More on the Density of Analytic Polynomials in Abstract Hardy Spaces

TL;DR

This paper investigates the density of analytic polynomials in abstract Hardy spaces, showing that certain weighted spaces lack norm convergence of Fejér means, but polynomial density still holds under separability.

Contribution

It demonstrates that weighted $L^1$ spaces can lack Fejér convergence, yet establishes polynomial density in Hardy spaces for all separable Banach function spaces.

Findings

Weighted $L^1$ spaces may not have Fejér mean convergence.

Polynomial density in Hardy spaces holds for all separable Banach spaces.

Boundedness of the Hardy-Littlewood maximal operator influences convergence properties.

Abstract

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 $X'$, 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 $X'$. In this note we show that there exists a separable weighted $L^1$ 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 $H[X]$ under the assumption that $X$ is merely separable.