Convergence and Summability in Classical and Martingale Hardy Spaces

The development of the theory of martingale Hardy spaces has been strongly influenced by the classical theory. Because of this fact it is inevitable to compare results of this theory to those on classical Hardy spaces. Many similarities between these theories have already been found and a lot of new results were obtained in the theory of martingale Hardy spaces which are open problems in classical case. By taking this into account, in the paper we have chosen a partly opposite approach by stating and proving some new results concerning partial sums two-dimensional Fourier series in martingale Hardy spaces for the classical Hardy space case.