Convolution theory in a space of generalized functions

Citation for published version (APA):

Janssen, A. J. E. M. (1978). Convolution theory in a space of generalized functions. (Eindhoven University of Technology : Dept of Mathematics : memorandum; Vol. 7807). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1978

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Memorandum 1978-07 June 1978

Convolution theory in a space of generalized functions

University of Technology Department of Mathematics PO Box 513, Eindhoven The Netherlands by A.J.E.M. Janssen

•

by

A.J.E.M. Janssen

Introduction.

This paper presents a convolution theory for the test function

*

space S of smooth functions and the space S of generalized functions . . . as introduced by De Bruijn (the terminology and notation is the one used in [B], where these spaces are defined). The space S can be regarded as an example of a test function space of the type studied in [GSJ, Ch. IV (actually, our space S can be identified with the space

st

of

*

[GS], Ch. IV, § 2.3). Since the spaces Sand S are adapted to the needs

of Fourier analysis (cf. [BJ, section 8 and 9, and [GS], Ch. IV, § 6), it was to be expected that it is possible to develop a satisfactory convolution theory for these spaces; it seems however that no such· theory has been published thus far.

Let us summarize the contents of this paper. Section 1 gives the

*

main definitions and theorems about the spaces Sand Stand some results about continuous linear transformations in these spaces are mentioned. This section is mainly included here for ease of reference.

Section 2 serves as a preparation. The convolution operators introduced here involve

follows. (I) I f g E S, (T f) (x) g then

smooth functions only, and they are the convolution operator T of S is

g 00

J

f(x - t)g(t)dt (x E Ie) - 0 0 defined defined as by

for f ~ S. Instead of the integral at the right hand side in (I) we can also write (Txf,g_), where Tx is the shift operator over distance x, and fL is the smooth function with values g_(t) = g(-t) for t E Ie.

Supported by the N(~tllCrlands Organization for the P.dvancement of Pure Research

<Z.w.o.).

•

In section 3 we generalize the notion of convolution operator.

*

The g in (I) is replaced by a generalized function: if G E S , then the convolution operator TG of S is defined by

(2) (x E (1:)

for f L S. Here G bears a similar relation to G as g_ does to g in

the previous paragraph. Special attention is paid to the case that TG maps S into S, and the class of all G ~ S* with this property is called

the convolution class

C.

For G E

C

we prove that TG has an adjoint, and

that TG can be extended in a natural way to a continuous linear operator of S*. We also discuss some alternative descriptions of the class

C.

4It

Section 4 presents a link between convolution theory and Fourier analysis. This involves what we call multiplication operators of Sand S*. If g E S, then the multiplication operator M of S is defined by M f

=

g·f

g g

for f E S, where the dot denotes pointwise multiplication; this

multi-plication operator can be extended in a natural way to a continuous linear

*

operator of S • We obtain a useful characterization of the class

C

in

terms of the Fourier transforms of its elements. Furthermore the convolution theorem is generalized in section 4, and a version of Titchmarsh's

theorem is proved. Finally we mention some results about the solutions F E S* of equations of type TGF

=

0, where G is a fixed element of

C.

Section 5 contains some additional material. There we prove that

the class of generalized functions of the form MgG with g E S, G E S* is ~

a proper subset of

C.

We make some remarks about convergence in

C,

and finally we pay some attention to convolution theory for the spaces of smooth and generalized functions of several variables.

Notation.

We use Church's lambda calculus notation, but instead of his A we have the symbol

y,

as suggested by Freudenthal: If S is a set, then

putting

r

XES in front of an expression (usually containing x) means to

indicate the function with domain S and with the function values given by the expression. For example, if g E S then Tg YfESTgf. In case it

LS clear from the context which set S is meant, we write ~x instead of I;J

*

1. The spaces Sand S •

1.1. We give a survey of the fundamental notions and theorems of De Bruijn's

I. 2.

theory of generalized functions (as far as relevant for this paper). Also, the main theorems of [J], appendix I about continuous linear operators of Sand S* are given. More details can be found in [B] and [J] .

The class S (of smooth functions) is the set of all analytic functions f of one complex variable that satisfy inequalities

If (t)

I

M exp(- 1f A (Re

tf

+ 'IT B (1m t) 2) (t E 0:)

,

where M > 0, A > 0, B >

a

depend on f. In S we take the usual inner product, denoted by ( )

.

Cf. [B], 2. I •

We consider a semigroup (N ) of linear operators of S (the Ct Ct>o

smoothing operators); they satisfy NCt+S =: NCtNS (Ct> 0, S > 0). These operators are integral operators (integration over lR) ; the kernels K (Ct > 0) are given by

Ct

•

(z EO:, tEO:

Cf. [B], section 4. 5 and 6. The operators N (Ct Ct

+

the larger space S consisting of all mappings f

> 0) lR have can be defined on -* 0: such that N f € S for Ct

~

tclR f(t)exp(-1f£t2) c I:. lOR) for every c > O. We

+

f t S , Ct >

a

(compare [B], section 20, where an equivalent definition

of S+ is used). Note that £2(lR) c S+.

1.3. We summarize some properties of N (Ct > 0)

Ct

(i) (N f,g)

=

(f,N g) for Ct > 0, f E S, g e S (cf. [BJ, 6.5).

ex ex

(ii) If f E S, Ct > 0, then there is at most one g E S with f

=

N g. Ct Also, if f E S, then there exists an Ct > 0, g E S with f

=

N g.

Ct And if f E S, and the numbers M > 0, A > 0, B >

a

are such that

then we can find an a > 0, M' > 0, A' > 0, B' > 0, only depending on A and B, such that the inequalities

(t E D:) hold for the unique g E S with f

=

N g (cf. [B], 10.1) •

a 1. 4.

(i)

We list some other linear operators of S (cf. [BJ, section 8 and 11).

The Fourier transform F and its inverse F*:

00

Ff

~

J

e-2niztf(t)dt,

= IZED: F*f ~

y

ZE o:(Ff) (-z)

- 0 0

(ii) The shift operators Ta (a E 0:) and

1),

(b E 0:):

T f

=

Y

o:f(z + a), R f

=

Y

e-2nibzf(z)

a ZE -b ZED:

(iii) The operators P and Q:

Pf ::

Y

ZEIC

ff(z)

21fi Qf

=

y

ZEu. II'zf(z) (f

E S) •

(f E S)

(f E S) •

1.5. A generalized function F is a mapping a E (0,00) + Fa E S such that

N Fo = F Q (a > 0, B > 0). We also write F(a) or N F instead of F •

a ~ a+~ a a

It follows from 1.3(ii) that F =

°

in case F _a =

°

for some a > 0

*

(F c S ).

