The basic space Exp (Omega)C(z,n)

From this, in particular,

it follows that ~v(z)~(z)(v->---~-~ ) i n ' E x p e ( C ~ ) i f

i) there exists a number r < R such that Uv(Z)6 a bounded set;

and only if

) (v = i, 2,...) and form in Expr(C~ )

2) uv(z) -+ 0 locally uniformly in C~, i.e., uniformly on any compact set ~ c C ~ . Remark.

It is possible to admit R = ,~.

Then

tim ind Exp7 (C~)= Exp (C~), i.e., it coincides with the space of all functions of exponential type equipped with the inductive-limit topology. 2.

The Basic Space Expo(Cz ~)

Let Q ~ C ~ be an arbitrary open set in the space of the variables ~ = ( ~ .... ,~n) ( t h e s e are future dual variables in the Fourier sense). Further, let ~ = (%~ .... ,Xn) be an arbitrary point. We denote by R(X) the radius of the maximal polycylinder~U~(~)={~:]~]--%i[~(%)~ , j = l,...,n} lying entirely in ~ . Definition 2.1.

We set

Exp~ (C~)= {e (z): where the values

%@~

Definition 2.2. tions are satisfied:

tt(z)=~eZz~z(z)},

run through a l l p o s s i b l e f i n i t e c o l l e c t i o n s We say that uv(z) + u(z) in space

and ~z (z)6 Exp.(x) (C~).

Exp~(C~)( if the following two condi-

i) there exists a finite collection of values %6~ such that for all v = I, 2,

( z ) = ~ eZ~%~(z), 2)

Thv(z)'-+~z(z)

~xv (z)E Exp~r (C~);

in Expe(~) (C~).

As already noted, the space Exp~(Cz) is the basic space for construction of the theory of p/d operators with symbol A(E) analytic in ~. However, before giving the definition of a p/d operator A(D) with analytic symbol, we present a description of the space Expa(C~) in familiar topological terms. We introduce the notation

e x* Expm~ ~(C~)={u (z): u (z) e-X*E Expmz ~(C~)}. Suppose now t h a t

@ eXzExp~cx) (C~) is the d i r e c t ( a l g e b r a i c ) sum of the spaces eX~ExpR(x)(C~).

By d e f i n i t i o n ( s e e , for example, [34, 40], e t c . ) the elements of t h i s sum are f i n i t e formal sums O uz (z), where uz (z)Eexz Exp~c~ (C.9, or, e q u i v a l e n t l y , c o l l e c t i o n s {uz (z)},, LEf21among which only a f i n i t e number of the f u n c t i o n s u~(z) are nonzero. and {v~(z)} in the case where

where the s ~ a t i o n the factored s u m

We i d e n t i f y the c o l l e c t i o n s

sign denotes the ordinary sum of functions. ~Exp~(~)(C~)/fW,

where JH-----[~tt~(z): ~ ( z ) ~ _ 0

{u~(z))

In other words, we consider I.

It is clear that the

cor-

J

respondence u(z) <-+ {u~(z)} given by the formula

(z) = ~ u~ (z), where {uk(z)} is an arbitrary representative of the coset, defines an algebraic isomorphism

Exp~ (C~.)~N~ @ e ~ ExpRr (C~)/~W~

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It is possible to proceed in another manner. Let F be an arbitrary finite collection of points in the domain ~. We consider the direct sum ~ r again identify the formal sums .~3U~(Z) and: @ ~ ( z ) , if x~r

9 ~r

~6r

(z).

(2. I)

Then

@ e x* Exp~(~) (C~)/Mr ~ Exp (r; C~), ~Er where Mr i s t h e s u b s p a c e of t h e d i r e c t i s i d e n t i c a l l y e q u a l t o z e r o , and

sum c o n s i s t i n g of c o l l e c t i o n s

Ex ,i; c:,=

(2.2) {ux(z)} f o r which ( 2 . 1 )

/ t

It is clear that algebraically

Expr (C~) m U Exp (P; Cn) and hence , Expa (C, )n ~ ~ { ~ a * Exp~)(C~)/Mr}. A topology can be introduced in the space Expa(C~) in various ways. can be done as follows.

For example, this

Let A = {~i, ~2,.-.} be a countable, dense set of points in ~ [it is clear that for a description of the space Exps(C~) it suffices to restrict attention to such a set of values %6~ ]. We set ~N={%1 ..... %NI and consider the sequence of spaces Exp(FN; C o constructed above, equipping them with the natural factor topology in correspondence with formula (2.2) and the topology of the spaces Exp~(~)(C~. Further, by adding to the representation of the function u(z)EExp(rN; C~)the required number of zero terms of the form e~/.0(j>N), we obtain the chain Of imbeddings

Exp (rl;C~)cExp (F2; C~)c . . . . where o b v i o u s l y each of t h e s p a c e s i s a c l o s e d s u b s p a c e in t h e n e x t .

Hence,

Expa (C~)=lira ind Exp (P~; C~) is a regular inductive limit. From familiar properties for example, [34]) it follows that the sequence uv(z) + is an index N such thatuv(z)CExp(FN; C~) for all v = i, In correspondence with Sec. 1 this means that all u,(z)

of regular inductive limits (see, 0 in Exp~(C~) if and only if there 2 .... and uv(z) § 0 in Exp(F~; C~). have the form

u~ (z) ----e~,~%~ (z) ~k... + e ~ N % ~ (z),

(2.3)

where ~v~j(z)cExp~j(C~), r j < R ( k i ) , j = 1. . . . . N, and infl]~v~j(z)[]r/-+0, where t h e infimum i s t a k e n over a l l r e p r e s e n t a t i o n s of u v ( z ) in t h e form ( 2 . 3 ) . As i s n o t hard t o s e e , from t h i s

i t f o l l o w s t h a t c o n v e r g e n c e in Exp~(Cz) d e f i n e d by t h e

topology introduced is majorized by the convergence of Definition 2.2. We use the latter convergence below although everything said below holds also for the inductive topology introduced. 3.

A Density Lemma

We shall need below the fact of the density of linear combinations of exponentials in the space Exp~(C~. LEMMA 3.1.

The linear hull of the exponentials exp~z, ~E~, is dense in the space Exp~(C~).

Proof. First of all, we note that a function u(z)6Expn(C~), just as any entire function, can be approximated by linear combinations of exponentials in the sense of locally uniform convergence in C~, i.e., uniform convergence on compact sets. Indeed, for any entire function its Taylor series converges in C~ locally uniformly. Further, for any ~ we have

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