If 
is square integrable over the real 
-axis, then any one of the following
implies the other two:
1. The Fourier transform ](/images/equations/TitchmarshTheorem/Inline3.svg)
is 0 for 
.
2. Replacing 
by 
,
the function 
is analytic in the complex plane 
for 
and approaches 
almost everywhere as 
. Furthermore, 
for some number 
and 
(i.e., the integral is bounded).
3. The real and imaginary parts of 
are Hilbert transforms of each other
(Bracewell 1999, Problem 8, p. 273).
