We give Fourier spectrum characterizations of functions in the Hardy spaces on tubes for For we show that F is the non-tangential boundary limit of a function in a Hardy space, where is an open cone of and is the related tube in if and only if the classical or the distributional Fourier transform of F is supported in where is the dual cone of This generalizes the results of Stein and Weiss for in the same context, as well as those of Qian et al. in one complex variable for Furthermore, we extend the Poisson and Cauchy integral representation formulas to the spaces on tubes for and with, respectively, the two types of representations.