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

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.