We give an elementary proof of the following theorem of Titchmarsh. Suppose
f
,
g
f,g
are integrable on the interval
(
0
,
2
T
)
\left ( {0,2T} \right )
and that the convolution
f
∗
g
(
t
)
=
∫
0
t
f
(
t
−
x
)
g
(
x
)
d
x
=
0
f * g\left ( t \right ) = \int _0^t {f\left ( {t - x} \right )g\left ( x \right )dx} = 0
on
(
0
,
2
T
)
\left ( {0,2T} \right )
. Then there are nonnegative numbers
α
,
β
\alpha ,\beta
with
α
+
β
≥
2
T
\alpha + \beta \geq 2T
for which
f
(
x
)
=
0
f\left ( x \right ) = 0
for almost all
x
x
in
(
0
,
α
)
\left ( {0,\alpha } \right )
and
g
(
x
)
=
0
g\left ( x \right ) = 0
for almost all
x
x
in
(
0
,
β
)
\left ( {0,\beta } \right )
.