) outer surface spherical monogenics of order (k-1) is equal to 2nM(m;k-1). But, as is well known (see also 11.1) N(m+1,k) = M(m,k) + M(m,k-1). 28.9 Now let us recall some fundamental results, which have been obtained by several authors, concerning the decomposition of analytic functions and functionals on Sm by means of surface spherical harmonics, the functions or functionals under consideration being [-valued. (i)

f E a(Sm) if and only if f(w) =

L

Sk(c,J), (Sk)kElN

k=O being a sequence of surface spherical harmonics for which C may be found such that for all k E fi

220

~

0 and 0

< r.
0 may be found with

~

a

and this for all k E ~ (see [17]). 00

(iii)

T E V'(Sm) if and only if T

k=O spherical harmonics for which C > 0 and S > 0 may be found such that for all k E

~

sup 1\(w)l ;:: C(1+k)s. wesm (see [14]). 00

(iv)

T

t

a'(Sm) if and only if T =

I

Sk' (Sk)kEl'J being a sequence of

k=O surface spherical harmonics such that for all E > 0, there exists CE > 0 with

and this for all k E ~. (see [7]). Using Theorem 28.8 and the estimates (28.4) we get: (ii ') f E E(r)(Sm;A) if and only if there exists a sequence (Pk,Qk)keN of surface spherical monogenics such that 00

I (Pk(w) + Qk(w)) k=O and satisfying the condition: for all s E ~. Cs f(w) =

ks sup ( IPk(w) 10 , IOk(w) 10 ) wesm and this for all k E ~.

;::

>

0 may be found with

Cs

221

(iii') T E v(r)(Sm;A) if and only if a sequence,.(Pk,Qk)kEI'l of surface spherical monogenics may be found such that T = L (Pk + Qk) and satisfying k=O the condition: there exist C ~ 0 and s ~ 0 such that for all kEN sup (I Pk(t>l) wESm

J

0,

IQk(r,))

J

0)

-:

C( 1+k)s.

(iv') T E a(r)(Sm;A) if and only if a sequence)Pk,Qk)kEN of surface spherical monogenics may be found such that T L (Pk + Qk) and satisfying k=O the conditon: for all s > 0 there exists C 0 such that for all k E ~ f.

0>

It should be emphasized that if T E a( 1)(Sm;A) then its expansion T=

L (Pk+Qk)

k=O is in fact a direct consequence of the isomorphism between a(lj(Sm;A) and M(r) (1Rm+ 1,sm;A) _,. 0 which associates to T its Cauchy transform T given by (28.1). Hereby notice that the sequence (Pk,Qk)keN of surface spherical monogenics is determined by the expressions (28.2). Observing that Theorem 28.7 already provided a monogenic version of statement (i) we have thus obtained ~presentations of the spaces a(r)(Sm;A), E(r)(Sm;A), v(r)(Sm;A) and a{r)(Sm;A) by means of spaces of sequences (Pk,Ok\EJ.J of surface spherical monogenics satisfying appropriate estimates. Of course these representations are topological isomorphisms when the spaces under consideration are endowed with their natural locally convex topology. 28.10 To close this section we define an inner product on Sm between analytic functions. Definition

Let f,g E a(r)(Sm;A).

(f,g) m = - 1S wm+1 the

222

inne~ p~oduct

Jr

Then we call

mf';w)do g(w) S ul

between f and g.

28.11

Remarks

28.11.1 Let f, g E a(r)(Sm;A) and l~t for some R1 > 1 f* and g* be the Cauch Kowalewski extensions off and g to B(O,R 1 ),B(O,R~ 1 ). Then for any open neig bourhood S of the origin having C1 -boundary as c B(O,R 1 ),B(O,R -1 1 ), 0

(f,g) m = - 11J)m+1 s

J

as

-

(28.5)

f*(u)do g*(u). u

r1oreover, if 00

f(ul) =

l:

pk f(l.&.l) +

k=O

l:

Qkf(w)

k=O

then 00

'\,

f(u.l)

l:

'\,

00

Pk f(w) +

I

k=O I

Qk f{l.&.i)

k=O 00

00

l k=Ol:

'\,

l:

Pkf(w) +

lJkT(w)

k=O

r\-

f(w)~.

Consequently, as on Sm, do = wdS , dS w

(f,g) m = - 1wm+1 S

w

being the elementary surface element,

J m f(w)g(w)dS • S

w

It thus follows that ~ 0 while = 0 implies f = 0. eo eo Hence all the requirements formulated in 7(i) - 7(v) are clearly satisfied and so ( , )Sm is indeed an inner product on a(r)(Sm;A). 28.11.2

If f,g E a(r)(Sm;A) admit the expansions 00

f(w) =

l:

00

Pk f(w) +

k=O.

l:

Qkf(w)

k=O

and 00

g(w)

l: k=O

00

Pkg(ul) +

l:

Qkg(w),

k=O 223

then the orthogonality relations between spherical monogenics (see 11.2.4 and 11.4.5.) imply that for any k 1 1,

Moreover, as for any pair Pk and Q1 of inner and outer spherical monogenics, k, lE.tl,

lim

0

R+oo

and

we obtain, using (28.5), that for all k, 1 E .tJ

Hence (f,g) m

s

29.

L2-Functions on the unit sphere- the Szeg6 and Bergmann kernels

29.1 In this section it is proved that each function f E L2(r)(Sm;A) may be expanded into a series of surface spherical monogenics. Moreover, using the Cauchy transform in L2 (r)(Sm;A), an orthogonal decomposition of this space is obtained. Finally the Szeg6 and Bergman kernels corresponding to the unit ball in Rm+ 1 are calculated explicitly. 29.2 Let f,g E L2 (r)(Sm;A) {see also 7.14) and define the inne~ p~oduct between them by 1

r

-

(f,g) m = - - Jsm ~(w)g{w)dSw' S wm+1 dS again being the elementary surface element. w

224

Furthermore put

Jsm dS w.

J m lf{w) lo2 dS w •

s

29.3 Theorem Let f E L2(r)(S m;A). surface spherical monogenics:

Then f may be expanded into a series of

00

I

f(w)

(Pkf(w) + Qkf(w)).

k=O r~oreover

Finally, if g E L2 (r)(Sm;A) admits the expansion 00

I

g{w) =

(Pkg{w) + Qkg{w)),

k=O then (f,g) m

s

Proof

Iff E L2 (r)(Sm;A) call for each k E fi

I

1 (k+m- 1\ Pkf(x) -- -w1 k+1 J m Kk m+ 1 w(x)do wf(w) ' m+ 1 ' S ' '

and

I

x E 1Rm+ 1 ,

Q f(x) = _1_ (k+m-1\ Kk,m+1 ,x(w) do f(w) x k wm+ 1 k+1 } 5m lxlm+k w ' Furthermore define rrk, rr-(k+ 1):L 2 (r)(Sm;A)

7

€

1Rm+1. o

L2(r)(Sm;A), k ~ 0, by

rrkf(w) = Pkf(w) and rr-(k+ 1 )f(w) = Qkf(w),

wE sm.

