Wavelet series expansion in Hardy spaces with approximate duals

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 \( 0<p\le 1 \). Based on these conditions, we obtain the wavelet series expansion in the Hardy space \(H^p(\mathbb{R})\) with the approximate duals. The important properties of our approach include the following: (i) our results work for any \( 0<p \leq 1 \); (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 \( \phi \).