@inbook{999feaa4c81147a5bc8df53a95380d04,

title = "More on the density of analytic polynomials in abstract Hardy spaces",

abstract = "Let {Fn} be the sequence of the Fej{\'e}r kernels on the unit circle T. The First author recently proved that if X is a separable Banach function space on T such that the Hardy–Littlewood maximal operator M is bounded on its associate space X′, then ‖ f * Fn - f ‖X → 0 for every f ϵ X as n → ∞. This implies that the set of analytic polynomials PA 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 L1 space X such that the sequence f * Fn does not always converge to f ϵ X in the norm of X. On the other hand, we prove that the set PA is dense in H[X] under the assumption that X is merely separable.",

keywords = "Abstract Hardy space, Analytic polynomial, Banach function space, Fej{\'e}r kernel",

author = "Alexei Karlovich and Eugene Shargorodsky",

note = "This work was partially supported by the Fundac¸{\~a}o para a Ci{\^e}ncia e a Tecnologia (Portuguese Foundation for Science and Technology) through the project UID/MAT/00297/2013 (Centro de Matem{\'a}tica e Aplicac{\~o}es).",

year = "2018",

month = jan,

day = "1",

doi = "10.1007/978-3-319-75996-8_16",

language = "English",

series = "Operator Theory: Advances and Applications",

publisher = "Springer International Publishing",

pages = "319--329",

booktitle = "Operator Theory: Advances and Applications",

}