### Abstract

We study the theory of multi-dimensional Fourier transforms, namely, the inversion formula and the convergence of Fourier transforms. We formulate the analogous results to those of Sections 2.1 – 2.4 for higher dimensions. In the first section, we introduce the Fourier transform for functions and for tempered distributions and give the most important results. Since these proofs are very similar to those of the one-dimensional ones, we omit the proofs. In Section 4.3, we consider four types of Dirichlet integrals of multi-dimensional Fourier transforms, i.e. the cubic, triangular, circular and rectangular Dirichlet integrals. Using the analogous results for the partial sums of multi-dimensional Fourier series proved in Section 4.2, we show that the Dirichlet integrals converge in the L_{p}(ℝ^{d}) -norm to the function (1 < p < ∞). The multi-dimensional version of Carleson’s theorem is also verified.

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

Publisher | Springer International Publishing |

Pages | 203-227 |

Number of pages | 25 |

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 |

### ASJC Scopus subject areas

- Applied Mathematics

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## Cite this

*Applied and Numerical Harmonic Analysis*(9783319568133 ed., pp. 203-227). (Applied and Numerical Harmonic Analysis; No. 9783319568133). Springer International Publishing. https://doi.org/10.1007/978-3-319-56814-0_4