### Abstract

Let {F_{n}} 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 * F_{n} - f ‖_{X} → 0 for every f ϵ X as n → ∞. This implies that the set of analytic polynomials 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 ϵ X in the norm of X. On the other hand, we prove that the set P_{A} is dense in H[X] under the assumption that X is merely separable.

Original language | English |
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Title of host publication | Operator Theory: Advances and Applications |

Publisher | Springer International Publishing |

Pages | 319-329 |

Number of pages | 11 |

DOIs | |

Publication status | Published - 1 Jan 2018 |

### Publication series

Name | Operator Theory: Advances and Applications |
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Publisher | Springer International Publishing |

Volume | 268 |

ISSN (Print) | 0255-0156 |

ISSN (Electronic) | 2296-4878 |

### Keywords

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

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## Cite this

*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