Two generalizations of Titchmarsh’s convolution theorem

Abstract

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