### Abstract

The summability of Fourier transforms can be generalized for higher dimensions basically in two ways. In this chapter, we study the ℓ_{q}-summability of higher dimensional Fourier transforms. As in the literature, we investigate the three cases q = 1, q = 2 and q = ∞. The other type of summability, the so-called rectangular summability, will be investigated in the next chapter. Both types are general summability methods defined by a function θ. We will generalize the results of Sects. 2.5 – 2.9. In the first section, we present the basic definitions of the ℓ_{q}-summability. In the next section, we prove the norm convergence of the θ-means. It is shown that the maximal operator of the θ-means is bounded from H_{p} ^{□}(ℝ^{d}) to L_{p}(ℝ^{d}) for any p > p_{0}, which implies the almost everywhere convergence. In Sect. 5.4, the convergence at Lebesgue points is investigated. Since the proofs are very different for different q’s, therefore each case needs new ideas. Using the result of the ℓ_{∞}-summability, in the last section we prove the one-dimensional strong summability results presented in Sect. 2.10.

Original language | English |
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Title of host publication | Applied and Numerical Harmonic Analysis |

Publisher | Springer International Publishing |

Pages | 229-382 |

Number of pages | 154 |

Edition | 9783319568133 |

DOIs | |

Publication status | Published - Jan 1 2017 |

### Publication series

Name | Applied and Numerical Harmonic Analysis |
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Number | 9783319568133 |

ISSN (Print) | 2296-5009 |

ISSN (Electronic) | 2296-5017 |

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### ASJC Scopus subject areas

- Applied Mathematics

### Cite this

_{q}-Summability of multi-dimensional fourier transforms. In

*Applied and Numerical Harmonic Analysis*(9783319568133 ed., pp. 229-382). (Applied and Numerical Harmonic Analysis; No. 9783319568133). Springer International Publishing. https://doi.org/10.1007/978-3-319-56814-0_5