225

m

Then for each s E l, ns is a bounded right A-linear operator on L2 (r)(S ;A) satisfying n~ = ns· Moreover for all f,g E L 2 (r)(Sm;A) and k ~ 0

Jsm "Pkf(w)g(,d)dS.)

(J

1- (k+m-1\\ k 1 ;f(w' )do- ,Kk , ( 1 we may consider its Taylor deve1opment about the origin wh1ch converges uniformly on -co r : B ,R ') , 1 < RI < TfT

where again K(x,t) =

(J)

'\,

00

L

K(rx,t)(u) =

Pk,r(u)

k=O ~

where Pk,r(u) = Pk K(rx,t)(u). ~ Hence, in view of (29.10), (m+1-2r)K(rx,t)(u) to ML 2(r)(B(0,1);A). But, in virtue of (29.8) 0

00

L

(2k+m+1)Pk,r(u) belongs

k=O

(<m+1-2r)K(rx,t)(u),f(ru)) 8 = (K(rx,tHuJ), fr(w))sm (29.12) Consequently, by means of (29.11) and (29.12) ~

m(

\

(29.13)

f(t) =lim r \(m+1-2r)K(rx,t)(u), f(ru>;s· r-+1

'
~ we obtain that in

0~

M(r)(B(O,

~);A) ~

~

rm(m+1-2r)K(rx,t)(u) -+ (m+1-2r)K(x,t)(u) if r-+ 1 0

so that also in L2(r)(B(0,1);A)


1) had to be treated differently. It was Vogt who gave in [V1] a complete description for the representation of V' (RN). f1eanwhile r1artineau studied in [t1ar] the boundary value problems connected with V'(R) and S'(R) in a more general framework. In fact he established a relationship between the existence of a distributional boundary value off E 0([~) and the extensibility off as a distribution in [, which in its turn could be characterized by means of growth conditions upon f. Moreover he put a link between the problem of representing V'(R) (resp. S'(R)) and the solution inV'(R 2 ) (resp. S'(R 2 )) of the equation a_S = T, T being given. In this way Martineau obtained the z algebraic isomorphisms already established by Tillmann. The question of course arose whether or not these isomorphisms were also topological when the spaces H0 , and HS' are provided with a suitable topological structure and when V'(R) and S'(R) carry the strong topology. This question was first answered by Konder in [Ko] while a refined study of the topology in H0 , was worked o~t by Vogt in [V2]. As was already mentioned, some problems arose with respect to the representation of V'(RN) (N > 1), especially concerning the surjectivity of the boundary value mapping. Vogt proved this surjectivity in [V1] by using the representation and boundary value theory for vector valued distributions and holomorphic functions. The case S'(RN;E),E being a locally convex space, was studied by r4eise in [t-1ei1], [Mei2] and [Mei3], while the cases S'(RN;E) and V'(RN;E) were investigated by Vogt in [V1], [V3], [V4]. The problem of the representation of distributions as boundary values of holomorphic functions 238

thus gave rise to intensive research during the last decades. As furtha· contributions to this theory we mention the papers by Luszczki - Zielesny, Carmichael and Vladimirov (see [LZ], [Ca1], [Ca2], [Ca3], [Ca4], [Ca5], [Vl1], [Vl2]). An excellent survey concerning its development up to 1977 may be found in [Mei4]. Applications of the theory of distributional boundary values of holomorphic functions to quantum field theory and network theory can be found in [BW] and [Bre]. It should be mentioned that the first investigations made in the beginning of the fifties were followed up only in the seventies by results concerning the representation of ultra-distributions and this in the works of Komatsu [Kom] and K6rner [Kor]. Recently Petzsche succeeded in giving a unified approach to the relationship between generalized functions (distributions, ultra-distributions, hyperfunctions) and boundary values of holomorphic functions, both in the one and several dimensional cases (see [Pe1]). Note that the concepts of hyperfunctions and microfunctions were worked out essentially by Sato, Kashiwara and Kawai (see e.g. [Sa], [SKK]). In the Sections 25-27 we deal with the problem of representing distributions in Rm by monogenic functions in Rm+ 1 ~m. Although the main idea goes back to the work of Martineau, our techniques are quite different. Note that the isomorphisms we finally obtain in 27.9 are of a pure algebraic nature. Nevertheless, just as in the case of one complex variable, it may be proved that they are also topological. The nature of our approach, namely to represent distributions in Rm as boundary values of functions in (m+1)-variables, should be compared with the one using several complex variables in order to distinguish its own character. In this context we also want to mention Langenbruch's papers [la1], [La2], [La3], [La4] and [la5], where distributions in Rm are represent~d as boundary values of nullsolutions of hypoelliptic differential equations; his methods again differ completely from our function theoretic ones. Finally note that ~yperfunctions may also be described in terms of monogenic functions as shown in Sommen's paper [So1]. It should be noted that hereby, just as in the case of one complex variable, the use of cohomology theory could be avoided. The theory of Fourier series has a long history and some basic questions already posed at the appearance of Fourier's work "Th~orie analytique de la chaleur" (1822) could only be answered by using Lebesgue's integration theory. The decomposition of a function f E L2 (S 1 ) , where the latter space may be 239

identified with L2 {[0,2rr]), may be found in most textbooks on real analysis (see e.g. [HS]). Also in the more dimensional case it is well known that the set of surface spherical harmonics forms a complete orthonormal system of eigenfunctions of the Laplace-Beltrami operator~*m+ 1 acting on the Hilbert space L2 (Sm) (see e.g. [Tr]). The problem of decomposing functions or functionals on Sm into a series of surface spherical harmonics, and hence the problem of generalizing the classical Fourier expansion for f E L2 (Sm), has been the subject of more recent research. As was already mentioned in Section 28, growth conditions upon the sequence (Sk)kEN of surface spherical harmonics have to be imposed in order to get series expansions of the functions or functionals under consideration. Such estimates were obtained by Seeley (see [Se1], [Se2]) for C and analytic functions, by Morimoto (see [Mor]) for distributions and by Hashizume-MinemuraOkamoto (see [HMO]) for analytic functionals. Furthermore Helgason showed in [Hel] that any harmonic function f in the unit ball admits a boundary value in a'(Sm), whence it may be decomposed into a series of surface spherical harmonics satisfying certain estimates. But also the converse is true, i.e. if a sequence (Sk)kEN of surface sphe~ical harmonics is given which satisfies the required estimates, then f(x) = L lxlk Sk(w) is harmonic in the unit k=O ball. The space of analytic functionals on Sm thus appears to be the largest space of functionals which may still be represented as boundary values of harmonic functions. The representation of a( 1 )(Sm;A) by M(r)(ffim+l,Sm;Al~a as sketched in 28.6 thus yields a refinement of Helgason's result. In this context it should be mentioned that Hashizume-Kowata-Minemura-Okamoto constructed in [HKMO] spaces of functionals on Sm which contain a'(Sm) as a proper subset. These spaces are connected with the Fourier-Borel transform (see also Chapter 5). Finally it should be no~ed that the representation of any f E L2 (r)(Sm;A) as the L2 -boundary value of f(w(1±E)) and the derived orthogonal decomposition of L2(r)(Sm;A) seem to be inherent in the function theory under consideration. Note also that the splitting of any surface spherical harmonic of order k into a sum of an inner and outer surface spherical monogenic, being themselves eigenfunctions of the spherical Cauchy-Riemann operator, has applications in the theory of electron spin (see [L]). Further investigations in this context have been carried out by Sommen in [So2]. The form of the Bergmann and Szeg6 kernels obtained in Section 29 for respectively the unit ball and sphere in ffim+l again illustrates the closeness 00 -