If F E S*, g C S, then the inner product (F,g) is defined as

follows: write g = N h with some a > 0, h E S (d. 1.3(ii», and put

a

(F,g) :- (F ,h) (this number depends only on F and g; cf. [B], section

a

17 and 18). We have (N F,g) = (F,N g) for a > 0, F

a a

*

We further define (g,F) :m (F,g) for F E S ,

*

E S , g E S.

g E S.

1.6. We give some examples of generalized functions.

(i) If f E S+, then the embedding of f (notation: emb(f» is defined

by

emb (f) :=Y·ONf. _{a> a}

I .7.

I .8.

+

Cf. [B], section 20. We have for f E S , g E S 00

(emb(f),g) =

f

f(t)g(t)dt.

- 0 0

It may be proved that f

=

0 (a.e.) if and only if emb(f)

=

O. (ii) For b t. 11:, the "delta function at bit is defined by

°b

:~

Yu>o

Ytca:KJt,b) •

and

Now (g'Ob)

=

g(b) for g E S (cf. [B], 17.3 and 27.18).

We next define convergence in S. Let (f) 1N be a sequence in S, n nE

let f E S. We write f ~ 0 if there are positive numbers A and B n

such that f (t)exp(TIA(Re

n

write f

~

f if f - f

~

t)2 - TIB(Im t)2)

~

0 uniformly in tEll:; we O. Similarly, we define f Ca )

~

0 (a

+

0) and

n n

f(a) ~ f (a t 0) if f(a) _{E S (a > 0), f E S. Cf. [B], section 23.}

The following theorem On S-convergence is useful.

Theorem. Let (f) 1N be a sequence in S •. The three following statements

n nE are equivalent.

(i) f

~

O. n

(ii) There exist a > 0 and g E S (n c IN) such that f

n n N a n' n g g

~

O.

-(iii) There exists an M > 0, A > 0, B > 0 such that

and f ~ 0 pointwise.

n

(t E 11:) ,

Proof. Equivalence of (i) and (ii) follows from [B], 23.1, and equivalence

I .9.

I • JO.

We proceed by defining convergence in S*. Let (F n) nE:lN be a sequence in S *

,

and let F E S • We write F * + S* 0 if N F ~ 0 for

S* s* n a n

every a >

o·

,

we write F + F if F - F + O. 5imilarly we define

n n

F(S)

§.

*

F (fS) s*

(13 '" 0) and (6 if F(t3) * (6 > 0), F E:

*

0 + F i- 0) E 5 S _•

[BJ, 24.2 states: a sequence _{(F n) nElN} 1.n S * 1.S S -convergent

*

if and only if. lim (F ,g)

n exists for every g E 5. n-l-«>

I t is not hard to prove from 1.8 that (F ,f ) + (F ,f) if n n

s*

§.

f,

F _n + F, f _{n where (Fn)nElN is a sequence in 5* and (fn) nElN is a} sequence 1.n 5.

We are going to study continuous linear transformations of S

*

and S •

Definition. A linear functional L of 5 is called continuous if Lf _n + 0 for every sequence (f) IN in S with f

~

O. A linear operator T of S

n nc n

is called continuous if Tf _{. n}

~

0 for every sequence (f) IN in 5 with _{n nE} f

~

O. The definitions of continuous linear functionals and operators

n

of S* are similar. (We use the word continuous instead of quasi-bounded, cf. [BJ, 22.2, and [j], appendix 1,2.2).

I. II.· Definition. A linear operator T of S is said to have an adjoint if for

* *

every g c S there is a g E S such that (Tf,g) =: (f ,g ) for every f E S.

*

.

.

* _{linearly on g I f we define}

-Such a g 1.S un1.que, and g depends E: S.

* *

T g := g for g E S

,

then T is a linear operator of 5, called the * adjoint of T,

*

*

*

Note that if T has an adjoint, then so has T , and (T) == T.

1.12. Example. We introduce some notation. If g E 5, then we define g := YZE~g(~),

g_ :=

~Z(~g(-z),g_

:==

YZE~g(-~),

Note that g E 5, g_ E: S, g_ E 5, and

that (g)_ = (8:)

= g_,

If F E S*, then we define

F

:=

Ya>OFa,

F_ :=~a>O(Fa)-'

F

:= ~ OCF) , Note that (by symmetry of the K 's; cf. 1.2)

F

E' 5*,

a>_ a -* a _ ~

F E S*, F_ E S, and that (F)_ (F_) = F , We have (F,g) == (F,g), (F_,g)

=

(F,g_), (F_,g) == (F,g_) for F E S*, g E S.

If-I is a continuous linear operato~f S, then we define 1 : == ~ _g~ sTg, T : = ~ S (Tg ) • 1 : = \J S (T -g ) • Now 1, T and 1 are

- gE - - - Igc - -

-continuous lineat operators of S with (1)

=

(~)

=

1_, and if T has an adjoint, then SO have T, T and 1 (1)*

=

~), (T_)* (T*)_,

--

*

*

(T_)

=

(T )_.

If T is a continuous linear operator Of~, then we define 1 :=

Y

_FES TF, T_:= YFES*(TF_)_,

T_

:= YFES*(TF_)_. Now

It

T and

T

*

-

-are continuous linear operators of Sand (T) • (T )

=

T •

1.13. Theorem. L is a continuous linear functional of S i f and Qnly if there

*

exists an F E S such that Lf

=

(f,F) (f E S). Such an F is unique.

Proof. Follows easily from [BJ, 22.2.

1.14. Theorem. Let T pe a linear operator of S. The four following statements are equivalent.

(i) T is continuous.

(ii) YfES(Tf)(x)is a continuous linear functional of S for every x E ~. (iii)TN has an adjoint for every a> O.

a

(iv) For every a > 0 there is a B > 0 and a bounded linear operator T)

o

of S (bounded with respect to inner product norm) such that TNa NBT

I.

Proof. This is proved 1n [J], appendix 1, 2.2 through 2.10.

o

~.

Remark. A useful alternative formulation of (iv) is: for every M > 0, A > 0,

B > 0 there exists M

O > 0, AO > 0, BO >

°

such that

(t E ~)

whenever f E Sand

(t E 0:) •

Equivalence of both conditions easily follows from the equivalence of (i) and (iv), and from [B], 6.3.

1.15. Theorem. If T is a linear operator of S with an adjoint, then it is

1.16.

