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

The Paley-Wiener Theorem; Hardy space

Beijing Normal University