240

of monogenic function theory to the theory of holomorphic functions of one complex variable. Bibliography [Bre] H. Bremermann, Distributions, Complex Variables and Fourier Transforms (Addison-Wesley, New York, 1965). [BW] E.J. Beltrami, M.R. Wohlers, Distributions and the Boundary Values of Analytic Functions (Academic Press, New York, 1966). [Ca1] R.D. Carmichael, Distributional boundary values in VL , Rend. Sem. Math. di Padova 43 (1970) 35-53. p [Ca2] , Distributional boundary values in Vlp II, Rend. Sem. Math. di Padova 45 (1971) 249-277. [Ca3] , Distributional boundary values of functions analytic in tubular radial domains, Indiana u. Math. J 20 (1971) 843-853. [Ca4] , Representation of distributions with compact support, ManuscPipta Math. 11 (1974) 305-338. [CaS] , Distributional boundary values in the dual spaces of spaces of typeS, Pacific J. Math. 56 (1975) 385-422. [Hel] S. Helgason, Eigenspaces of the Laplacian: Integral representations and irreducibility, J. Functional Analysis 17 (1974) 328-353. [HKMO] M. Hashizume, A. Kowata, K. Minemura, K. Okamoto, An integral representation of the ~aplacian on Euclidean space, Hiroshima Math. J. 2 (1972) 535-545. [HMO] M. Hashizume, K. Minemura, K. Okamoto, Harmonic functions on Hermitian hyperbolic spaces, Hiroshima Math. J. 3 (1973) 81-108. [HS] E. Hewitt, K. Stromberg, Real and Abstract Analysis (Springer Verlag, Berlin, 1965). [H3] G. K6the, Die Randverteilungen analytischer Funktionen, Mach. z. 57 (1952) 13-33. [Ko] P.P. Kender, Funktionentheoretische Charakterisierung der Topologie in DistributionenrMume, Math. z. 123 (1971) 241-263. [Kom] H. Komatsu, Ultradistributions I. Structure theorems and a characterization, J. Fac. Sci. Tokyo 20 (1973) 25-105. [Kor] J. K6rner, Romieusche Ultradistributionen als Randverteilungen holomorpher Funktionen (Thesis, UniversitMt Kiel, 1975).

241

[LJ

P. Lounesto, Spinor valued regular functions in hypercomplex analysis (Thesis, Helsinki University of Technology, 1979). [La1] M. Langenbruch, Randverteilungen von Null6sungen hypoelliptischer Differentialgleichungen, ManusrJl'ipta Uath. 26 (1978) 17-35. [La2] P-Funktionale und Randwerte zu hypoelliptischen Differentialoperatoren, Math. Ann. 239 (1979) 55-74. [La3] , Fortsetzung von Randwerten zu hypoelliptischen Differentialoperatoren und partiellen Differentialgleichungen, J. Reine AngeUJ. f.1ath. 311/312 (1979) 57-79. [La4] , Darstellung von Distributionen endlicher Ordnung als Randwerte zu hypoelliptischen Differentialoperatoren, Math. Ann. 248 ( 1980) 1-1 7. [La5] -------------, Dualraum und Topologie der (lokal) langsam wachsenden Nu116sungen hypoe 11 i pti scher Differentia 1operatoren, f.1anuacripta Math. 32 ( 1980) 29-49. [LZ] Z. Luszczki, Z. Zielesny, Distributionen der RMume VL und Randverteilungen analytischer Funktionen, Colloq. Math. 8 (1961) 12~-131. [Mar] A. Martineau, Distributions et valeurs au bord des fonctions holomorphes, in Theory of Distributions, Proc. Intern. Summer Inst., Inst. Gulbenkian de Ciencia, Lisboa (1974) 113-326. [Mei1] R. Meise, Darstellung temperierter vektorwertiger Distributionen durch holomorphe Funktionen I, Math. Ann. 198 (1972) 147-159. [Mei2] , Darstellung temperierter vektorwertiger Distributionen durch holomorphe Funktionen II, Math. Ann. 198 (1972) 161-178. [Mei3] , Ra6me holomorpher Vektorfunktionen mit Wachstumbedingungen und topologische Tensorprodukte, Math. Ann. 199 (1972) 293-312. [14ei4] , Representation of distributions and ultradi stributions by holomorphic functions, in Functional Analysis: Surveys and Recent Results, Proc. Paderborn Conference on Functional Analysis (North-Holland, Amsterdam, 1977) 189-208. [Mor] M. Morimoto, Analytic functionals on the sphere and their FourierBorel transformations, to appear in a volume of the Banach Center Publication. [Pel] H.J. Petzsche, Verallgemeinerte Funktionen und Randwerte holomorpher Funktionen (Habilitation, UniversitMt D6sseldorf, 1981).

