Real and Complex Analysis (Rudin)

372 REAL AND COMPLEX ANALYSIS
Let US rewrite (1) in the form
f(x + iy) = 100
F(t)e-tYeitx dt,
(2)
regard y as fixed, and apply Plancherel's theorem. We obtain
- If(x+iy)12 dx= IF(tWe- 2tY dt:::;; IF(t)1 2 dt
1 foo 100
100
2n - 00 0 0
for every y > O. [Note that our notation now differs from that in Chap. 9. There
the underlying measure was Lebesgue measure divided by fo. Here we just use
Lebesgue measure. This accounts for the factor 1/(2n) in (3).] This shows:
(a) Iffis of the form (1), thenfis holomorphic in II+ and its restrictions to horizontal
lines in II + form a bounded set in J3( - 00, 00).
Our second class consists of allf of the form
f(z) = fAAF(t)eitz dt (4)
where 0 < A < 00 and F E J3( - A, A). These functions f are entire (the proof is
the same as above), and they satisfy a growth condition:
I f(z) I:::;; fAAI F(t) I e-ty dt :::;; eA1yI fAAI F(t) I dt. (5)
If C is this last integral, then C < 00, and (5) implies that
I f(z) I :::;; CeA1z1. (6)
[Entire functions which satisfy (6) are said to be of exponential type.] Thus:
(b) Every f of the form (4) is an entire function which satisfies (6) and whose
restriction to the real axis lies in 13 (by the Plancherel theorem).
It is a remarkable fact that the converses of (a) and (b) are true. This is the
content of Theorems 19.2 and 19.3.
Two Theorems of Paley and Wiener
(3)
19.2 Theorem SupposefE H(II+) and
1 foo
sup -2 If(x + iy)1 2 dx = C < 00.
O