  For alpha>0 we consider the system l_k^{(alpha-1)/2}(x) of the Laguerre
functions which are eigenfunctions of the differential operator Lf
=-\frac{d^2}{dx^2}f-\frac{alpha}{x}\frac{d}{dx}f+x^2 f. We define an atomic
Hardy space H^1_{at}(X), which is a subspace of L^1((0,infty), x^alpha dx).
Then we prove that the space H^1_{at}(X) is also characterized by the Riesz
transform Rf=\sqrt{\pi}\frac{d}{dx}L^{-1/2}f in the sense that f\in H^1_{at}(X)
if and only if f,Rf \in L^1((0,infty),x^alpha dx).
