### Abstract

A general summability method, the so-called θ-summability is considered for multi-dimensional Fourier series. Equivalent conditions are derived for the uniform and L _{1}-norm convergence of the θ-means σ _{n} ^{θ} f to the function f. If f is in a homogeneous Banach space, then the preceeding convergence holds in the norm of the space. In case θ is an element of Feichtinger's Segal algebra S _{0}(ℝ^{d})then these convergence results hold. Some new sufficient conditions are given for θ to be in S_{0}(ℝ ^{d}). A long list of concrete special cases of the θ-summation is listed. The same results are also provided in the context of Fourier transforms, indicating how proofs have to be changed in this case.

Original language | English |
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Pages (from-to) | 333-349 |

Number of pages | 17 |

Journal | Monatshefte fur Mathematik |

Volume | 148 |

Issue number | 4 |

DOIs | |

Publication status | Published - Aug 1 2006 |

### Keywords

- Amalgam spaces
- Besov-
- Feichtinger's algebra
- Fractional Sobolev spaces
- Homogeneous Banach space
- Sobolev-
- Wiener algebra
- θ-summability of Fourier series

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

^{d}and norm summability of fourier series and fourier transforms.

*Monatshefte fur Mathematik*,

*148*(4), 333-349. https://doi.org/10.1007/s00605-005-0358-4