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Two generalizations of Titchmarsh's convolution theorem


Author: Raouf Doss
Journal: Proc. Amer. Math. Soc. 108 (1990), 893-897
MSC: Primary 42A85; Secondary 45E10, 45J05
DOI: https://doi.org/10.1090/S0002-9939-1990-1004416-8
MathSciNet review: 1004416
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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$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1990-1004416-8
Article copyright: © Copyright 1990 American Mathematical Society

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