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Subsequent to our recent work on Fourier spectrum characterization of Hardyspaces $H^p(\mathbb{R})$ for the index range $1\leq p\leq \infty,$ in thispaper we prove further results on rational Approximation, integralrepresentation and Fourier spectrum characterization of functions in the Hardyspaces $H^p(\mathbb{R}), 0 < p\leq \infty,$ with particular interest in theindex range $ 0< p \leq 1.$ We show that the set of rational functions in $H^p(\mathbb{C}_{+1}) $ with the single pole $-i$ is dense in $H^p(\mathbb{C}_{+1}) $ for $0<p<\infty.$ Secondly, for $0<p<1$, throughrational function approximation we show that any function $f$ in$L^p(\mathbb{R})$ can be decomposed into a sum $g+h$, where $g$ and $h$ are, inthe $L^p(\mathbb{R})$ convergence sense, the non-tangential boundary limits offunctions in, respectively, $ H^p(\mathbb{C}_{+1})$ and$H^{p}(\mathbb{C}_{-1}),$ where $H^p(\mathbb{C}_k)\ (k=\pm 1) $ are the Hardyspaces in the half plane $ \mathbb{C}_k=\{z=x+iy: ky>0\}$. We give Laplaceintegral representation formulas for functions in the Hardy spaces $H^p,$$0<p\leq2.$ Besides one in the integral representation formula we give analternative version of Fourier spectrum characterization for functions in theboundary Hardy spaces $H^p$ for $0<p\leq 1.$

Rational Approximation, Hardy Space - Decomposition of Functions in  $L_p, p<1$: Further Results in Relation to Fourier Spectrum Characterization  of Hardy Spaces