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The authors obtain some multiplier theorems on $H^p$ spaces analogous to the classical $L^p$ multiplier theorems of de Leeuw. The main result is that a multiplier operator ${\left(Tf\right)}^{(}x)=\lambda \left(x\right)f̂\left(x\right)$ $\left(\lambda \in C\left({ℝ}^n\right)\right)$ is bounded on $H^p\left({ℝ}^n\right)$ if and only if the restriction ${\lambda \left(\epsilon m\right)}_{m\in \Lambda }$ is an $H^p\left(T^n\right)$ bounded multiplier uniformly for ε>0, where Λ is the integer lattice in ${ℝ}^n$.