""

*

possible to extend T to a continuous linear operator T of S such

,-..,)

,....,

*

*

that T(emb(f»

=

emb(Tf)(f € S), (TF,f)

=

(F,T f)(F E S , f E S). Here emb(f) for f E S is to be read as emb(f

O)' where fO is the restriction of f to ]R (cf. 1.2).

Proof. This is [J], appendix I, theorem 3.2.

o

We denote the extended operator again by T. For examples, see 1.4.

We finally devote some attention to (generalized) functions of several variables. The previous definitions and theorems can be given and proved (with the proper modifications) without any restriction for the more dimensional case. For instance, the class Sn (where n E m)

is defined as the set of all complex-valued functions f of n complex variables that are analytic in all variables, for which there exist positive numbers M, A and B such that

n 2 2 $ M exp(~

I

(-A(Re t k) + B(Im tk)

»

k=1 n for (tl, ••• ,t_n) € ~ •

As an example of a smooth function of n variables we have

fl <&I ® fn :=y(t) ....

'tn)E~n

f)(tl)···fn(t n) ,where f1 E S, ••• ,fn €

s~

n+ n*

The classes Sand S (of embeddable and generalized functions respectively) are introduced in a similar way (the smoothing operators N are defined as the n-fold tensor products of N (~> 0». Cf. [B],

~,n ~

section 7 and 21.

As an example of a generalized function of n variables we have F} ® ••• ® Fn := YON F) ® ••• ® N F ,

~> ~ ~ n

where FI E S* , • • • , _F E _S

*

n

The notions of convergence and continuity are adapted correspondingly, and theorems 1.13, 1.14, 1.15 hold for the present case.

1.17. The following theorem is important (we state it only for the case n ... 2).

1 , 18.

Theorem. If T. (i = 1,2) are continuous linear operators of S, then J.

the mapping II S T

2, defined by (T_I ® _T2)f := Y(Zl'Z2)T₁(Y

tt (T2(Yt2f(tl,t2»(Z2»)(ZI) for f E S2, is a continuous linear operator of S2. If T. (i

=

1,2)

J. have adjoints , then so has

Z

*

*

in S ), and (T

I ® T2) = TI

TI ® T2 (with respect to the inner product

*

® T2' If furthermore T

I, T2 and TI ® TZ are

* * 2* .

extended to linear operators of S ,S and S (accordJ.ng to 1.15),

*

*

then we have (T

I ® T2)(F1 ® F2) = TIFI ® T2F2 for F] E S , F2 E S • Proof. This follows from [J], appendix I, 2.13 and 3,12.

2

An example of an operator of S (not of the type discussed in 2*

1.17) that can be extended to a continuous linear operator of S is the following one. Define

t ] - t2

---

) (f E S) •

De

. 2

It is not hard to see that

Zu

J.S a continuous linear operator of S that

*

satisfies

Zu

=

Zu .

2. PrepaE,ation.

2. I • We introduce in this section convolution operators defined on S

in which only smooth functions appear. Some simple results are derived.

2.2. Definition. For g E S the convolution operator T J.S defined by

g

T f _g :=

Y

J

f(x - t)g(t)dt

XEI[ (f E S) •

- 0 0

Note that Tgf is the ordinary convolution of f and Yt8(t) (it will have some notational convenience J.n the subsequent sections to take Ytg(t) instead of g).

•

To avoid confusion with the translation operators T (a € ~) a

of 1. 4 (ii), we shall always denote convolution operators by Tf' T, T

h, •• .,

*

*

g

*

T

F• TG, TH , ••• , where f E S, g E S, h E S , ••• , F E S , G E S , H E S , ••• , whereas translation operators are denoted by T ,Tb,T , ••• ,T ,T ,T , ••• _{a c x y z}

with a E [ , b E ~, C E ~ , ••• , X E ~, Y E [, Z E [, ••••

2.3. Theorem. If g E S, then we have (i) (ii) T maps S g T has an g (cL t.1Z)

linearly and continuously

*

adjoint, viz. T

=

T-g

g-.

into S.

-,

and T I f h EO S, then TgTh

=

ThTg and Tgh ... Thg • ::I T- (T ) = T g g' g - ~ (iii)

(iv) I f h E S, then F(T-h) '" Fg·Ff (pointwise multiplication).

g

Proof. If f E S, then it is easily seen that T f is an analytic function,

• g

and we therefore concentrate on the estimation. Let M1,A1,B₁, M₂,A_Z,B₂ be positive numbers such that

If(x + iy)1 S Ml exp(-

~Alx2

+

~Bly2)

(x E lR, y E lR) ,

I

g(x + iy)

I

s

M2 exp(- ~A2x 2 + ~B2Y 2 ) (x E lR, Y E lR)

Using the optimal shift technique as displayed in the proof of [B], theorem 8.1, we obtain

(x € lR, y ElR) •

This proves smoothness of Tg f, _{and it also shows continuity of Tg •} I t is trivial that Tg is linear.

(ii) and (iii) follow from elementary calculations which we shall

•

3. Convolution operators and generalized functions.

3. I. In this section we define convolution operators TF with F E S •

*

We pay special attention to operators TF that map S into S, and it shall be proved that such operators have an adjoint (so that we can extend them to linear operators of S* according to 1.15). Furthermore we shall derive a number of useful properties of these convolution operators.

3.2. Definition. We define for F E S* the mapping TF (cf. 1.4 and 1.12) by TFf :=

Y

_XEu.. ~(T f. F )

x ' - (f E: S) • We shall write T

f instead of Temb(f) in case f E S+ (cf. 1.2). Note that in case f E S definition 2.2 and the present one yield the same operator T

f •

3.3. Theorem. If F E S* and f E S, then TFf is an analytic function.

Proof. It is sufficient to prove analyticity of the function ~XEII:(Txf,F_) in a point Xo E 11:. It is easy to prove (by using Cauchy's theorem and a continuous version of 1.8) that

f(x O + x + h) - f(xO + x) _{.§. Y f' (x + x)} Y _{XEII: h XEII: 0} (h -+ 0) •

~

Hence, by I. 9, T f - T f xo + h _Xo lim ( h ,F _)

=

(~X€lI:f'(xO + x) ,F_) , h->-O

and this shows analyticity of yXEII:(Txf,F_) in xo.

Remark. We shall prove in 5.4 that YZEII: exp(- TI£z2)(T

Ff)(z) E S for

*

every £ > 0, F E S , f € S.

