Wavelet series expansion in Hardy spaces with approximate duals

In this paper, we provide sufficient conditions for the functions \( \psi \) $\psi$ and \( \phi \) $\varphi$ to be the approximate duals in the Hardy space \(H^p(\mathbb{R})\) $H^p(\mathbb{R})$ for all \( 0<p\le 1 \) $0<p\le 1$ .Based on these conditions, we obtain the wavelet series expansion in the Hardyspace \(H^p(\mathbb{R})\) $H^p(\mathbb{R})$ with the approximate duals. The important properties of our approachinclude the following: (i) our results work for any \( 0<p \leq 1 \) $0<p\le 1$ ; (ii) we do notassume that the functions \( \psi \) $\psi$ and \( \phi \) $\varphi$ are exact duals; (iii) we provide a tractablebound for the operator norm of the associated wavelet frame operator so that itis possible to check the suitability of the functions \( \psi \) $\psi$ and \( \phi \) $\varphi$ .