Real and Complex Analysis (Rudin)

374 REAL AND COMPLEX ANALYSIS
Note that this holds for every t and that the sequence {ex)} does not depend
on t.
Let us define
1 fl%i .
gj,y, t) = -
2n -I%i
Then we deduce from (5) and (10) that
f(x + iy)e-·tx dx.
(11)
lim [eYgiy, t) - egi1, t)] = 0 (-00 < t < 00). (12)
Write f,(x) for f(x + iy). Then f, E I3( - 00, 00), by hypothesis, and the
Plancherel theorem asserts that
J~rr:, L:,/,(t) - gJ{Y, t)12 dt = 0, (13)
where/, is the Fourier transform offy. A subsequence of {giY, t)} converges
therefore pointwise to /,(t), for almost all t (Theorem 3.12). If we define
it now follows from (12) that
(14)
F(t) = el,(t). (15)
Note that (14) does not involve y and that (15) holds for every y E (0,00).
Plancherel's theorem can be applied to (15):
OO
f foo 1 foo
_ooe- 2ty 1 F(tW dt = _ool/'(tW dt = 2n _oolfy(xW dx ~ C.
If we let y-+ 00, (16) shows that F(t) = 0 a.e. in (- 00,0).
If we let y-+ 0, (16) shows that
(16)
1001 F(t) 12 dt ~ C.
(17)
It now follows from (15) that/, E Ll if y > O. Hence Theorem 9.14 gives
fy(x) = L: !,(t)e itx dt (18)
or
f(z) = 1
00 F(t)e-yteitx dt = 1
00 F(t)eit % dt (19)
This is (2), and now (3) follows from (17) and formula 19Y3).
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