More on the Density of Analytic Polynomials in Abstract Hardy Spaces

Let $\{F_n\}$ be the sequence of the Fej'er kernels on the unit circle $\mathbb{T}$. 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 $X^{\mathrm{\prime }}$, then $\mathrm{\parallel }f\ast F_n-f{\mathrm{\parallel }}_X\to 0$ for every $f\in X$ as $n\to \mathrm{\infty }$. 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 $X^{\mathrm{\prime }}$. In this note we show that there exists a separable weighted $L^1$ space $X$ such that the sequence $f\ast F_n$ does not always converge to $f\in X$ in the norm of $X$. On the other hand, we prove that the set ${\mathcal{P}}_A$ is dense in $H[X]$ under the assumption that $X$ is merely separable.