More on the density of analytic polynomials in abstract Hardy spaces

Alexei Karlovich, Eugene Shargorodsky

Research output: Chapter in Book/Report/Conference proceedingChapter

1 Citation (Scopus)

Abstract

Let {Fn} be the sequence of the Fejér 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.

Original languageEnglish
Title of host publicationOperator Theory: Advances and Applications
PublisherSpringer International Publishing
Pages319-329
Number of pages11
DOIs
Publication statusPublished - 1 Jan 2018

Publication series

NameOperator Theory: Advances and Applications
PublisherSpringer International Publishing
Volume268
ISSN (Print)0255-0156
ISSN (Electronic)2296-4878

Keywords

  • Abstract Hardy space
  • Analytic polynomial
  • Banach function space
  • Fejér kernel

Fingerprint Dive into the research topics of 'More on the density of analytic polynomials in abstract Hardy spaces'. Together they form a unique fingerprint.

  • Cite this

    Karlovich, A., & Shargorodsky, E. (2018). More on the density of analytic polynomials in abstract Hardy spaces. In Operator Theory: Advances and Applications (pp. 319-329). (Operator Theory: Advances and Applications; Vol. 268). Springer International Publishing. https://doi.org/10.1007/978-3-319-75996-8_16