Martingale Hardy spaces and summability of the one dimensional Vilenkin-Fourier series

Abstract: The classical theory of Fourier series deals with decomposition of afunction into sinusoidal waves. Unlike these continuous waves the Vilenkin(Walsh) functions are rectangular waves. Such waves have already been usedfrequently in the theory of signal transmission, multiplexing, filtering,image enhancement, codic theory, digital signal processing and patternrecognition. The development of the theory of Vilenkin-Fourier series hasbeen strongly influenced by the classical theory of trigonometric series.Because of this it is inevitable to compare results of Vilenkin series tothose on trigonometric series. There are many similarities between thesetheories, but there exist differences also. Much of these can be explainedby modern abstract harmonic analysis, which studies orthonormal systems from thepoint of view of the structure of a topological group.In this PhD thesis we discuss, develop and apply this fascinating theoryconnected to modern harmonic analysis. In particular we make new estimationsof Vilenkin-Fourier coefficients and prove some new results concerningboundedness of maximal operators of partial sums. Moreover, we derivenecessary and sufficient conditions for the modulus of continuity so thatnorm convergence of the partial sums is valid and develop new methods toprove Hardy type inequalities for the partial sums with respect to theVilenkin systems. We also do the similar investigation for the Fej\'er means.Furthermore, we investigate some N\"orlund means but only in the case when their coefficients are monotone. Some well-know examples of N\"orlund means areFej\'er means, Ces\`aro means and N\"orlund logarithmic means. In addition, weconsider Riesz logarithmic means, which are not example of N\"orlundmeans. It is also proved that these results are the best possible in aspecial sense. As applications both some well-known and new results arepointed out.This PhD is written as a monograph consisting of four Chapters:Preliminaries, Fourier coefficients and partial sums of Vilenkin-Fourierseries on martingale Hardy spaces, Vilenkin-Fej\'er means onmartingale Hardy spaces, Vilenkin-N\"orlund means on martingale Hardyspaces. It is based on 15 papers with the candidate as author or coauthor,but also some new results are presented for the first time.In Chapter 1 we first present some definitions and notations, which arecrucial for our further investigations. After that we also define somesummabilitity methods and remind about some classical facts and results. We investigate some well-known results and prove new estimates for the kernels of these summabilitity methods, which are very important to prove our main results. Moreover, we define martingale Hardy spaces and construct martingales, which help us to prove sharpness of our main results in the later chapters.Chapter 2 is devoted to present and prove some new and known results about Vilenkin-Fourier coefficients and partial sums of martingales in Hardy spaces.First, we show that Fourier coefficients of martingales are not uniformly bounded when $0