## Approximation properties of Cesàro means of Vilenkin-Fourier series

dc.contributor.advisor | Goginava, Ushangi | |

dc.contributor.author | Tepnadze, Tsisino | |

dc.date.accessioned | 2022-04-12T08:52:18Z | |

dc.date.available | 2022-04-12T08:52:18Z | |

dc.date.issued | 2022-05-05 | |

dc.description.abstract | <p>This PhD thesis focuses on the investigation of approximation properties of Cesàro means of the Vilenkin-Fourier series. In particular, we obtain some new inequalities related to the rate of <i>L<sup>p</sup></i> approximation by Cesàro means of the Vilenkin-Fourier series of functions from <i>L<sup>p</sup></i>. These inequalities imply sufficient conditions for the convergence of Cesàro means of the Vilenkin-Fourier series in the <i>L<sup>p</sup></i>−metric in terms of the modulus of continuity. Furthermore, we also proved the sharpness of these conditions. In particular, we find a continuous function under some condition of the modulus of continuity, for which Cesàro means of the Vilenkin-Fourier series diverge in the <i>L<sup>p</sup></i>− metric. <p>This PhD thesis consists of three main Chapters, based on five papers. At first, we have an Introduction, where we give a general overview of fundamental definitions and notations, followed by historical and new results, on which our study is based and inspired. We also give a formulation of our main results in this general frame and review some auxiliary results, that are significant for the proofs of our new theorems in the next main chapters. <p>In Chapter 1, we investigate the approximation properties of Cesàro means of negative order of the one-dimensional Vilenkin-Fourier Series. In particular, we derive sufficient conditions for the convergence of the means <i>σ<sup>−α</sup><sub> n</sub> (f, x)</i> to <i>f(x)</i> in the <i>L<sup>p</sup></i>− metric in terms of the modulus of continuity. Moreover, we prove the sharpness of these conditions. <p>Chapter 2 is focused on a new approach to investigate the rate of <i>L<sup>p</sup></i> approximation by Cesàro means of negative order of the two-dimensional Vilenkin-Fourier Series of functions from <i>L<sup>p</sup></i>. In particular, we derived a necessary and sufficient condition for the convergence of Cesàro (C,−α,−β) means with α,β ∊ (0, 1) in terms of the modulus of continuity. Some corresponding sharpness results are proved also in this case. <p>Chapter 3 is devoted to deriving some new results concerning the behavior of Cesàro (C,−α) means of the quadratic partial sums of double Vilenkin-Fourier series. The new results are sharp also in this case. | en_US |

dc.description.doctoraltype | ph.d. | en_US |

dc.description.popularabstract | This PhD thesis focuses on the investigation of approximation properties of Cesàro means of the Vilenkin-Fourier series. In particular, we obtain some new inequalities related to the rate of $L^{p}$ approximation by Cesàro means of the Vilenkin-Fourier series of functions from $L^{p}$. These inequalities imply sufficient conditions for the convergence of Cesàro means of the Vilenkin-Fourier series in the $L^{p}-$metric in terms of the modulus of continuity. Furthermore, we also proved the sharpness of these conditions. In particular, we find a continuous function under some condition of the modulus of continuity, for which Cesàro means of the Vilenkin-Fourier series diverge in the $L^{p}-$ metric. | en_US |

dc.identifier.isbn | 978-82-783-237-8 | |

dc.identifier.isbn | 978-82-783-238-5 | |

dc.identifier.uri | https://hdl.handle.net/10037/24764 | |

dc.language.iso | eng | en_US |

dc.publisher | UiT Norges arktiske universitet | en_US |

dc.publisher | UiT The Arctic University of Norway | en_US |

dc.rights.accessRights | openAccess | en_US |

dc.rights.holder | Copyright 2022 The Author(s) | |

dc.subject.courseID | DOKTOR-008 | |

dc.subject | VDP::Mathematics and natural science: 400::Mathematics: 410 | en_US |

dc.subject | VDP::Matematikk og Naturvitenskap: 400::Matematikk: 410 | en_US |

dc.title | Approximation properties of Cesàro means of Vilenkin-Fourier series | en_US |

dc.type | Doctoral thesis | en_US |

dc.type | Doktorgradsavhandling | en_US |