3.4. Definition. The class

C

is defined as the set of all generalized functions

F

for which

tF{S)

c

S.

Remark. This definition is somewhat uneasy to handle, but we shall give alternative descriptions of the class

C

later on.

3.5. Examples.

(i) If f E S, then emb(f) E

C.

(ii) (iii) (iv) If a E ~, F

=

C , then T

=

T - (cf. 1.4(ii», so

c

a E

C.

a F - a If F

=

Pc

O' then TF

=

P, so PcO E

C

(cf. 1.4(iii».

If f is an integrable function defined on lR with a compact support, the~ emb(f) E

C.

(v) I f P is a measure on

:m.,

and if there is an I:; > 0 such that

r

dP(t)

It1;o:x

2

.. O(exp(- 'IT I:; X

»

(x ~ 0) ,

then F, defined by

OQ

(F,f) :=

J

~dP(t)

(f E S) - 0 0

(d. 1.13), belongs to C.

3.6. The following lemma turns out to be very useful in this section.

(a. ... 0).

Proof. We first note that TF is a continuous linear operator: if x E ~, then

~fES (TFf)(x) is a continuous linear functional of S (cf. 1.10 and 1.14).

For a. > 0, a E ~ we have (T

F( )f)(a) a.

=

(N T f,F ) and using the a. a -formula N T

=

exp(- na2 cosh a. • sinh o.)T h R. . h N (that

a. a a cos a. 1a s 1n a. a.

•

2

(TF(a)f)(a) = exp(- na cosh a • sinh a)(T _{a cos a l.a Sln a a} h R. . h N f ,F )

-2

= exp (- 'Ira cosh a • sinh a) (TF(R. . h N f)) (a cosh a) l.a Sln a. a.

We are gOlng to estimate R . N f for a€ ~, a. > O. It is ia slnh a. a.

not hard to prove from smoo.thness of f that there is an M > 0, A > 0,

B > 0 such that

1

2 2

(N f)(t)1 ~ M exp(- nACRe t) + 'lrB(Im t) )

a (t € ~, a. > 0) •

(This may be proved, by using the optimal shift technique of the proof of [BJ,~1). 2 2 2

Hence, using the inequality 12 atl $ lal + (Re t) + (1m t) (a € ~, t € ! (R. . h N f)(t)1 = lexp(2nat sinh a)(N f)(t)! $

l.a Sl.n a. a a

for every a E ~, t E~, a > O. This shows that for sufficiently small a > 0

I

_{(Ria sinh aNa£) (t)}

I

_{~ M exp(-}

_2'

n _{A(Re t)} 2 _{+ 2'1rB(Im t) )exp(1T a} 2

I

12 . _{sl.nh a} for every a E ~, t E ~.

Now we use continuity of T

F• It follows from 1.14, remark that there are numbers MO > 0, AO > 0, BO > 0 such that

I

(TF(R. _l.a . h N f»(a cosh a)

I

~

Sl.n a a

for sufficiently small a > 0 and all a E ~. Hence

for sufficiently small a > 0 and all a E ~. It is easy to see now that there exists a

1 > 0, M} > 0, A} > 0, BI > 0

•

Since TF(a)f ~ TFf pointwise we easily conclude from a continuous version

S

of 1.8 that TF(a/ --l'- TFf • 0

3.7. A lot of properties of the TF's with FEe easily follow now.

-

-Theorem. Let F E

C.

Then F E

C,

F_ E

C

and F E e (cf. 1.]2), and

furthermore TF and TF_ are adjoint operators.

Proof. We shall only prove that F _ E C (the other cases can be t r e a t e d e similarly). Let g E S. Using the relation Tit._i

=

(Thg_)_ that holds by

-2.3(ii) for h E S, we obtain by 3.6

if a } O. Furthermore T(F(a»_g ~ TF_g pointwise. SO TF g and hence F E C.

We further have for f E S, g E S by 2.3(ii) and 3.6

so TF and TF are adjoint operators.

Remark 1. According to 1.15 we can extend TF to a linear operator of S* in case F E C. We denote this extended operator again by T

F• The following properties are satisfied

(i) _TF(emb (f» :::: emb(TFf) (f E S)

,

(ii) G E S * (n E IN) , G S* --l'-

o ..

s*

TFGn --l'-

o ,

n n

(iii) (TFG,g) :::: (G,T

F g) (G E S , g * E S)

.

Remark 2. For F E C the following relations hold:

o

•

Proof. First assume that F E

C,

G E emb(S), and let f E S. We have

(the limits are in S-sense by 3.6). The general case is reduced to the above one by noting that

TFTGf

=

_{lim TFTG( )f}

=

_{lim TG( )TFf}

=

TGTFf

a+O a a+O a

(the limits are again in S-sense).

3.9. Another theorem of the above type is the following one,

Theorem. Let F E

C,

G E

C.

We have TpG = TGF , TT-G

=

_TFTG,

F

Proof. Let us first note that TFG and TGF are well defined by 3.7, remark I.

To show TFG

=

TGF first assume that F E

C,

G E emb(S). Then

De

(the limits are in S*-sense). Here we used 2.3(iii» and the fact that ~

S* .

-emb (F(a» -+ F(a + 0). The general case can be reduced to this one by noting that

(the limits are again in S*-sense), where the latter equality follows from

(a

+

0)

holding for f E S (cf. 1.9). Now we show TT-G :::: TFT

G, and we use therefore the relation

F

that follows at once from the definition of T

K• We easily see from 3.7, remark 2 that (T-FG)_

=

T- G so

F_ - '

Remark 1. The preceding theorem states (among other things) that TF maps C into C in case F E C. For if G c C then TT G = TFT_G, and TFTG

S into S, _{hence TFG} E C. F

maps

*

Remark 2. Let F E S • We mention the possibility to extend the linear

mapping TF (which maps S into the class of all entire functions) to a linear mapping of the space

C

into S*. This is done by putting

-TFf := TfF for f E C. In case F E C this definition coincides with the

one given in 3.7, remark 1.

o

Remark 3. In 3.7, remark 1 we have extended the operator TF to a linear operator of S* (in case F E

C).

If F E S however, there is a more direct

*

-way to define this operator on S J namely by putting TFG

=

TeF for

G L S* (this

1CF

has been defined in 3.2, and is an entire function). ~

It is not obvious yet that this alternative definition yields the same operator, i.e. that emb(TcF) equals TFG (as it is defined in 3.7, remark 1) for G E S*. The proof is pretty hard, and will be postponed until 5.5.

