Density of Analytic Polynomials in Abstract Hardy Spaces

Let $X$ be a separable Banach function space on the unit circle $\mathbb{T}$ and $H[X]$ be the abstract Hardy space built upon $X$. We show that the set of analytic polynomials is dense in $H[X]$ if the Hardy-Littlewood maximal operator is bounded on the associate space $X^{\mathrm{\prime }}$. This result is specified to the case of variable Lebesgue spaces.