An elementary proof of Titchmarsh’s convolution theorem

Abstract:

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