# On the Partial Sums and Marcinkiewicz and Fej\'er Means on the One- and Two-dimensional One-parameter Martingale Hardy Spaces

## TL;DR

This thesis investigates convergence and summability of Walsh-Fourier series in martingale Hardy spaces, providing new estimates, conditions for convergence, and strong convergence results for partial sums and means in one- and two-dimensional cases.

## Contribution

It offers novel convergence criteria, divergence estimates, and strong convergence proofs for Walsh-Fourier series in martingale Hardy spaces, extending understanding in one- and two-dimensional settings.

## Key findings

- Estimation of convergence and divergence of Walsh-Fourier partial sums.
- Necessary and sufficient conditions for convergence in Hardy spaces.
- Strong convergence results for Fejér and Marcinkiewicz means.

## Abstract

In this PhD thesis we are dealing with convergence and summability of partial sums, Fej\'er and Marcinkiewicz means with respect to one- and two-dimensional Walsh-Fourier series on the martingale Hardy spaces. This thesis is focus to achieve the following main results: To find estimation of convergence and divergence of the subsequences of partial sums of the one-dimensional Walsh-Fourier series on the martingale Hardy spaces $H_p(G)$, when $0<p\leq1$. To find necessary and sufficient conditions in terms of modulus of continuity of martingale Hardy spaces, for which subsequences of partial sums of the one-dimensional Walsh-Fourier series convergence in $H_p(G)$ norm, when $0<p\leq1$. To find estimation of convergence and divergence of the subsequences of Fej\'er means of the one-dimensional Walsh-Fourier series on the martingale Hardy spaces $H_p(G)$, when $0<p\leq1/2$. To find necessary and sufficient conditions in terms of modulus of continuity of martingale Hardy spaces, for which subsequences of Fej\'er means of the one-dimensional Walsh-Fourier series converge in $H_{p}(G)$ norm, when $0<p\leq1/2$. To prove strong convergence of one-dimensional Fej\'er means with respect to Walsh system on the martingale Hardy spaces $H_{p}(G)$, when $0<p\leq 1/2$. To prove strong convergence of diagonal partial sums with respect to the two-dimensional Walsh-Fourier series on the martingale Hardy spaces $H_{p}(G^2)$, when $0<p<1$. To prove strong convergence of Marcinkiewicz means with respect to the two-dimensional Walsh-Fourier series in $H_{2/3}(G^2)$ norm. To find necessary and sufficient conditions in terms of modulus of continuity of Hardy spaces, for which Marcinkiewicz means of the two-dimensional Walsh-Fourier series converge in $H_{2/3}(G^2)$ norm.