Subsequent to our recent work on Fourier spectrum characterization of Hardy spaces H(R) for the index range 1 ≤ p≤ ∞, in this paper we prove further results on rational Approximation, integral representation and Fourier spectrum characterization of functions for the Hardy spaces H(R) , 0 < p≤ ∞, with particular interest in the index range 0 < p≤ 1. We show that the set of rational functions in H(C) with the single pole - i is dense in H(C) for 0 < p< ∞. Secondly, for 0 < p< 1 , through rational function approximation we show that any function f in L(R) can be decomposed into a sum g+ h, where g and h are, in the L(R) convergence sense, the non-tangential boundary limits of functions in, respectively, H(C) and H(C) , where Hp(Ck)(k=±1) are the Hardy spaces in the half plane C= {z= x+ iy: ky> 0}. We give Laplace integral representation formulas for functions in the Hardy spaces H, 0 < p≤ 2. Besides one in the integral representation formula we give an alternative version of Fourier spectrum characterization for functions in the boundary Hardy spaces H for 0 < p≤ 1.