## Abstract

Beurling's algebra A* = {f : ∑^{∞}_{k=0}sup_{k≤|m|} | f̂ (m)| < ∞ }is considered. A* arises quite naturally in problems of summability of the Fourier series at Lehesgue points, whereas Wiener's algebra A of functions with absolutely convergent Fourier series arises when studying the norm convergence of linear means. Certainly, both algebras are used in some other areas. A* has many properties similar to those of A, but there are certain essential distinctions. A* is a regular Banach algebra, its space of maximal ideals coincides with [-π, π], and its dual space is indicated. Analogs of Herz's and Wiener-Ditkin's theorems hold. Quantitative parameters in an analog of the Beurling-Pollard theorem differ from those for A. Several inclusion results comparing the algebra A* with certain Banach spaces of smooth functions are given. Some special properties of the analogous space for Fourier transforms on the real axis are presented. The paper ends with a summary of some open problems.

Original language | English |
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Pages (from-to) | 103-129 |

Number of pages | 27 |

Journal | Journal of Fourier Analysis and Applications |

Volume | 3 |

Issue number | 2 |

DOIs | |

State | Published - 1997 |

## Keywords

- Absolute convergence
- Banach algebra
- Synthesis