Thus, the Fourier transform of the entire functionexpaz ~ is a regular exponential functional defined by the kernel exp(...
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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-.. 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