Wavelet series expansion in Hardy spaces with approximate duals

In this paper, we provide sufficient conditions for the functions ψ and ϕ to be the approximate duals in the Hardy space H^p(R) for all 0<p≤1.Based on these conditions, we obtain the wavelet series expansion in the Hardyspace H^p(R) with the approximate duals. The important properties of our approachinclude the following: (i) our results work for any 0<p≤1; (ii) we do notassume that the functions ψ and ϕ 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 ψ and ϕ.