  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 $w\in X$ and $w^{-1}\in
X'$, 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
$M(H[X(w)])\le 2$. Moreover, if $X$ is reflexive, then $H[X(w)]$ has the BCAP
and the DCAP with the approximation constants $M(H[X(w)])\le 2$ and
$M^*(H[X(w)])\le 2$, respectively. In the case of classical weighted Hardy
space $H^p(w) = H[L^p(w)]$ with $1<p<\infty$, one has a sharper result:
$M(H^p(w))\le 2^{|1-2/p|}$ and $M^*(H^p(w))\le 2^{|1-2/p|}$.
