More on the density of analytic polynomials in abstract Hardy spaces

Abstract.

Let $\left\{F_n\right\}$ be the sequence of the Fejér 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 $X^\prime$, then $\| f * F_n - f \|_X \to 0$ for every $f \in X\; \mathrm{as}\; n \to \infty$. This implies that the set of analytic polynomials $\mathcal{P}_A$ is dense in the abstract Hardy space $H \left[X \right]$ built upon a separable Banach function space X such that M is bounded on $X^\prime$. In this note we show that there exists a separable weighted L¹ space X such that the sequence $f * F_n$ 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 $H \left[X \right]$ under the assumption that X is merely separable.

This work was partially supported by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through the project UID/MAT/00297/2013 (Centro de Matemática e Aplicações).