Remark 4. Let F E C, G E S*. We can define the convolution F

*

G of

F and G by putting F

*

G

=

TFG. If we restrict ourselves to F E

C,

G E

C,

then this convolution product has the usual properties. We

mention commutativity (follows from the preceding theorem), and associativity: i f H E S*, then F * (G

*

H) = (F

*

G)

*

H , for

•

F

*

(G

*

H)

=

TF-(G

*

H)

=

T-T-H _{F G}

=

We mention furthennoreTitchmarsh's theorem for C : if FEe, GEe, then F

*

G = 0 ~ F c 0 v G

=

O. This will be proved in the next section.

3.10. We consider an alternative description of the class

C

which is

3. 1 I •

*

related to Weyl correspondence (cf. [BJ, 26). If F E S , then we can define for every g E S the continuous linear functional (cf. 1.10 and

1 • 18)

of S (E is the function emb(~t I » . For every g E S we can find (by 1.13)

'*

exactly one ~g E S such that

(f € S) •

This ~ is a linear mapping of S into S*.

If F E

C,

g E S, then we can prove that ~g E emb(S). It suffices

therefore to show that ~g

=

emb(TGg), where G is the element of S* that satisfies (G,f) == (F,Y /2f(x/2» for f E S (cf. 1.13; note that

x

G L C). This equation holds in case F E emb(S), and the general case

S s*

can be handled by using TF(a)g -r TFg'~(a)g -~ ~g i f a -I- O. The converse of the above statement is also true, i.e. if ~g E emb(S) for every g L S, then F E

C.

We shall prove this in 5.6.

One of the main features of the convolution operators ~s the fact that they commute with the time shifts T (a E ~). That this fact

a

actually characterizes the convolution operators is expressed in the following theorem.

Theorem. Let T be a continuous linear operator of S that satisfies TTa "" TaT for every a E lR. Then there is aGE C such that T == T

•

Proof. We note that YfES(Tf)(O) is a continuous linear functional of S.

*

This means that there exists an H E S such that

(Tf) (0) = (f ,H) (f E S) •

Now we have for every f E S, X E ~

(Tf)(x) = (T Tf)(O)

=

(TT f)(O)

x x

This proves the theorem with G .,. H •

(T f ,H) •

x

Gorollar:y. Let T be a continuous linear operator of S that connnutes with all time shifts Ta (a E 1R) and all frequency shifts

1\

(b E ~).

Then there is aCE ~ such that T

=

cI. This is· proved as follows. We infer from 3.11 that there is an H E

C

such that (Tf)(x)

=

(T f,H)

x

(f E S, X E ~ ) ; we have H

=

T

* .

00' S~nce T, and ence T commutes W1t h

*

.

h

11 f .

*

*

T* 1: ( )

a requency sh1fts we have

1\

T

°

₀

=

T

1\°

₀ =

*

U

o

b E 1R • So, by

a theorem that will be proved in 4.11, remark, T 00 is a mUltiple of 00.

Hence T is a mUltiple of I.

Remark. Let T be a continuous linear operator of S satisfying TP

=

PT

D

(cf. 1.4(iii». It may be proved from 3.11 that there is aGE

C

with T

=

T G• 4. Fourier transform and generalized convolution operators •

4. I. This section ~s devoted to the Fourier transform in its relation to convolution theory. We shall generalize the convolution theorem, and we shall give a characterization of the class

C

in terms of Fourier transforms. Some remarks are made on the equation TfF

=

a

with fEe,

*

F E S •

4.2. Definition. Let h be a mapping of ~ into ~. We define the multiplication operator ~ by

~f :=

Y

",h(z)f(z)

ZE ... (f E S) •

We also write h.f instead of Mbf.

4.3. Lemma. I f h : II: -+

~

satisfiesV

rY

".h(z)exp(- 1f£z2) E SJ, then

~

is

e>(f Z€1Lo -11

•

Proof. Almost trivial.

Remark. The ~ of the above lemma can be extended in the familiar way to a continuous linear operator of S*, which is again denoted by~. We shall also write h.F instead of ~F if F E S*.

4.4. Definition. Let

M

be the class of all generalized functions F for

D

which there exists an analytic g satisfying V

OcY

~g(z)exp(-TI£z2)

E sJ

E> ZE~· -I

such that F

=

emb(g) (cf. 1.6(i». On M we define the mapping emb

by putting emb -I (F) '" g if F E M, F

=

emb(g), where g satisfies the above •

-1

description (note that such a g is unique, hence the mapping emb is well defined on M).

4.5. The following characterization of

C

~s very useful.

Theorem. F

E

C

~ FF

EM.

Proof. Let F E

C.

We have for every f E S

i f a i- 0 by [BJ, Theorem 9.1, 2.3(iv) and lemma 3.6. I t easily follows that g != ~ZEO: ~tfl «FF)(a»(z) ~s an analytic function that satisfies

~ _~g(z)exp(- TIEz2) E S for E > O. Furthermore

FF

=

emb(g) since we

z(;~

have for every h c: S

(FF,h) = lim «FF) (a) ,h) ai-O 00 00 lim

I

«FF)(a»(t)h(t)dt =

I

g(t)h(t)dt ai-O - 0 0 - 0 0 Hence FF E

M.

Now assume that FF E

M.

It suffices to show that for every f E S the function

Y

~(T f,F ) E S. Note therefore that for every z E 0:

ZE.., Z

-lim (T f,F (a»

=

(T f,F ) ,

-•

and that the proof will be complete if we can show that this limit is achieved in S-sense.

We have by 2.3(iv) for every a >

°

F(TF(a)f)

=

(FF)(a)-Ff

Now let M > 0, A > 0, B > 0 be such that 1 (Ff)(x + iy)1

~

M exp(-

~Ax2

+

~By2)

for every x E JR, Y E JR. Since FF E M, we infer the existence of a g

that satisfies V

OcY

o:g(z)exp(-~Ez2)

E S] such that F

=

emb(g). • E> ZE

This means that there exists an MI > 0, BI > 0 such that

for every x E JR, Y E JR. It is not hard to show that there are numbers

M2 > 0, B2 > 0, a_O >

°

such that

for every x E JR, Y E JR. Since N g = (FF) (a) for every a > 0, we see that a

(FF)(a) f

§

g. f if a +

°

(here we have used a continuous version of 1.8),

so F(TF(a)f)

~

g.Ff, and hence TF(a)f

~

F*(g.Ff) if a

+

°

(cf. 1.4(i». This shows that TFf

=

F*(g.Ff), and hence that TFf E S • 0

-I

Corollary. I f

FEe,

f E S, then F (TFf) = emb (FF). Ff. This follows easily from the second part of the proof of the above theorem.

