Two generalizations of Titchmarsh’s convolution theorem

Abstract:

Titchmarsh’s convolution theorem states that if the functions $f,g$ vanish on $( - \infty ,0)$ and if the convolution $f * g(t) = 0$ on an interval $(0,T)$, then there are two numbers $\alpha ,\beta \geq 0$ such that $\alpha + \beta = T,f = 0$ a.e. on $(0,\alpha )$, and $g = 0$ a.e. on $(0,\beta )$. $T$ may be infinite. For the case $T = \infty$ we prove that if $f * g = 0$ on $R$ and one of the two functions $f,g$ is 0 on $( - \infty ,0)$, then either $f$ or $g$ is 0 a.e. on $R$. Next we consider the integro-differential-difference equation $f * g(t) + \sum {{\lambda _{p\sigma }}{f^{(p)}}(t - {a_{p\sigma }}) = 0}$ for $t$ in $(0,T)$, where ${a_{\rho \sigma }} \geq 0,{\lambda _{p\sigma }}$ are constants. Conclusions similar to Titchmarsh’s hold with the additional information that $\alpha \geq T - {a_{\rho \sigma }}$ whenever ${\lambda _{\rho \sigma }} \ne 0$.