Tsitsino Tepnadze

Tsitsino Tepnadze disputerer for ph.d.-graden i ingeniørvitenskap og vil offentlig forsvare avhandlingen / Tsitsino Tepnadze will defend her thesis for the PhD degree in Engineering Science: 

“Approximation properties of Cesàro means of Vilenkin-Fourier series”

Avhandlingen er tilgjengelig her / The doctoral thesis is available here.

Auditoriet er åpent for publikum. Disputasen vil også bli strømmet. Opptak av disputasen vil være tilgjengelig i en måned.

The auditorium is open to the public. The defense will be streamed. A recording of the defense will be available for one month.

Prøveforelesningen starter kl. 10:15 / The trial lecture starts at 10:15. Tittel / title:

“Fundamental and Applied Aspects of Fourier Analysis”

Disputasen starter kl. 12:15 / The defense starts at 12:15.

Prøveforelesning og disputas strømmes her / The trial lecture and defense will be streamed here.

Sammendrag av avhandlingen / Summary of the thesis:

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 Lp approximation by Cesàro means of the Vilenkin-Fourier series of functions from Lp. These inequalities imply sufficient conditions for the convergence of Cesàro means of the Vilenkin-Fourier series in the Lp−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 Lp− metric.

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.

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 σ−α n (f, x) to f(x) in the Lp− metric in terms of the modulus of continuity. Moreover, we prove the sharpness of these conditions.

Chapter 2 is focused on a new approach to investigate the rate of Lp approximation by Cesàro means of negative order of the two-dimensional Vilenkin-Fourier Series of functions from Lp. 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.

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.

Veiledere / Supervisors:

Hovedveileder / Main supervisor:

Professor Ushangi Goginava, Ivane Javakhishvili Tbilisi State University.

Medveiledere / Co-supervisors:

Professor Lars-Erik Persson, Department of Computer Science and Computational Engineering, UiT The Arctic University of Norway,

Professor Annette Meidell, Department of Computer Science and Computational Engineering, UiT The Arctic University of Norway. 

Bedømmelseskomité / Evaluation committee:

Professor Giorgi Oniani, School of Mathematics, Kutaisi International University (1st opponent),

Associate professor Sorina Barza, Department of Mathematics and Computer Science, Karlstad University (2nd opponent),

Associate professor Aleksander Pedersen, Department of Computer Science and Computational Engineering, UiT The Arctic University of Norway (administrator of the committee).

Prøveforelesning og disputas ledes av prodekan for forskning, Svein-Erik Sveen / The trial lecture and defense is led by Vice Dean of research, Svein-Erik Sveen.

De som ønsker å opponere ex auditorio kan sende e-post til Svein-Erik Sveen / Opponents ex auditorio should contact Svein-Erik Sveen.

When: 05.05.22 at 10.15–16.00
Where: Auditorium 1 (D1080)
Location / Campus: Digital, Narvik
Target group: Employees, Students, Guests, Invited
Contact: Audun Reigstad
E-mail: audun.reigstad@uit.no
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