4.6. Exampl'es. (i) (ii) . 2 IJ ~~z Let F := I ~e • ZE", 1 • 2 of

Y

(_i)-2e-~~z ZEO:

It is not hard to check that FF is the embedding (principal root), and it follows that F E

C.

1

Let h be the function defined by h := --2 x( )' Its Fourier

T T T -T,T

IJ sin 2~AL

•

2 (iii) Let (d) _nnE. ~ be a complex sequence that satisfies Id _n

I

= O(e-sn )

00

(n E 'll) for some €: > 0, and let F := Ln=_oodno

n (cf. [B], 27.24.2~ii». Then FF is the embedding of the analytic function

Y

Loo d e-2~~nz

z n=-O<) n

and it is not hard to show that F E

C,

*

-}

4.7. Theorem. I f F E

C,

G E S , then F(TFG) = emb (FF)-FG.

Proof. The case with G E emb(S) follows from 4.5, corollary, and "the general case is deduced from this one by noting that

F(TFG)

=

lim F(TFG(a» .. a4-0

"" lim emb-1(FF),emb«FG)(a» ""

emb-I(FF}F~,

ai-O

*

where the limit is achieved in S -sense.

*

+

4.8. Theorem. If F E S, F E emb(S ), GEe, and if TCF "" 0, then F

=

°

or

G '" O.

-1 -}

Proof. By 4.7 we have emb (FG)'FF

=

O! Write g = emb (FG), and let f E S+ be such that emb(f) "" FF. Then g'FF "" emb(g·f), for we have

for h E S (go F,h) = ( F,g'h) .. (emb(f),g'h) 00 00

o

=

f

f(t)g(t)h(t)dt ==

J

g(t)f(t)h1t)dt =(emb(g·f) ,h) • - 0 0 - 0 0

It follows from 1.6(i) that gof = 0 (a.e.). We conclude by analyticity

of g that gaO or that f =

°

(a.e.), and so G "" 0 or F = 0. 0 4.9. We enter somewhat further into questions of the type: if f E

C,

F E S* and TfF "" 0, then what can we tell about F. As we see from 4.7 such questions can be translated into (g ==

Fi,G

= FF): i f gEM, G E S*,

-I

and if emb (g)'G = 0, then what can we tell about G. In [S] these problems have been solved for the space K' (dual space of K), but we cannot use the techniques employed there, since we have (by lack of non-trivial elements of S of compact support) not the occasion to consider the elements of S* locally, as it is done in [S] for the

elements of K' (cf. [8J, Ch. V, § 4).

We shall prove here some simple results in this direction, and we shall mention some further theorems without proof.

-I

*

4.10. Theorem. Let gEM, and assume that emb (g) has no zero's. If G E S -I

and emb (g)'G

=

0, then G

=

O.

Proof. Put h := emb-I(g). We infer from the fact that Vzh(z)exp(-

~Ez2)

E S for every E > 0, and the fact that h has no zero's, that there are

complex numbers a

O' a1 and a2 with Re(a2) $ 0 such that

(Z E 0:) •

If a > 0 is such that coth a> la21, and if we denote for tEO:

v

i

-~ 2 2 -1

k t := izEo:(sinh a) exp(sinh a«z + t )cosh a - 2zt»h (z),

then we have k

t E S, h'kt

=

0a(t) (cf, 1.6(ii». Hence for tEa:

(cf. [B], 27.18», so G

=

0, and therefore G == 0 by 1.5.

a D

•

4.11. Theorem. I f G E: 8*, QG = 0 (cf. 1.4(iii», then there is aCE a: such

tha_

G

=

coO' This c is uniquely determined by G.

Proof. Let a

...

> 0 be fixed. We infer from [BJ, (11.9) and (11.11) that N Q = cos h a QN + i sinh a PN

a a a

so we find

cosh a QG + i sinhaPG == 0 •

a a

The solution of this differential equation 1S given by

G

=

Y o:G (O)exp(- nz2 coth a) == G (O)(sinh

a)~(OO)N

•

4. 12.

Hence, withe:= Ga(O)(sinh ex)!, (G - cOO)a :: G

a - c(oO)a follows from 1.5 that G :: coO'

Uniqueness of c is trivial.

O. It

*

Remark. It follows easily from the above theorem that an F E S that satisfies T F == F (a E lR) is the embedding of a constant function.

a 1

*

For we have PF == lim ~ (ThF - F)

=

O.(the limit is in S -sense), hence h-"'O 1f~

QFF

=

FPF :: 0. Th1s means that FF '= coO with some c E ~, and hence F .. embCY tE~C). Also, i f F E S*, ~F ... F (b E lR) , then, F :: coO

for some c E ~.

o

Theorem 4.11 can be generalized as follows. Let n E lN J a

1 E ~, ••• ,an E I

~ n vk

*

VI E IN, ••• , vn E lN, and let h:= IZE~ TIk=l(z - ak) • If G E S

satisfies h'G .. 0, then there are complex numbers dkR. (R, CI 0,. I " , v

k - I; k

=

l, •.• ,n) such that n F ==

L

k=1 vk-I

I

dk£ £=0

The numbers dkR. are uniquely determined by F.

-I

We still can go further. Assume that h E emb (M), and let h have its zero's in a

l,a2, ... with multiplicities v1,v2' •••

.

Let V be n the set of all elements F E S * of the form (*), and let V be the union

of all V ' s. We assume that

l::~::1 ,1~ltO

_{vk/lakl2 <} ro _{Then every} n

* * . .

F IS S that satisfies h'F == 0 is S -hm1t of a sequence in V, and every F E S* that is S*-limit of elements of V satisfies h'F

=

O.

We note that every element h of emb-I(M) has order

~

2, and this means that the limit exponent of h does not exceed 2 (cf. [BiJ, Chi VI,

§ 4) : if a

l,a2, ••• and vI,v2' ••• are as in the above,then

00 1 12+£

~k=I,laklto _{vkl ak} < ro oofor every £ > O. ;n case that the order

of h is less than 2, we have l::k=I,lakltO vk/l~1 < "', so the above theorem applies to h. We do not know how to handle the general case in

e

5. Some further remarks on convolution theoEY'

5.1. In this section we give some further theorems and definitions

5.2.

about the class C. We shall also pay attention to convergence in C, and to convolution operators in Sn (n E 1M) •

We are g01ng to show that g'G E C if g E S, G E S •

*

This means that T-F E M if f E S, F E S

*

(cf. 4.5 and 4.7), and it will turn out

f

+

that T-F .,. emb(TFf) (in particular T-f E S ). The following lemma is

f F

useful in the proofs of the above statements.

