Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 $X'$, 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.

30
The categories of this paper are: q-alg
28
The categories of this paper are: alg-geom
19
The categories of this paper are: eess
35
The categories of this paper are: mtrl-th
26
The categories of this paper are: bayes-an
7
The categories of this paper are: hep-th
34
The categories of this paper are: chem-ph
23
The categories of this paper are: solv-int
13
The categories of this paper are: quant-ph
3
The categories of this paper are: physics
21
The categories of this paper are: adap-org
0
The categories of this paper are: hep-ph
8
The categories of this paper are: hep-ex
27
The categories of this paper are: comp-gas
15
The categories of this paper are: nucl-ex
33
The categories of this paper are: plasm-ph