The author establishes the inequality \(\sum_{n=0}^\infty {|c_n(f)|\over (n+1)^{29/36}}\leq C|f|_{H^1(\mathbb{R})}\) for \(f\) being in the Hardy space \(H^1(\mathbb{R})\) and \(c_n(f)\) being the coefficients of the Hermite-Fourier expansion, as well as the inequality \(\sum_{n=0}^\infty {|c_n^\alpha(f)|\over (n+1)}\leq C|f|_{H^1(0,\infty)}\) for \(f\) being in the Hardy space \(H^1(0,\infty)\) and \(c_n^\alpha(f)\) being the coefficients of the Laguerre-Fourier expansion. The proofs are based on the atomic decomposition characterization of Hardy spaces. Furthermore, Paley type theorems are deduced.