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The first author recently proved that if $X$ is a separable Banach function space on $\\mathbb{T}$ such that the Hardy-Littlewood maximal operator $M$ is bounded on its associate space $X'$, then $\\|f*F_n-f\\|_X\\to 0$ for every $f\\in X$ as $n\\to\\infty$. This implies that the set of analytic polynomials $\\mathcal{P}_A$ is dense in the abstract Hardy space $H[X]$ built upon a separable Banach function space $X$ such that $M$ is bounded on $X'$. 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