Property of complex unitarity

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