242

[Sa] M. Sa to, Theory of hyperfunctions, I and I I, J. FarJ. Sci. Univ. 1'-:;kyo 8 (1959/60) 139-193 and 387-436. [SKK] M. Sato, M. Kashiwara, T. Kawai, Hyperfunctions and pseudodifferential equations, in Lecture Notes in Mathematics 287 (Springer-Verlag, Berlin, 1973) 265-529. [Se1] R.T. Seeley, Spherical harmonics, Af.J:: :.Jortthly 73, part II, no 4 (1966) 115-121. [Se2] - - - - - , Eigenfunction expansions of analytic functions, Pr>oc. A/?l .. i". l·iath. Soc. 21 ( 1969) 734-738. [So1] F. Sommen, Hyperfunctions with values in a Clifford algebra, to appear in ,)'·::"!lor:. :Jt.·vin. [So2] Spherical monogenic functions and analytic functionals on the unit sphere, to appear in To~:;1o Jo;B•nal of 1-iath. [T1] H.G. Tillmann, Randverteilungen analytischer Funktionen und Distributionen, f.1ath. z. 59 (1953) 61-83. [T2] , Distributionen als Randverteilungen analytischer Funktionen I I, l-1cith. z. 76 ( 1961) 5-21. [T3] , Darstellung der Schwartzschen Distributionen durch analytische Funktionen, Math. z. 77 (1961) 106-124. H. Triebel, HBhere Analysis (VEB Deutscher Verlag der Wissenschaften, [Tr] Berlin, 1972). [Vl1] V.S. Vladimirov, On functions holomorphic in tubular cones, Izv. Akad. Nauk :JSSR Scr>. Matlz,;m. 27 ( 1963) 75-100. [V12] , On the construction of envelopes of holomorphy for regions of a special type and their applications, Amu•. Math. Soc. Tmnslation 48 (series 2) (1966) 107-150. [V1] D. Vogt, Distributionen auf dem ffiN als Randverteilungen holomorpher Funktionen, J. Reine Angcw. Matlt. 261 (1973) 134-145. [V2] ------ Randverteilungen holomorpher Funktionen und die Topologie von V', Nath. Ann. 196 (1972) 281-292. [V3] , Temperierter vektorwertiger Distributionen und langsam wachsende holomorphe Funktionen, !-lath. z. 132 (1973) 227-237. [V4] Vektorwertige Distributionen als Randverteilungen holomorpher Funktionen, f.Janu;;cr-ipta Math. 17 (1975) 267-290.

243

5 Transform analysis in Euclidean space

The first two sections of this last chapter are devoted to the Fourier transform and the generalized Laplace transform in higher dimension within the framework of monogenic functions and A-distributions. First we treat of the Fourier transform in S(ffim;A), V(ffim;A) and E(ffim;A) and their respective duals of A-distributions. This leads a.o. to a generalization of the GelfandShilov Z-space in one complex variable, which consists of all entire functions of the form f(z) =

JR

e-itz ¢(t)dt

for some¢ E V(R). Moreover an analogue of the classical Paley-Wiener-Schwar~ theorem is obtained (§30). Next the generalized Laplace transform of tempered A-distributions is studied, extending the complex Laplace transform im

z

> 0

im

z




0;

is surjective, bounded and open, while ker BV =OR([). In Section 32 we introduce a generalized version of the Fourier-Borel transform, which maps monogenic functions in annular domains into monogenic functions in the complement of the origin. This transform extends the transform o = o+ + o in the complex case, where T ~ 0

T being an analytic functional with carrier in an annular domain. the classical result

Moreover

Exp'([) being the class of Helgason's entire functionals, is generalized, 0 leading up to an integral form for the Laurent series expansion of a monogenic function in an annular domain. In the last section the Radon transform P:

O'(B(0,1))-. O(B(0,1))

given by 1

P(T)(u) = , ~

~

is generalized to higher dimensions using a monogenic version P(u,z),(u,z) E Rm+ 1 x [m of the function (1-u~)- 1 , (u,~) E [ x [. This results into an optimal version of the Cauchy-Kowalewski theorem (see 14.2) concerning the monogenic extension of an A-valued analytic function in some open subset of Rm 30.

Hypercomplex Fourier transforms

30.1 In this section we first introduce the Fourier transform of rapidly decreasing A-valued C -functions; the classical Fourier kernel e-itx,(t,x) E m m R x R is replaced here by the function E(t,x), (t,x) E R x R introduced in 15.4. 00

~-.

~~

245

Next we introduce the Fourier transform of A-valued testfunctions by means of the exponential function E(t,x), (t,x) E ~m x ~m+ 1 , introduced in 15.4, which itself is a natural generalization of e-itz, (t,z) E ~ x [and which fort fixed is monogenic in the whole of ~m+ 1 • This leads to a generalization of the Gel 'fand-Shilov Z-space in one complex variable. Finally the generalized Fourier transform of A-distributions with compact support is studied. A.

The Fourier transform in S(~m;A) and S*(~m;A)

30.2

Let ~ E S(r)(~m;A); then we define 1 , ••• ,m

and

The function F¢ is called the

Four•ier' tmncfor•m

of ¢·

30.3 Just as in the classical theory it is obtained that Fj is a topological automorphism of S(r)(~m;A) and

Theorem

F.-1 q,(t 1 , ••• ,t. 1,x.,t. 1 , ••• , t ) = -21 J JJ J+ m IT Proof

Let ¢

=

J+ooexp(-t.x.e.)~(t)dt -+ .• _ 00

J J J

1eA~A' where ¢A E S(~m;~) for all A E PN.

J

Then for each

component ¢A the classical Fourier inversion formula holds since exp(t.x.e.) J J J

= e0 cos(t.x.) + e.sin(t.x.). J J J J J

So for each A E PN, 1

~ Ti

J+ooexp(-t.x.e.)F.¢Adx. -co

J J J

J

J

1

[ F.~

J

LTI

J+ooexp(-t.x.e.)tAdt.Jl -oo

J J J

J

1,A(t).

As F. and F~ 1 are right A-linear the above inversion formula also holds for J J -1 ¢· Furthermore Fj and Fj are continuous. o 246

30.4

F- 1¢(x) 30.5

F is a topological automorphism of s(r)(Rm;A) and

Corollary

JRm E(t,;)~(t)dt. If we define for~ E s(l)(Rm;A)

Remark

= J+'~'q;(t)exp( t .x .e. )dt., -oo

J J J

J

j

1 , ••• ,m

and

then of course Fj{j S( l ) (Rm; A) • 30.6

1 , ••• ,m) and Fare topological automorphisms of

Definitions

(i) LetT E s(l)(Rm;A); its Fourier transform is the tempered left Adistribution FT given by = , ~ E S{l)(Rm;A). (ii)

LetS E s(r)(Rm;A); its Fourier transform is given by <SF,¢> = <S,F¢>, ¢ E S(r)(Rm;A).

. of both S(l)s * (Rm;A) It can be proved that F is a topological automorph1sm and S(l)b(Rm;A). 30.7

Introducing the r>eflection oper>ator>u Si,

1, ••• ,m, given by

S/ ("Jt) = f (X 1 , ••• , Xi _1 , -Xi , Xi+ 1 , ••• , Xm) where f stands for a function or an A-distribution in Rm, some calculation formulae for the Fourier transform can be stated: (i)

F(3t_f)("it) = -xiei s 1 ••• si_ 1 Ff(x) 1

247

(i i )

(iii )

F(e.f)(;) =e.1 s 1••• S.1- 1s.1+ 1••• 5m Ff(x) 1 ~

~

F(eiat_f)(x) =X; Si+ 1••• Sm Ff(x) 1

(iv) (v)

F(t;f)(x) = -ei aX; s 1••• S;_ 1 Ff(x) ~

~

F(t;e;f)(x) =ax. Si+ 1••• Sm Ff(x) 1

(vi)

~~

~

~

F(f(t+a))(x)=exp(-a 1x1e 1) ••• exp(-a 2x2e 2s 1) ••• exp(-amxmems 1••• Sm- 1)Ff(x)

where

B.

