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Two generalizations of Titchmarsh's convolution theorem

1990
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Proceedings of the American Mathematical Society
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Titchmarsh's convolution theorem states that if the functions /, g vanish on (-co, 0) and if the convolution /* g(t) = 0 on an interval (0, T), then there are two numbers a , ß > 0 such that a + ß = T, f = 0 a.e. on (0, a), and g = 0 a.e. on (0,ß). T may be infinite. For the case T = oo we prove that if / * g = 0 on R and one of the two functions /, g is 0 on (-OO.0), then either / or g is 0 a.e. on R. Next we consider the iritegro-differential-difference equation / * g(t) + ^ZXpaßf>(t -apa) =

doi:10.1090/s0002-9939-1990-1004416-8
fatcat:cx3z6ek5xnf3ddcczjiklpo7r4