Karlovych, OleksiyShargorodsky, Eugene2024-09-242024-09-242024-010019-3577PURE: 93322698PURE UUID: 3e45327d-0f31-4e4e-a173-2527862fc40dScopus: 85175318139WOS: 001158799000001http://hdl.handle.net/10362/172330Publisher Copyright: © 2023 The Author(s)The aim of the paper is to highlight some open problems concerning approximation properties of Hardy spaces. We also present some results on the bounded compact and the dual compact approximation properties (shortly, BCAP and DCAP) of such spaces, to provide background for the open problems. Namely, we consider abstract Hardy spaces H[X(w)] built upon translation-invariant Banach function spaces X with weights w such that w∈X and w−1∈X′, where X′ is the associate space of X. We prove that if X is separable, then H[X(w)] has the BCAP with the approximation constant M(H[X(w)])≤2. Moreover, if X is reflexive, then H[X(w)] has the BCAP and the DCAP with the approximation constants M(H[X(w)])≤2 and M∗(H[X(w)])≤2, respectively. In the case of classical weighted Hardy space Hp(w)=H[Lp(w)] with 1<p<∞, one has a sharper result: M(Hp(w))≤2|1−2/p| and M∗(Hp(w))≤2|1−2/p|.16336797engBounded compact and dual compact approximation propertiesTranslation-invariant Banach function spaceWeighted Hardy spaceGeneral Mathematicsjournal article10.1016/j.indag.2023.10.004https://www.scopus.com/pages/publications/85175318139