The generalized Fourier transform in V(Rm;A)

30.8 Let¢ E V(r)(Rm;A); then we define

f

~

~

~

F¢(x) = Rm E(t,x)¢(t)dt,

m+1 x ER •

It is clear that F¢ is left entire; moreover it is the unique left monogenic extension of F¢Cit)

=

~

~

~

E(t,x)¢(t)dt

which belongs to S(r)(Rm;A). Hence the following inversion formula is obtained:

30.9 Call Z(r)(m;A) the module of all functions F¢ where¢ runs through V(r)(Rm;A). Obviously Z(r)(m;A) is a submodule of M(r)(Rm+ 1;A). This module Z(r)(m;A) is now characterized by means of estimates in the following theorem, the proof of which is rather straightforward. Theorem Let¢ E V(r)(Rm;A) have its support contained in B(O,R). Then for each ~s E ~ m and s > 0 a constant C 6 > 0 may be found such that for all m+1 s '" X E R

248

s

30.10 Remark In virtue of Cauchy's Representation Theorem, for any E ~m. m+1 m a E~ and c > 0 there exists c~R.a,c > 0 such that for all ~ E V(r)(R ;A)

In the following theorem such estimates are used to determine completely the space Z(r)(m;A). 30.11 Theorem Let f E M(r)(Rm+ 1;A) be such that for a certain R > 0 and for any ! E ~m. a E ~m+ 1 and c > 0, there exists c~ > 0 such that f3,c:t.E

IXB aaf(x) I X

0

0. Choose s > 0 arbitrarily and take o > 0; applying Cauchy's Theorem yields

From the assumed estimates it follows that for a certain constant C > 0 ( R+c) IX I o 1+1XIm+1 o ~ m and as A is bounded for x E R we get lf(x)lo:;;:Ce

249

In an analogous way it can be shown that for any ,.,

0

I r m B(t,x)f(x)dxlo ~ C"e(R+c-ltl ),'.. J]R

Now iftElRm.....S(O,R+d then R+s-!tl < 0 and taking limits for,~++oo in the + above inequalities leads to ~(t) = 0. So supp~ c B(O,R) and it is possible to consider its generalized Fourier transform F¢(x). The function f-F¢ is left entire and it vanishes in JR~+ 1 • Hence it is identically zero in lRm+ 1 or f(x) = F¢(x). o 30.12 Now we are able to endow the module Z(r)(m;A) with a locally convex topology. Let k,s E ~and call Z(r),k,s(m,A) the submodule of Z(r)(m;A) consisting of those left entire functions f which are such that for each E ~m and a E ~m+ 1 there exists a constant C+ > 0 such that

e

s.a

+

lxB

(

a~f(x) lo ~ c!.ae k+s

-1)

lxol.

Provided with the proper system of seminorms {Pj,l :(j,l) E ~ 2 }, where sup e-(k+s lu.l:;:;l

-1

+

Z(r),k,s(m;A) becomes a right Frechet A-module. Z(r)(m;A) = lim ind lim proj kE:tl

sEJl

a~f(x)lo•

)lxollxB

Then we put

Z(r),k,s(m;A).

Notice that z(r)(m;A) is an inductive limit of right Frechet A-modules. 30.13 The following topological result may be proved along the same lines as in the classical case of one complex variable. Theorem The Fourier transform

is a topological isomorphism. 30.14 250

Denote by z(r)(m;A) the dual module of Z(r)(m;A), i.e. the left A-

module of bounded right A-linear functionals on z( )(m;A). By means of this . * m r dual module a Four1er transform on V(r)(R ;A) may be defined. Indeed, if T E Vtr)(Rm;A) then its Fourier transform is defined as to be the element of z(r)(m;A) which acts on functions f E Z(r)(m;A) as follows: = , for all f E Z(r)(m;A). It is clear that this transformation F is a topological isomorphism between * m;A) and Z(r) * (Rm;A ) • bot h prov1"d ed either with the weak* or strong V(r)(R topology. 30.15

If¢ E V( l)(ffim;A) then we define ¢F- 1 (x) = J ¢(t)E(t,x)dt. Rm

It is clear that ¢F- 1 is right entire. Call z(l)(m;A) the module of functions ¢F- 1 where ¢ runs through V(l)(Rm;A). This module Z(l)(m;A) may be characterized in the same way as was done for z(r)(m;A). Moreover a locally convex topology may be defined on it such that Z(l)(m;A) and z~ 1 )(m;A) are topologically isomorphic to v(l)(P.m;A) and v( 1 )(ffim·A) respectively. C.

. E(l)(ffi * m;A) The generalized Fourier trans f orm 1n

30.16 LetT E E( 1 )(ffim;A); then of course T E S(l)(ffim;A) and soFT is defined and belongs to s(1 ) (Rm;A); moreover for any ¢ E s(l) (ffim;A) holds





f""\J

-+

-+

¢(x)T(x)dx where we have put

251

So if we consider the function "' T(x)

..... = , x E Rm+ 1 ,

we have at once that .....

FT(x}.

Hence it is quite natural to define 30.17 Definition function

* (Rm;A) is the The generalized Fourier transform ofT E E(l)

... m As T is left A-linear and bounded on E(l)(R ;A) and as E(t,x) is analytic in Rm x Rm+ 1 it is easy to show that FT is left entire; it is in fact the unique '\, ..... left monogenic extension of T(x). 30.18 Theorem LetT E E{ 1 )(Rm;A) and let R > 0 be such that supp T c B(O,R). Then there exist C > 0 and k E J~ such that for all x E Rm+ 1

Proof The desired inequality follows immediately from the definition of FT. 30.19 The converse of Theorem 30.18 may be regarded as the analogue of the famous Paley-Wiener-Schwartz Theorem. The proof, which is rather technical and may be developed analogously to the proof of Theorem 30.11, is omitted. Theorem (Paley-Wiener-Schwart~) Let f E M(r)(R m+1 ;A) and let R that for some C > 0 and k E ~

0 be such

. * m+1 ;A) with supp T c B(O,R) Then there ex1sts T E E(l)(R such that f =FT. 30.20 Let k E ~and let R > 0; call ~(r),k,R the module of all left entire functions f satisfying an estimate of the form

252

c

for some C > 0. Equipped with the norm

~

(r),k,R is a right Banach A-module.

Putting

~(r)(m;A) = li~.~nd ~(r),k,R

the following result is obtained. 30.21

Theorem The generalized Fourier transformation

* m;A) .... ~(r)(m;A) F:E(l),b(lR is a topological isomorphism. Proof Use Theorem 30.3 and 30.19 and an analogous reasoning as in the case of one complex variable. c 31.

