Fourier transform of analytic functions

tt (t, z ) = e x p

(tlX+Xz)

and

~-~ t n n-4 ~kzk u(t, z)=exp(t/~.+~z)-- Z_.X~nf ~ k! ' n=l

k=0

corresponding to the actions of the p/d operator I/D on expkz according to the formulas

[I/D] exp ~z --: %--1exp ~z and If / D] exp ~z -- ~-f(exp ~z -- 1). CHAPTER 2 THE COMPLEX FOURIER METHOD _i.

Fourier Transform of Analytic Functions G

n

L e t c C z be a Runge domain and let u(z) be an arbitrary analytic function in G. Let, as previously, r = (~l, .... ~n) be the dual variables, and let 8 = (8/8~i ..... B/8~ n) be the differentiation symbol in the space C~. In correspondence with Sec. 4, to U(Z) there corresponds the p/d operator

u (-- 0): Exp; (C~) -~ Exp; (C~), where Expa(C~)n is the space of exponential functions v(~) associated with the domain G .... :-C n"

Exp; (C~) i s t h e d u a l s p a c e . Definition 1.1. nential functional

The F o u r i e r t r a n s f o r m [ F u ] ( ~ ) - - ~ ( ~ ) o f a f u n c t i o n u(z)E(Y(G) i s t h e expo~ g ) = ~ (-- 0) 6 (~).

The v a l u e o f ~(~) on a t e s t

f u n c t i o n v(~)~Expo(C~) i s d e f i n e d by t h e f o r m u l a

9 < ~ ( 0 , v (~) > =

THEOREM 1 . 1 .

< 6 g ) , u (o) v (~) > = ~ (o) v (o).

The mapping '

r/

f :&(G) -~ Expa (C•) is one-to-one,

(*)

and t h e i n v e r s e mapping i s d e f i n e d by t h e f o r m u l a u ( z ) = < ~(~), expz$ >, zO.O.

(1.1)

Proof_____=. We s h a l l f i r s t e s t a b l i s h the inversion formula (1.1). I n d e e d , i f fi = u ( - 8 ) 6 ( r i s t h e F o u r i e r t r a n s f o r m of some f u n c t i o n u(z)E~Y(G), t h e n f o r any f u n c t i o n exp z~, where z~G, we have = = ~ ~(~)~(d~), ~ (~)~C(f).

(2.3)

K

Since K is compact, there obviously exists a finite family of Borel sets K i (i = l,...,n) such that: I)A~A'7-----~ (i=/=j); 2) U f~=A'; one polycylinder of "analyticity" UR(li ). u(d~) to the set K i.

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3) any set K i is entirely contained in at least We denote by ~i(d~) the restriction of the measure

Then by properties i),

2) ~(d~)=~(d~)+...+VN(d~),

and hence