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 = <6 (~), u (0) exp zr = u (z), which was required. We further note that each exponential functional is the Fourier transform of one and only one function analytic in G, Which is precisely the assertion of Theorem 6.1 of Cha p . i on the representation of an arbitrary functional h(~)6Expd(C$) in the form h(~)---A(--O)6(~), where A(z)~s Thus, the mapping (*) is one-to-one which was required. The Fourier transform of analytic functions as an exponential functional satisfies properties analogous to the properties o f the classical operational calculus. Among them we note two properties in which there occurs a change of the domain with which the space of Fourier transform, is associated. The Similarity Theorem.

Let ~(z)6~(O), a=(~x .....

[Ftt (az)l (~)= aT'.., aV'a (~1/al. . . . . where

a-10

=

an).

~n/a.)~Exp'~-,G

tZ

(C~),

{z :azGOb

The D i s p l a c e m e n t Theorem. L e t tt(z)6ly(G), a=(a~ . . . . . an), Expa+~ (C0, where G-ra----{Z:z--afiO} . '

Then

Then [Ft,t(z'a)](~)=exp(--iaz)ft(~)~

I

Example 1. L e t tt(z)~expaz2--exp ( a t z ~ + . . . -t-a~z~), where a~. . . . . an a r e complex numbers. C o n s i d e r i n g f o r m u l a ( 6 . 1 ) o f Chap. 1, we h a v e IF (exp az=)l (4)=

(21,'-~)-" (<...a,,) -*/2 exp ( -

r

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Thus, the Fourier transform of the entire functionexpaz ~ is a regular exponential functional defined by the kernel exp(--~=/4=) and the system of contours ( - - ~ , V~). Example 2. Let u(z) = z -m, where z6C ~, and m is a natural number. with Example 2, Sec. 4, Chap. i we have

In cor'respondence

( 1'~-~ [Fz-~] ~ - - , ~ - ' [Fz-'l =(-~)~ (~)-- (m--l)t ($) (m--l)i ~m-ln($)' where n(~) is the natural primitive of the delta function. 2.

Fourier Transform of Exponential Functions

We shall first of all establish the connection with the classical Borel transform. Let u(z):Cn-+C I be a function of exponential type r = (r I .... ,rn). In correspondence with Definition i.i the function u(z) has a Fourier transform fi(~) which is an exponential functional defined on the space Exp(C~) of all functions of exponential type. We shall show that in the present case N(~) can be extended to an analytic functional whose kernel is the Borel transform Bu(~). Indeed, let ~(~)s where U R is an arbitrary polycylinder of radius R > r, i.e., Rj > rj, l-.<]-.
~"

"'~ ! ~t= O~u(O)o I

[~[=0

~'

'"~ ~Bu(s)CP(s)ds,

(2.1)

where F is the hull of any polycylinder U~, r < r < R. The last relation shows that u(~) is a continuous functional on CY(U~) with the topology of uniform convergence on compact sets. Moreover, formula (2.1.) defines a regular analytic functional in the polycylinder U R and in the entire space C~. The kernel of this regular functional is the Borel transform Bu(~). We note that by the inversion formula

u (z)----( ~ (~), exp z~ ) = ( 2! ~ l B~ (~) exp z~d~, P

which coincides with the classical Borel formula. Conversely, since any functional h(~)EgY'(C~) defines a compact measure (see, for example, HSrmander [39], Napalkov [25], etc.), the function u(z)= (h(~), expz~ ),z6Cn, is a function of exponential type. It remains to note that u(~) = h(~), and the correspondence between h(5) and u(~) is one-to-one. In summary we thus obtain the (known) Assertion 2. i.

There is the algebraic isomorphism

F : Exp (C~)+~O ' (C~), whereby the inverse mapping is defined by the classical inversion formula of Borel. We now turn to the general case.

u(z)~Expa(C~)be

Let

an arbitrary function.

Then

(0 = ~ ~ (z), where in c o r r e s p o n d e n c e w i t h f o r m u l a ( 2 . 1 ) uik(5) a r e r e g u l a r a n a l y t i c c y l i n d e r s [JR(h) and, as a c o n s e q u e n c e , in t h e domain ~. Thus,

functionals

F :Expn (C~) -+ (7' (~2). C o n v e r s e l y , l e t h(~)~d?'(~). Then, as a l r e a d y n o t e d , t h e r e a c o u n t a b l y a d d i t i v e measure ~(d~) c o n c e n t r a t e d on K such t h a t

in p o l y (2.2)

exist

a compact s e t K c ~ and

< h(~), ~(~) > = ~ ~(~)~(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