The generalized Laplace transform in

s(1 )(1Rm;A)

31.1 First we introduce the Laplace transform ~L- 1 of a testfunction ~ E S(l)(lRm;A) vanishing in a closed ball centered at the origin. As ~L- 1 is right monogenic in lR~+ 1 we may investigate its S(l)-boundary value for xo .... 0±.

~F

-1 ....

(x)

= JRm

~(t)E(t,x)dt

....

= JlRm

~(t)[B(t,x)-A(t,x)]dt

+

0

....

....

o++

....

o-++

belongs to S(l)(lRm;A). Now take ~ E.S(l)(lRm;A) with ~ ~L- 1 in JR~+ 1 given by

-+

0 in Bm(O,R) and consider the function

253

¢L

-1

JIRm (x)

-+ ¢(t)B(t,;)e-ltixodt,

{J!Rm ¢(t)A(t,;)elt[xodt,

X

E

1Rm+1 •

1 . c 1ear th a t ¢L- 1 1s . 1n . 1Rm+ h 11 prove th at ¢L- 1 . r1g . ht monogen1c It 1s 1 ; we s a -1 admits ¢F as S(l )-boundary value. Thereto we need a lemma on estimates of ¢L- 1 , the proof of which is omitted since it is based on straightforward estimates.

e

31.3 Lemma Let¢ E S(l)(!Rm;A) vanish in Bm(O,R). Then for each E Nm, m~ > 0 such that for all a E Jl and s > 0 there exists a constant C-+ B,u.' r: m+1 1R X E f

31.4

Proposition 1im ¢L X +0+

-1

The boundary values

+ (x±x 0)

0

exist in S(l)s(!Rm;A) and BV¢L- 1(x) = lim X +0+ 0

Proof It follows from the previous Lemma 31.3 that for each S E Jlm and a E Jlm+ 1 there exists a constant C+ > 0 such that for all x0 E]0,1]

s.a.

-1 I+s X dCJ. ¢ L (X) I x o

Hence for each

254

Z E Nm

~

C+

s.a. •

and x~ 1 ), x~ 2 ) E ]0,1]

Hence

exists in S(l)s(lRm;A).

Analogously

exists in S(l)s(lRm;A) and moreover

converges to 4-

0

-+ -+

-+

¢(t)E(t,x)dt for x0

-+

0+.

¢F

-1

-~

(x)

o

31.5 A converse of Lemma 31.3 runs as follows. m+1 Lemma Let f E ~1(ll(lR# ;A) be such that there exists R > 0 for which, given any E ~m. a E ~m 1 and 0 < E < R, a constant C > 0 may be found such S,a,E that

s

Then there exists a testfunction that f ¢L- 1

-->-

->-

J~ ¢(x)LT(x+x 0 )dx = . r~oreover'

as ->-

-.. -..-.. -JtJx Odx-.. -.. J m¢(x)B -.. (t,x)dx -..-..-.. JJRm ¢(x)B E (t,x)e JR E

In an analogous manner we get that

... r -.. -.. -.. -.. ! ¢(x)A (t,x):ix > JlRm ¢(x)LT(x-x o )dx = ->) JlRm¢(x)[LT(x+x o

1 im X -.-0+

->-

->-

- LT(x-x )]dx o

0

-jo.

-+-+

-jo.-jo.

-+

= "L

258

lRm

c

E.

+

-~

= .
0, k, r 0 < f: ..-: R, a constant CE: > 0 may be found such that in JR~+ 1

E

f'l and any

* (lRm;A ) such that Then there exists a unique tempered A-distri bution T E S(l) T is zero in Bm(O,R) and LT = f. Proof As for x0 E]- 1,0[U]0,1[

f admits an s( 1 )(JRm;A)-boundary value for x0 exists a unique T E s(1 )(JRm;A) such that FT

= BVf = lim X

0

+0+

+

0 (see 26.8).

Hence there

[f(x+x ) - f(x-x 0 )]. O

First we prove that TiBm(O,R) = 0. Take a testfunction ~ E V(l)(JRm~A) wit~ suppcp c: Bm(O,R) and take E: > 0 sufficiently small such that suppcp c: Bm(O,R-2,J Then 0 such that 3 1 1 (R- 7d IXol I( 1+ Ix 12 ) r+m+ 0 be fixed and letT

E

1Rm+ 1

X E

1Rm+ 1

X E

1Rm+1

LT(x), x

{

0

,

E

s( 1)(1Rm;A) vanish in Bm(O,R).

+

and 0

L_T(x)

{

LT ( X )

'

+

x

,

E

IRm+ 1•

1 d Bot h f unct1ons are 1e ft monogen1c 1n IRm+ 1 an 0

0

0

Moreover, just as LT does, they both satisfy an estimate of the form appearing in Theorem 31.9. Then by Theorem 31.12 there exist unique tempered diso( O,R ) and such that tributions P+T, P_T E S(* l) (lRm;A ) vanishing in B

and LP T = LT. Furthermore P!T = P+T,

P~T

P_T,

P+T + P T = T and P+ P- T

= P- P+T = 0.

Moreover it can be shown that

263

m j =1

s1 ••• s.J- 1t.)T. J

s(

This decomposition ofT E 1)(Rm;A) is illustrated as follows in the cases where m = 1 and m = 2. If m = 1 then it is an easy matter to check that

notice that here p_ is the restriction operator toR+. For m = 2, let + -+t = (t 1 ,t 2 ) E R2 and call e the polar angle of t. Then it is obtained that

-+-

P+Tt = ~ (1±cose)T(t 1 ,t 2 ) ± ~(sine)T(-t 1 ,t 2 )· In complex analysis the Laplace transform can be defined as follows. e.g. f be a continuous function of slow growth in R2 and put

Let

-+-

p±1,±1f = fi{t E IR 2

:

+t 1 > 0, :;t 2 > 0}.

Then for (o 1 ,o2 ) E {1 ,-1} 2 one can define

this function is holomorphic in transform which is defined by L 01•02

is holomorphic in

{[~) 2

f(~ 1 ,~ 2 )

{(~ 1 ,~ 2 )

if sgn

E [ 2 : sgn

Im~ 1.

Im~i

oi}.

So the Laplace

= o1.,

and can be split up into four parts:

which correspond to the Laplace transforms of the restriction of f to the 'octants'; hence one could say that this Laplace transform is of a 'cartesian nature'. In the hypercomplex approach here presented, the Laplace transform always consists of two parts Lf

= LP +f + LP·- f

where P± are the above introduced 'orientation operators' in the Euclidean 264

space, which rather have a 'spherical nature'. 'natures' coincide. 32.

