Two generalizations of Titchmarsh's convolution theorem

Raouf Doss
1990 Proceedings of the American Mathematical Society  
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) =
more » ... ^ZXpaßf>(t -apa) = 0 for / in (0, T), where apa > 0, Xpa are constants. Conclusions similar to Titchmarsh's hold with the additional information that a > T -apa whenever V ?í o. For /, geLl(R) the convolution f*g is defined as f*g(t)= [ f(t-x)g(x)dx. JR Titchmarsh's theorem states that if /, g = 0 on the interval (-oo,0) and if f*g(t) = 0 forte(0,T), then there are numbers a, ß > 0 with a + ß = T for which f(x) = 0 for almost all x in (0,a) and g(x) = 0 for almost all x in (0,/? ). T may be infinite. There are many different proofs of this famous theorem; most of them, like Titchmarsh's [11], Crum [2], Dufresnoy [4], Boas [1], Koosis [7], and Lax [8], are based on the theory of analytic or harmonic functions; others, like Mikusinski [10], use real variable methods; still others, like Helson [5], Doss [3], rely on harmonic analysis. For an extension to functions of several variables see Lions [9] and to functions taking values in certain Banach algebras see Mikusinski
doi:10.1090/s0002-9939-1990-1004416-8 fatcat:cx3z6ek5xnf3ddcczjiklpo7r4