### Abstract

This paper provides a fairly general approach to summability questions for multi-dimensional Fourier transforms. It is based on the use of Wiener amalgam spaces W(L_{p}, l_{q})(ℝ^{d}), Herz spaces and weighted versions of Feichtinger 's algebra and covers a wide range of concrete special cases (20 of them are listed at the end of the paper). It is proved that under some conditions the maximal operator of the θ-means σ_{T}^{θ}f can be estimated pointwise by the Hardy-Littlewood maximal function. From this it follows that σ_{T}^{θ}f → f a.e. for all ∈ W(L _{1},l_{∞})(ℝ^{d}), hence f ∈ L _{p}(ℝ^{d}) for any 1 ≤ p ≤ ∞. Moreover, σ_{T}^{θ}f(x) converges to f(x) at each Lebesgue point of f ∈ L_{1}(ℝ^{d}) (resp. f ∈ W(L _{1},l_{∞})(ℝ^{d})) if and only if the Fourier transform of θ is in a suitable Herz space. In case θ is in a Besov space or in a weighted Feichtinger's algebra or in a Sobolev-type space then the a.e. convergence is obtained. Some sufficient conditions are given for θ to be in the weighted Feichtinger's algebra. The same results are presented for multi-dimensional Fourier series.

Original language | English |
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Pages (from-to) | 509-536 |

Number of pages | 28 |

Journal | Mathematical Proceedings of the Cambridge Philosophical Society |

Volume | 140 |

Issue number | 3 |

DOIs | |

Publication status | Published - May 1 2006 |

### ASJC Scopus subject areas

- Mathematics(all)