  This article studies the Fourier spectrum characterization of functions in
the Clifford algebra-valued Hardy spaces $H^p(\mathbf R^{n+1}_+), 1\leq p\leq
\infty.$ Namely, for $f\in L^p(\mathbf R^n)$, Clifford algebra-valued, $f$ is
further the non-tangential boundary limit of some function in $H^p(\mathbf
R^{n+1}_+),$ $1\leq p\leq \infty,$ if and only if $\hat{f}=\chi_+\hat{f},$
where $\chi_+(\underline{\xi})=\frac{1}{2}(1+i\frac{\underline \xi}{|\underline
\xi|}),$ where the Fourier transformation and the above relation are suitably
interpreted (for some cases in the distribution sense). These results further
develop the relevant context of Alan McIntosh. As a particular case of our
results, the vector-valued Clifford Hardy space functions are identical with
the conjugate harmonic systems in the work of Stein and Weiss. The latter
proved the corresponding results in terms of the single integral form for the
cases $1\leq p<\infty.$