Only in the case m

1 both

A generalized Fourier-Borel transform

32.1 In this section we study a generalized version of the Fourier-Borel transform of certain classes of analytic functionals with support in an annular domain or the complement of an annular domain in ~m+ 1 • The transform a = a+ + a_ under consideration reduces in the complex case to

o_

T being a complex analytic functional with carrier in an annular domain. In the present monogenic approach the function~ eu/z, (u,z) E [ x [ 0 is replaced by the function H(u,y), already introduced in 15.8.3~ which for y E ~~+ 1 fixed is left and right monogenic in y E ~~+ 1 with lim H(u,y) = 0. Y-><x>

Notice that first some modules of monogenic functions and analytic functionals in annular domains and the complement of annular domains are introduced, the images under o of which will be determined.

32.2 Let R1 and R2 be positive constants. The notation

First assume that 0


0 there exists c6 > 0 for which (32.1) hold. Then S(r)(R 1 ,R 2 ) is a right (FS)-A-module fqr the system of seminorms P = {pj : j E ~} where

where

k 1 -k IPkCu)l • Rz-k (1+ ~J) 1 -k IQk (ul ) 1 }. p.(s) =sup sup {Rl(1+ -J.) 0 0 J k€14 tuESm

267

Furthermore the mapping from M(r)(R 1 ,R 2 ) to S(r)(R 1 ,R 2 ) which maps a function f into its associated sequence of surface spherical monogenics, is a topological isomorphism. Moreover if (Pkf(u)), 0,~f(uJ\EINES~r)(R 1 ,R 2 ) is the sequence corresponding to f E M(r)(Rl'R 2 ) , then to f E H(l)(R2, R~1) the sequence (Qkf(w), P'kf( 0 1))kEINES(l)(R;1, R~ 1 ) is associated. m+1 32.5 As for each ~2 clR open, M(r)(~;A) and r~(l)(";A) are (FS)-A-modules when endowed with the topology of uniform convergence on the compact subsets of~. it is meaningful to introduce the following modules. Definition 1 i m i nd O 0

is chosen sufficiently small.

32.9 The duality between M(r)(R1,R 2 ) and M(l )(R 2 ,RI) may be regarded as an 'inner product dual1ty between M(~)(R1,R 2 ) and M(r)(R1-1 R2-1 ) in the following sense. -1 -1 '\, Take g E M(r)(R1,R 2 ) and f E M(r)(R1 , R2 ) ; then f E t1(l )(R 2 ,RI). Hence it is quite natural to define an inner product between f and g by putting •

1

-

'\,

(f,g)

.

269

Furthermore this duality may be expressed in terms of the associated sequence spaces. ForgE M(r)(R 1 ,R 2 ) the associated sequence (Pkg(,,J), Qkg(u.J\Ei'J is in -1 -1 . . ~(r)(R.:;R 2 ~i for f E M(r)(R 1 ~ R2 ) the sequence (Pkf(,D), Qkf( 0 J))k.Q-J 1s 1n S(r)(R 1 ,R 2 ), while for~ E M(l)(R 2 ,R 1 ) the associated sequence (Qkf(w), 'V PkfC

J

H(y,u)dou T(u)

a[B(R 2+n),B(Rl-n)] r

J H(y,u) dou T(u), aB(R 1 -n) which is a left entire function. On the other hand if R1 > R2 we introduce the functions

275

H(u,y),

1 ( 2R +R ), y E Rm+1 1 2 0

IU I

< ~

luI

>~

f 1

H+(u,y)

0

1 ( 2R 1+R 2 ), y E Rm+1 0

and u E Rm+1' IYI

0

f 1

H_(u ,y)

H( u ,y) ,

u

€

m+1 R • IY I

1

< ~(R 1 +2R 2 )

1

> ~(Rl+2R2).

Then clearly H+(u,y) E M(l)(R 2 ,R 1) with respect to the variable u, while H_(u,y) E M(l )(R 2 ,R 1) with respect toy. So we are able to define in R~+ 1 :

(

J

H+(u,y) dou T(u)

a[B(Rl-n),B(R2+n)]

J

H(u,y) dou T(u)

aB(R 2+n)

=-I

H(u,y) dou T(u) as{R1+R2) 2

which is a left monogenic function in R~+ 1 with lim o+(T)(y) = 0. y~ ogously we put in Rm+1': o_(T)(y) =

0 a constant C

>

0 may be found such that in

E

0

which means that a(T) E Exp(r)(R 1 ,R 2).

The case R1

>

R2 is treated similarly.1

32.18 Actually the inclusions stated in the above Theorem 32.17 are equalities between the considered function modules. This will be proved in 32.19, but first the images of M(r)(R 1 ,R 2 ) and M(r)(R 1 ,R 2 ) under the transformation a are characterized in terms of their associated sequence spaces. _Theorem Let f E ~4(r)(oo,O) = M(r)(JR~+ 1 ;A) have (Pkf(c,l),Qkf(w))kElN as sequence representation in S(r)(oo,O). Then (i) f E a(M(r)(R 1 ,R 2 )) if and only if for every n such that

>

0 there exists en> 0

279

{ (ii) that

IPkf(w)lo

:; c11

IQkf(w) lo

:; c

( 1+n)K

k!

-k Rl

( 1+~)k

k Rz;

k.

f E o(M(r)(R 1 ,R 2 )) if and only if there exist C

>

0 and 6 > 0 such

Proof Assume that R1 :;; Rz • Take T E M(r)(R 1 ,R 2 ) and put f(y) (IR~+ 1 ;A). Then

I

f(y) = aB( =

I

2

H(u,y) dau T(u) +n)

L ~!

aB(R 2 +n) k=O

I

o(T)(y)

E

M(r)

H(y,u) dou T(u)

aB(Rl-n) qk(u,y)dou T(u)aB( 00

L

r

t

00

I

1 TI qk(y,u) dou T(u)

-n) k=O

1

IT pk T(y)

k=O wm+1 PkT(w ) and Qkf ( w) = ~ ~+1 which means that Pkf(w) =- ~ QkT ( w). The statement (i) then follows from the sequence representation (PkT(w), QkT(w)) E S(r)(R 1 ,R2 ). The cases T E M(r)(R 1 ,R2 ) with R1 > R2 and T E M(r)(R 1 ,R 2 ) are treated along the same lines. c 32.19 Finally we show that a is a topological isomorphism between M(r)(R 10 R2 ) (resp. M(r)(R 1 ,R2 )) and Exp(r)(R 1 ,R2 ) (resp. Exp(r)(R 1 ,R2 )), all modules under consideration being endowed with their respective natural topologies. Theorem The following topological isomorphisms hold:

280

Proof It is clear that (ii) follows from (i) by taking induct·ive limits. Now by Theorem 32.17 it is already known that

In view of Theorem 32.18 we still have to prove the necessary condition of Theorem 32.11.3 in order to conclude that

m+1 So let therefore f E Exp(r) (R 10 R2 ) . i.e. f E t-1(r) (R0 ;A) and for every E: there exists C > 0 such that E: lf(x) lo:;;; CE: exp [(1+d

>

(~ + 1~1 )].

