A density lemma

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

2752

z==#{e~zl~=o , and hence any monomial z ~ can be locally uniformly approximated by the finitedifference relation corresponding to derivative 0~r 7 which is obviously a linear combination of exponentials. From this it follows immediately that the function u(z) itself can be approximated by linear combinations of exponentials. We shall show that this scheme of arguments leads to a sequence of linear combinations of exponentials

UN (Z) such t h a t ~ ] N 6 ~ ( I < j K N ,

=

C1Ne ~lNz -{- . . . -{- C NNe :NNz

N - ~ l , 2 . . . . ) and UN(Z) + u ( z )

in t h e t o p o l o g y o f t h e s p a c e Expa(C~).

For this we prove two assertions. Assertion 3.1. If a function ~(z)~Exp~(C~ , then its Taylor series converges in the topology ofExpe(C~ ). Proof. Indeed, from the familiar formulas defining a hypersurface of conjugate types rz,...,r n of an entire function of exponential type it follows that the hypersurface of conjugate types r~,...,r~ of the function

1~I=O is the same as for the function _%

q~(z)-~- ~ r

~,

(3.1)

~=D~i(O)/M

I~1----0

[~*(z)]~<Mexpr*]zl

Thus, there exist a nuaaber r* < R and a constant M > 0 such than for all z6Cn. From this it follows that for any N = i, 2 .... the partial sums SN(Z) of the series (3.1) satisfy the estimate

]S~(z)[<

~

[ ~ ] [ z l [ ~' . . . ] z ~ l ~ < 3 " l e x p r * [ z ] ,

I,ml
which [in correspondence with the definition of convergence in Expe(C~)] Assertion 3.2. Let A~f(0;z)/~(~ER 1) be the finite-difference f(~, z) - exp~z. Then as ~ § 0

A~'/ (0, z)/;~' -.

is what was required.

relation (in ~) of order

z '~

locally uniformly in Czn. Proof. Indeed, in correspondence with known results on estimating the error of formulas of numerical differentiation we have

A=/(O, z ) / ; ~ = a ~ / ( ~ ,

z),

(3.2)

where a~(al ..... ~n), ~-(~ ..... ~n) is some intermediate point, i.e., ~6(0, c~i~i)..... ~n6( 0, ~$n). Hence, for the function f(~, z) - exp ~z we obtain the inequality

] zO~ a=f (0, z) ~< I I 0%[ ~; ]] z, ]~t+' exp oh 1;izi [, I t=l

( 3. 3 )

whence it is clear that as ~ + 0 As/(o,

locally

u n i f o r m l y i n C%

The a s s e r t i o n

z)/~-~z

~

is proved.

We s h a l l now c o m p l e t e t h e p r o o f of t h e lemma. Let u(z)@Expa (C~).

This means that

u (z) = ~ e~z~ (z), where LGQ r u n s t h r o u g h a f i n i t e s e t o f v a l u e s , and t h e functions~x(z)cExp~{n)(C~n). I t i s t h u s c l e a r t h a t i t s u f f i c e s t o p r o v e t h e a s s e r t i o n o f t h e 1emma f o r e a c h t e r m h x p L z ~ ( z ) s e p a r a t e l y . For this we note that in correspondence with Assertions 3.1 and 3.2 there exists a sequence = ~N + 0 (N + ~) such that the sequence of linear combinations of exponentials (the notation i s clear) 2753

AN.~. (z) ~

~ : (P~xA=f (0, l~zl
z)/;~

converges to the function ~(z) locally uniformly in Cz". We shall now show that there are numbers M > 0 and r ~'~ < R(%) such that ]AN,~(z)l~
Icq<:N

i=l

From this, using again the fact that the hypersurfaces of conjuate types of the function ~(z) and the function

a r e t h e same, we f i n d

that

I~l=0 f o r a n y r > 0 and a n y zEC n

I

(z) I

r I z I+

p

z

[],

provided only that the quantities ~ = CN are sufficiently close to zero. arbitrary and r < R(%), this implies the validity of the inequality [A~.x (z) [ < 3 4 exp r*

Since g > 0 is

[z ],

where M > 0 and r* < R(~). It remains to note that for the function e~zAN,~(z)all the exponents in the exponential functions have the form ~ + 8~N, where I~I < N, and hence ~N can be chosen so that ~ - ~ N ~ Thus, as N + i e~A~.~ (z) -+ e ~

~)

in the topology of the space Exp~(~)(C~n). The lemma is completely proved. 4.

Algebra of P/D Operators with Constant Analytic Symbols

LetA(~)EG(~), i.e., A(~) is an analytic function in the domain ~ c C ~ . We assign to the function A(~) a p/d operator A(D), D = (3/8z I .... ,3/8z n) acting in the space EXps(Czn)Definition 4.1.

Let u(z)CExps(Cz") and

a (z)-- ~ e ~ z (z) be some representation of it.

where

aa(~)----O~A (~)/a]

[as usual,

(4.1)

Then

a = (~l .....

~)

are multiindices

of differentiation].

Remark 4 . 1 . S i n c e t h e f u n c t i o n u ( z ) 6 E x p ~ ( C , " ) c a n be r e p r e s e n t e d i n t h e f o r m ( 4 . 1 ) i n v a r i o u s ways t h e q u e s t i o n a r i s e s o f w h e t h e r t h e o p e r a t o r A(D) i s w e l l d e f i n e d . This question i s c o n n e c t e d w i t h t h e c h a r a c t e r o f t h e domain ft. I t t u r n s o u t t h a t i f fl i s an a r b i t r a r y open s e t , t h e n t h e v a l u e s A ( D ) u ( z ) c a n , i n f a c t , be d i f f e r e n t , generally speaking, i.e., the o p e r a t o r A(D) i s m u l t i v a l u e d . On t h e o t h e r hand i f fl i s a Runge d o m a i n , t t h e n t h e v a l u e A ( D ) u ( z ) d o e s n o t d e p e n d on t h e r e p r e s e n t a t i o n ( 4 . 1 ) and h e n c e D e f i n i t i o n 3.1 i s good. The proof of this fact is carried out in Sec. 5. In view of what has been said, everywhere below (with the exception of Sec. 8) we shall assume without special mention that ~ is a Runge domain. THEOREM 4.1.

The mapping A (D):Expa (C~n)-~ ExpQ (C~n)

(*)

is defined and continuous. fWe recall that a domain is called a Runge domain if any function analytic in this domain can be locally uniformly approximated by a sequence of polynomials or, equivalently, by a sequence of linear combinations of exponentials. 2754