Real and Complex Analysis (Rudin)

HOLOMORPHIC FOURIER TRANSFORMS 373
and
(3)
Note: The function F we are looking for is to have the property thatf(x + iy) is
the Fourier transform of F(t)e- yt (we regard y as a positive constant). Let us
apply the inversion formula (whether or not this is correct does not matter; we
are trying to motivate the proof that follows): The desired F should be of the
form
1 foo 1 f
F(t) = eY • - f(x + iy)e-it" dx = - f(z)e-1tz: dz.
21t _ 00 21t
The last integral is over a horizontal line in n +, and if this argument is correct at
all, the integral will not depend on the particular line we happen to choose. This
suggests that the Cauchy theorem should be invoked.
(4)
PROOF Fix y, 0 < y < 00. For each !X > 0 let r", be the rectangular path with
vertices at ± IX + i and ± IX + iy. By Cauchy's theorem
r f(z)e - itz: dz = o.
(5)
Jr.
We consider only real values of t. Let (p) be the integral off(z)e-1tz: over
the straight line interval from P + i to P + iy (P real). Put I = [y, 1] if y < 1,
1= [1, y] if 1 < y. Then
Put
I (P)12 = Ilf(p +iu)e-it(/I+iU) dul2 ~ llf(p + iuW du le2tu duo (6)
A(P) = II f(P + iu) 12 duo
Then (1) shows, by Fubini's theorem, that
21t
1 faO
_ 00 A(P) dP ~ Cm(I).
(7)
(8)
Hence there is a sequence {!Xj} such that !Xr-+ 00 and
By (6), this implies that
A(!Xj) + A( -!Xj)--+ 0
U--+ 00).
asj--+ 00.
(9)
(10)