Fourier transform of exponential functions

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.

2762

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

N

N

i=I K i

and hence tt{z)~Expn( C nz); moreover

By p r o p e r t y 3) each f u n c t i o n u~(z)CExp~(~i)(C~) Hence~ the mapping (2.2) is one-to-one. TheOREM 2.2.

t=1

u(~) = h ( ~ ) .

We have thus established

If ~ is a Runge domain, then the mapping 9F

:

Expa (C~)-+ 0" (.<2)

is one-to-one, and the inverse mapping F -l determined by the inversion formula is the FourierBorel transform of an analytic functional. Remark. Theorem 2.2 is naturally called a complex Paley-Wiener theorem, since each analytic functional is determined by a compact measure. Thus, the Fourier transform of an entire function u(z) is defined by a compact measure contained in some Runge domain ~ C ~ , if and only if ~(z)6Exp~(C~). 3.

Pro e p _ ~

of Complex Unitarit Z

Let ~ (z)~G (G) and ~ (~)EExp~ (-C~). Then ~ (~)6 Exp~ (C~), and ~ (z)~G~(G).

We have

THEOREM 3.1. For any functions u(z)ECY(G) and ~(~)~Expo(C~)there is the formula

/, ~ (;), ~ (;) > = < ~ (z), ~ (z) >. Proof.

( 3. i)

Indeed, since ~(~)~Expo(C~), it follows that

~, (;) = , ~ e~p~ (;), where %~(~)~Exp~(~)(C~), and ~EG

runs through a finite set of values.

Then

( u (;), q~(;) > -- < u (-- O) 5 (;), q~(;)) = ~ < 5 (;), u (0) e~q~ (;) > = ~ < ~ (~),

~ D% (Z) CO- ~,I) ~ [ea;~px (~)1 > = ~&, I~1=o 2 ~ D~u (~,)O~P,~ (0). lczi=O On the other hand, def

< u(z), ~(z) ) = (~(z),u(z) ) = ~

(e-XZ~P~(--D)5(z), u(z) > ---~!

;~

~,

0 % (~,).

,L I ~ ] = 0

Comparing the expressions obtained, we arrive at formula (3.1)o

The theorem is proved.

Remark. Formula (3.1) obviously generalizes one of the forms of the Parseval equality. It is clear that this formula can also be given the following equivalent form: for any functionals f (~)~0" (g2) and f~(z)EExp~ (Czn) there is the equality

'< f (E), [Y-~hi (;)) ----- < [F-'f] (zL ~ (z)l >, where :[Y-~h](~)_----( h (z),exp ~z ) is the Fourier-Borel (Fourier-Laplace) transform. In summary, as already noted in Introduction, the spaces of exponential functions and exponential functionals and also the spaces of analytic functions and analytic functionals are connected by the diagram

Expa (Cfl)-~eU ' (~2)

I.

F

't*

Exp~ (Cz")~-e 62), where (~'~) denotes the operation of passing to the dual space. Here the elements q~(z)fiExp.q (Czn), ~ (~)@(Y'(~),~ ($)@~Y(Q) and ~ (z)'6Exp'~(Czn) are connected by the unitarity formula (3. I). 4._.~lications

to P/D Equations

Before describing the complex Fourier method, we note that an additional property of the Fourier transform important for applications follows directly from the definition of a 2763