Lemma. Let f E S, g E emb -I (M), and let M

J > 0, At > O,B. > 0, M2 > 0, A2 2 0, B2 >

°

be such that

If(z)1 M exp(- 'ITA. (Re z)2 + 'ITBI (1m z) ) 2

,

I

Ig(z)1 :::;

M2 exp(- 'ITA2 (Re z)2 + 'ITB2 (1m z) ) 2

for every Z E ~. To every £ with

°

< £ < Al + A2 there exists a C >

°

*

and a

a

>

°

(only depending on B

I, B2 and e) such that for every F E S , Y t: ~ (II II denotes inner product norm in S)

Proof. We have for every y E ~, Z E ~

where P is defined by P(z,y) '" - (AI + A 2) (Re z)2 + (B1 + B2) (1m z)2 + 2AII Re z Re yl

•

2 2 + 2B) 11m z 1m yl - A)(Re y) + B1(Im y) (z E ~, Y E ~).

I I

2 -) 2

Let £ satisfy 0 < E < A) + A

2• Applying the inequality 2 ab ::; ya + y b (valid for a €

::m.,

b €

::m.,

y > 0) to 21Re Z Re yl and 211m z 1m yl with

-)

y

= )

+ (A_Z - £)A) and y

=

1 respectively, we obtain

for y € ~, Z E~. Now put for everyy E ~

A - £ h :=

~

",g(z)(T f) (z)exp(1I'A

I 2 (Re- y)2 - 211'Bl" (1m y)2) •

y Z E... Y A I + A2 - £

Then we have h € S, and y

Ihy(z)1 ::; MIM2 exp(-1I'£(Re z)2 + 1I'(2B

1 + B2)(lm z)2) (z EO ~) •

I t follows from 1.3(ii) that we can find a C > 0 and a

a

> 0 (only depending on E and 2B) + B₂) such that for every y € II: there exists an

9-Y with h

=

N R- II R- II s M

1M2C. So we have for every y E:

a:

y

a

y y

•

E S

A - £ 2 2

l(g'T f,F)1 ::; M₁M₂C1IF(S)lIexp(-1I'A₁ A +2A _£(Re y) + 2rrB_t (1m y) )

y I 2

if F E S* (here we apply I (hy,F) I = I (NSR-y,F)I

=

I (R-y,F(S»I s M

1M2C1IF(S) 1\ •

*

5.3. Theorem. If g E: S, G E:

S ,

then g'G E

C.

Proof. Let f E S. Then T Gf

=

Y ~(-g'T f,G ). Analyticity of T Gf

g' yEv. - Y - g.

follows from theorem 3.3, and it follows easily from lemma 5.2 that

T Gf € S. g'

*

5.4. Theorem. If f € S, F E S , then emb(Tpf) E

M.

Proof. Take g

=

YzEII:I and P_(instead of min 5.2 to conclude that TFf € S+. It further follows from lemma 5.2 and theorem 3.3 that

o

5.5. We can prove now the statement in 3.9, remark 3.

5.6.

*

Theorem. If F E S , f E S then TfF

=

emb(TFf) •

Proof. We first prove the formula with F E

C.

We have in that case by

theorem 3.9, theorem 3.7 and 3.7, remark for every g E S

hence TfF

=

emb(TFf) by the uniqueness part of theorem 1.13.

The general case is reduced to the above one as follows. Denote he :=

YZE~exp(-~ez2),

and define Fo := ho'F for 6 > O. Now Fo E

C

by

5.3, and we have TfF

o

=

emb(TF f) for

a

> O. The proof will be complete

*

a

s*

if we can show that TfF

_o

§ TfF, emb(T

F f) + emb(TFf) if

a

~

O. We

*

a

note therefore that Fa § F if

a

~

0, so, by 3.7 remark 1, we have TfF

_o

§* TfF if 0 + O. Furthermore we have (T

F f)(y) + (TFf)(y) if 0 + 0 for every y E

~,

and it is easily proved now &ith the aid of lemma 5.2

s*

and 1.9 that emb(T

F

_a

f) + emb(TFf) if 0 ~

o.

n

Remark. Note that not every element of

C

can be obtained as the product

*

of agE S and a G L S (cf. theorem 5.3), and so not every element of

*

M

can be obtained as TfF with some f E S, F E S •

. 2

Example. k := emb(Y o:e~1Z) E

C

cannot be of the form goG with some ZE

g E S, G E S*. For if so, then we consider the sequence (f) m defined

V -~iz2-~(z+n)2 n nE

by f := I z<:o:e (n E m) • Now we have _nS ....

f _n 'g+O, but (f _n 'g,G) = (f ,g'G) -_n (f ,k) _n == 1 (n Em).

We are going to prove the statement at the end of 3.10. With the notation used there we have to show that F E C in case KFg E emb(S) for every g E. S. Let f ( S, g E S. We have (KFg,f) ₌₌ (E ®

F,

_ZU(f ®

g»

by

. 5.7.

that (E ®

F,

ZU(f ®

g»

(C,T f), where G ~s the generalized function

g-that satisfies (G,h) g (F,Y i:2h(xi:2» for h E S (cf. 1.13). Hence, by

x

3.7, remark I and 5.5,

(~g,f) = (C,T f)

=

(T-gG,f)

=

(emb(TGg),f) •

g-This means that KFg = emb(TGg), and by analyticity of TGg we concl~de

that TGg E S. Hence G E C, and so F E C •

We make some remarks on convergence of convolution operators.

Definition. Let f E C

n (n E IN), f E C. We wri te fn

!;.

0 if T f g

~

0 for every g E S;we write f

£

n f if f n - f

£

O.

n

I f f E C (n E IN) , then the following statements are equivalent.

n (i) f

!;.

0 n

f

£

0 n (ii) (iii) ( f )

£

0 , n -(iv) (v) (vi)

The proofs are almost trivial.

5.8. Theorem. Let fn E C (n e IN). We have 3

feC[fn

!;.

fJ

~

"ge S[ (Tfng)nEJN is S-convergent].

