  In this paper, we provide sufficient conditions for the functions $\psi$ and
$\phi$ to be the approximate duals in the Hardy space $H^p(\mathbb{R})$ for all
$0<p\leq1$. Based on these conditions, we obtain the wavelet series expansion
in the Hardy space with the approximate duals. The important properties of our
approach include the following: (i) our results work for any $0<p\leq1$; (ii)
we do not assume that the functions $\psi$ and $\phi$ are exact duals; (iii) we
provide a tractable bound for the operator norm of the associated wavelet frame
operator so that it is possible to check the suitability of the functions
$\psi$ and $\phi$.
