Abstract
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 \( \phi \).
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This work was supported in part by the National Research Foundation of Korea (NRF) [Grant Numbers 2015R1A5A1009350 and 2021R1A2C1007598].
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Hur, Y., Lim, H. Wavelet series expansion in Hardy spaces with approximate duals. Anal Math (2024). https://doi.org/10.1007/s10476-024-00022-z
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DOI: https://doi.org/10.1007/s10476-024-00022-z