Proof. If f E C is such that fn

£

f, then we have Tf g - Tfg

for every g E S. n

S

=

T_f -fg _{-+ 0}

n

Now assume that (Tf g)nEJN is S-convergent for every g E S. n

Denote Tg

=

lim T

f g for g c S. It follows from [J], appendix 1, 2.12

that T is a continuous linear operator of S, and TT = T T for every

a a

a E lR. So by 3.11, there is an fEe such that T

=

T

f• It follows at

e

once that f + f.

0

n

5.9. Theorem. Let fEe, f E. e (n E IN), gEe, g E e (n E IN) , and

e

n

e

~ n

assume that f + f, g + g. Then f * g ~ f * g (cf. 3.9, remark 4).

n n n n

Proof. Let u E S, _{and denote h :=} i_emb -I (Fr ) (n _{E IN) , k := emb} -1 _(Fg) - _{(n E}

-1 - -1 _ n n n n

h :"" emb (Ff) , k := emb (Fg), v := Fu. By 5.7(v) it suffices to show that h 'k 'v § h-k.·v

.

Therefore we note that h ·k 'V

~

h'k'v pointwise,

e

n n n n

and that there is an M > 0, A > 0, B >

a

such that

Ih

_n (z)k (z)v(z)1 _n ~ M exp (-7TA(Re z) 2 + 7TB(Im z) ) 2 (z E

a:,

n E IN)

as one easily sees from the fact that k 'v

~

k·v, h 'w

~

h·w (w E S).

n n

So , by 1 8 • J h ' k ' v n n

~

h·k·v. 0

Remark. The above theorem is a special case of the following one. If (T ) _{n nElN '} (U) _{n nE} IN are sequences of continuous linear operators of S for which (T g) IN' (U g) IN are S-convergent for every g E S, then

n nE n nE

T := Y gES S, and we

5.10. Examples.

lim T g, U :=

Y

Slim U g are continuous linear operators of

-+<X> n ~ gE n

Rave TUg ~ TUg fo¥~very g E S.

n n

(i) I f (f) IN is an S-convergent sequence

n nE. in S, then (emb(f» n nE IN is an

:IN:

e-convergent sequence in e. If (g) IN is

n nE a e-convergent sequence in e,

(ii)

then (gn)nclN is an S*-convergent sequence in S*.

e

If fEe, then emb(N f) + f if a +

a

(we have of course a similar

a

definition of e-convergence for this case as in 5.7). This is lemma 3.6.

2

(iii) If (d) ~ is a complex sequence satisfying d '" a(e-en ) (n E '4)

n llEu n

00

for some £ > 0, then E d 0 is a e-convergent series in the

5. 11.

(iy)

(v)

sense that YXEo.:

E:=_N

dng(n + x)

§

YXEo.:

L:=_~

dng(n + x) (N ~ 00, M + 00) for every g E S.

If g :=

Y

~

y!

exp(-~yz2)

(y > 0), then emb(g )

£

00 (y

~

00).

Y ZE~ Y

1

C

If h := --2 x( )

«

> 0), then emb(h ) + 00

«

~ 0). More

l' < -< , < _I < C

generally: if h E

C,

and (emb (Fh»(O) = I, then VAh ~ 00 (A + 00), where VAh is the generalized function that satisfies (V,h,f)

=

(h,Y .f(x/A»(f E S) for A > 0 (cf. 1.13). This may

1\ XEIL _)

be proved by using FV

A = A V _IF (A > 0), the equivalence of 5.7(i) and S.7(v), and 1.8. A

* S*

IN) , G E S , G E (n E IN) , and assume

C

n

Let g E S, g E S (n E

S n ~*

that g _n + g, G _n ~ G. Then g. _n

*

G ~ g·G. For it easily follows

n

from 1.9 that g 'G

§

g·G. So if fESt then we have T f

~

T f

n n _gn'Gn g'G

pointwise, and it may be proved from lemma 5.2 that T G f ~ T Gf.

g • . g'

*

C

n n

We also have: if g E S, G E S , then g'G(a) ~ g'G (a ~ 0) •

We finally make some remarks about convolution theory for (generalized) functions of several variables. It is possible to develop the theory as it is presented here almost entirely for the more dimensional case (an

exception should be made for the results of 4.12). We shall restrict our-selves here to the case of functions of two variables.

2*

The definition of TK with K E S becomes TKf

=

Y( _x,y ).2 (T ® T f,K_)

ElL X Y (f E S) ,

where K is the generalized function

Y oY( )

_{a> z,w E~}

~2

(N _a, 2K) (-z,-w)

(cf. 1.16). In order to prove the two-dimensional version of theorem 3.3, we can use a theorem of Hartogs ([BTJ, Ch. III, § 4, Satz )5) about the

analyticity of functions of several variables.

We introduce the set

C

2 as the class of all generalized functions K for which TK maps s2 into itself. The crucial lemma 3.6 still holds for the present case, and its proof differs only from that of lemma 3.6 in notational respect. This enables us to prove the two-dimensional versions of the theorems of section 3 and 4. We mention in particular theorems

3.5 and 3.7 (the definition of the class M2 is obvious) •

An important example of an element of C 2 is the tensor product of two elements of C. Let gl E C, g2 E C. We claim that gl ® g2

2

and that T e e T ® T (cf. 1.]7). To prove this note that

g, g2 gl g2

(F ® F)(gl ® g2)

=

Fg_I ® Fg

2 £

M2,

hence, by the two-dimensional version of theorem 3.5, g) ® g2 £ C2• Furthermore we have

T ® (h] ® _h2)

=

T hI ® T h2 for hI £ S, h2 E S, and the

8 1 82 . g) g2

be completed in the style of [J], appendix I, 2.13.

proof can

References.

[B] De Bruijr., N.G., A theory of generalized functions, with applications to Wigner distribution and Weyl correspondence; Nieuw Archief voor Wiskunde (3), XXI, 1973, pp. 205-280.

[Bi] Bieberbach, L., Lehrbuch der Funktionentheorie, Band II. Moderne Funktionentheorie; Chelsea Publishing Company, New York, 1945. [BT] Behnke, H., Thullen, P., Theorie der Funktionen mehrerer komplexer

Veranderlichen;Chelsea Publishing Company, New York, 1933. [GS] Gelfand, I.M., Schilow, G.E., Verallgemeinerte Funktionen

(Distributionen), II; VEB, Deutscher Verlag der Wissenschaften, Berlin, 1962, series: Hochschulbucher fur Mathematik.

[J] Janssen, A.J.E.M., Generalized Stochastic Processes,

T.H. Report-76-WSK-07; Technological University Eindhoven, December 1976.

[S] Schwartz, L., Theorie des Distributions, Tome 1; Hermann, Paris, 1957.