Expending f into its Laurent series f(x) =

I

lxlk Pkf(w) +

k=O

I

lxl-(k+m) Qkf(w)

k=O

where, independently of r E ]O,+oo[,

it is obtained that forE:

:; ; C'

E:

>

0 and for all r

>

r0

>

0

1 exp(1+E: ) -R r • (1+k) m --.. rll. 1

For all k E N such that r 0


J

while

I

o(E(u-x)) (y)

J

E(u-x) dou H(u,y) -

aB(R 1-n)

H(x,y)

E(u-x)dou H(y,u)

;)B( R2+n)

H(y,x)

+

and hence f(x)

= ,

0

-

x E B(O,R 1 ),B(O,R 2 ),

yielding, when~ is an A-valued measure representing T E Exp( 1)(R2 ,R 1 ), f(x) =

I

H(x,y)dpy

+

1Rm+1

r J

o

H(y,x) dpy' x E B(O,R 1 ),B(O,R 2 ).

1Rm+1

In terms of the Laurent series expansion of f this leads to

and

L

oo

k=O

Qkf(x) =

I

H(y,x)

d~y'

x E lRm+1 'B(O,R 2

).

1Rm+1

Those two integrals might be called the "Taylor and Law•ent integrals" of f E M(r)(R 1 ,R 2 ) (R 1 > R2 ). A same reasoning may be followed in the case where R1 s R2 leading up to the same expressions for the "Taylor and Laurent integrals".

283

33.

Connections with complex analysis- a generalized Radon transform

33.1 In [8] Hayman proved that every harmonic function, and hence every holomorphic function, in the open disc {r; E [ : lsi < R} admits a multiple Taylor series expansion about the origin converging absolutely in the domain {(x,y) E JIF : JxJ + IYI


0 small enough such that B(O,r) cU. Taking into account that ¢(u,y) is left and right monogenic in both variables in the region lui

k

k=O F(u,z).

c

33.10 Now we define a generalized Radon transform. Definition Let D* Then we put

c

[m be a domain of holomorphy and let T E

o( 1 ) CJ*;Ac).

PT(u) = , u belonging to a suitable open subset r2 of 1Rm+ 1 depending on ::z*.So observe that if P(T) is well defined it is left monogenic in 12. Notice also that P transforms analytic functionals in complex domains into left monogenic functions. Furthermore P(TA) = P(T)>..

* ( g* ;Ac) and all >.. E Ac • for all T E O(l) Finally note that form = 1 and by the 'classical' identifications, P reduces to P( T)( u)

= • ( u •d z;,

E [

X

[

1-uz;,

which implies, by K6the's duality theory [12], that Pis then a topological isomorphism from O'(B(0,1)) to 0(8(0,1)). 33.11 Our aim now is to characterize P(O*(l)(n*;Ac)) in the particular cases .... 'V-+ where g* is either A(R) or A(R) or still B*(O,R). We first prove .... * - .... c ). Then P(T) is left monogenic in rr(R) and its Theorem LetT E O(l)(A(R);A multiple Taylor series expansion about the origin converges absolutely in .... rr(R).

* - .... c ). Proof Take T E O(l)(A(R);A that

290

Then for each

E >

0 there exists C

E

>

0 such

-+

c

m

->-

for all 1> E 0(1) (~(R+2t) ;A ) , where R + [~ stands for (R 1+E•· •• ,Rm+E) E R • Now let u E n(R) be fixed and choose~> 0 such that u E n(J + 2l ); then -+ -+ c -+ of course P(u,z) E O(l)(AlR + 2t);A ). So for all u E n(R} the function P(u,!) belongs to O(l)(f.(R);Ac) and P(T)(u) = z -+

is well defined. Hence P(T) is left monogenic in rr(R). -+ u E n(R) as above we have P(T)(u) =

-

c

it may be proved that if TEOQ) (t,(R) ;A )

P(T) is left monogenic in rr(RJ or rr*(R) series expansion about the origin conrespectively.

converse of Theorem 33.11 we need the following ->'\,->[13]). Let g* be either A(R) or A(R) or still '\,->->1 Q* equal to A(R) or A(RJ or B*(O,R- ). Then the given by

1

1-

>

is a topological isomorphism. If T E 0( 1 )(~*;Ac) then there exist TA E 0'(~*), A c {1 , ••• ,m}, such that T = L eATA in 0( 1 )(~*;Ac), and it is an easy matter to check that the transform~tion which maps Ton L eAP*(TA), and which is still denoted by P*, is a topological isomorphism fro~ the Frechet Ac-module O{l)(~*;Ac) onto the Frechet Ac-module O(r)(n*;Ac). 33.14 Theorem Let f be left monogenic in a neighbourhood of the origin such ..... '\,-+ that its multiple Taylor series converges absolutely in either rr(R) or rr(R) or still IT'ii(R). Then there exists T E OCl)(A(R),Ac), respectively T E 0(1 ) (A(R);Ac) or T E o( 1)(B*(R);Ac), such that P( T)

= f.

Proof Let f be left monogenic such that its Taylor series expansion about the origin converges absolutely in n(R). Then f(t) = f(x)IR~+ 1 admits an absolutely converging multiple Taylor series expansion in m

rr(R)JR:+ 1

=

{tERm :

L

Rj Jxj I

< 1}.

j =1

Hence its complex extension f(!) is holomorphic in A(R) and so by the con* )(A(R);A - . . . c ) such that siderations made in 33.13 there exists T E O(l 292

p*(T) (!)

f(t)

or j =1 J ax j j =1 J j =1 J

_a

= ~+1, a refinement is obtained of t4orimoto's result for the case E'(Sm- 1). Observe too that when m = 2, the entire functionals in 0:'{0} introduced by "' . Helgason in [Hel] represent the space B(S 1 ). The theory of analytic functionals and their Fourier-Borel transforms is of fundamental importance in solving existence and approximation problems for the equation P(Dz)u = h, h being holomorphic in an open convex subset ~ of O:m and P(Dz) being a differential polynomial in az , ••• ,az • The Fourier-Borel transform itself was 1 m introduced by Martineau in his basic work [Mar1]. To fix the ideas, let T be an analytic functional in 0: and call FT(z) = . Then F transforms analytic functionals into entire functions of exponential type, where the growth conditions of FT are determined by the carrier ofT, and conversely if f is an entire function of exponential type, then f = FT for some analytic functional T in 0: having convex carrier. In [Mor1] Morimoto extended the Fourier-Borel transform to analytic functionals having non convex carrier. LetT E 0'(~) where~= B(O,R 1 ),B(O,R2 ), and put F,T(u,v) = ..(uz+! )>, z z (u,v) E 0: x 0:. Then in this way a transform from 0'(0.) to spaces of solutions of the equation at2:v = A. 2 F is obtained. Morimoto also showed that for each T, FA.T is completely determined by its restrictions to the coordinate axes, i.e. F(u,O) and F(O,v), and hence by 1\

298