### Abstract

A general summability method, the so-called θ-summability is considered for multi-dimensional Fourier transforms and Fourier series. A new inequality for the Hardy-Littlewood maximal function is verified. It is proved that if the Fourier transform of θ is in a Herz space, then the restricted maximal operator of the θ-means of a distribution is of weak type (1, 1), provided that the supremum in the maximal operator is taken over a cone-like set. From this it follows that σ_{T}^{θ} f → f over a cone-like set a.e. for all f ∈ L_{1} (R^{d}). Moreover, σ_{T}^{θ} f (x) converges to f (x) over a cone-like set at each Lebesgue point of f ∈ L_{1} (R^{d}) if and only if the Fourier transform of θ is in a suitable Herz space. These theorems are extended to Wiener amalgam spaces as well. The Riesz and Weierstrass summations are investigated as special cases of the θ-summation.

Original language | English |
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Pages (from-to) | 42-54 |

Number of pages | 13 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 344 |

Issue number | 1 |

DOIs | |

Publication status | Published - Aug 1 2008 |

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### Keywords

- Cone-like sets
- Hardy-Littlewood maximal function
- Herz spaces
- Lebesgue points
- Restricted convergence
- Wiener amalgam spaces
- θ-Summation of Fourier series

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics