## Abstract

Let W(R) be the Wiener Banach algebra of functions representable by the Fourier integrals of Lebesgue integrable functions. It is proved in the paper that, in particular, a trigonometric series ∑k=-∞∞ckeikt is the Fourier series of an integrable function if and only if there exists a ϕ∈ W(R) such that ϕ(k) = c_{k}, k∈ Z. If f∈ W(R) , then the piecewise linear continuous function ℓ_{f} defined by ℓ_{f}(k) = f(k) , k∈ Z, belongs to W(R) as well. Moreover, ‖ℓf‖W0≤‖f‖W0. Similar relations are established for more advanced Wiener algebras. These results are supplemented by numerous applications. In particular, new necessary and sufficient conditions are proved for a trigonometric series to be a Fourier series and new properties of W are established.

Original language | English |
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Pages (from-to) | 185-206 |

Number of pages | 22 |

Journal | Constructive Approximation |

Volume | 54 |

Issue number | 2 |

DOIs | |

State | Published - Oct 2021 |

### Bibliographical note

Publisher Copyright:© 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC part of Springer Nature.

## Keywords

- Bernstein inequality
- Fourier series of a measure
- Fourier transform of a measure
- Hilbert transform
- Poisson summation formula
- Wiener algebras, entire function of exponential type