We give Fourier spectrum characterizations of functions in the Hardy H spaces on tubes for 1 ≤ p≤ ∞. For F∈ L(R) , we show that F is the non-tangential boundary limit of a function in a Hardy space, H(T) , where Γ is an open cone of R and T is the related tube in C, if and only if the classical or the distributional Fourier transform of F is supported in Γ , where Γ is the dual cone of Γ. This generalizes the results of Stein and Weiss for p= 2 in the same context, as well as those of Qian et al. in one complex variable for 1 ≤ p≤ ∞. Furthermore, we extend the Poisson and Cauchy integral representation formulas to the H spaces on tubes for p∈ [ 1 , ∞] and p∈ [ 1 , ∞) , with, respectively, the two types of representations.