Approximation properties of Cesàro means of Vilenkin-Fourier series

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^p</i> approximation by Cesàro means of the Vilenkin-Fourier series of functions from <i>L^p</i>. These inequalities imply sufficient conditions for the convergence of Cesàro means of the Vilenkin-Fourier series in the <i>L^p</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^p</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;^−&alpha;_n (f, x)</i> to <i>f(x)</i> in the <i>L^p</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^p</i> approximation by Cesàro means of negative order of the two-dimensional Vilenkin-Fourier Series of functions from <i>L^p</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.