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dc.contributor.advisorGoginava, Ushangi
dc.contributor.authorTepnadze, Tsisino
dc.date.accessioned2022-04-12T08:52:18Z
dc.date.available2022-04-12T08:52:18Z
dc.date.issued2022-05-05
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>&sigma;<sup>−&alpha;</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,−&alpha;,−&beta;) means with &alpha;,&beta; &#8714; (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,−&alpha;) means of the quadratic partial sums of double Vilenkin-Fourier series. The new results are sharp also in this case.en_US
dc.description.doctoraltypeph.d.en_US
dc.description.popularabstractThis 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.isbn978-82-783-237-8
dc.identifier.isbn978-82-783-238-5
dc.identifier.urihttps://hdl.handle.net/10037/24764
dc.language.isoengen_US
dc.publisherUiT Norges arktiske universiteten_US
dc.publisherUiT The Arctic University of Norwayen_US
dc.rights.accessRightsopenAccessen_US
dc.rights.holderCopyright 2022 The Author(s)
dc.rights.urihttps://creativecommons.org/licenses/by-nc-sa/4.0en_US
dc.rightsAttribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0)en_US
dc.subjectVDP::Mathematics and natural science: 400::Mathematics: 410en_US
dc.subjectVDP::Matematikk og Naturvitenskap: 400::Matematikk: 410en_US
dc.titleApproximation properties of Cesàro means of Vilenkin-Fourier seriesen_US
dc.typeDoctoral thesisen_US
dc.typeDoktorgradsavhandlingen_US


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