On the partial sums of Walsh–Fourier series
Tom 141 / 2015
Colloquium Mathematicum 141 (2015), 227-242 MSC: Primary 42C10. DOI: 10.4064/cm141-2-7
We investigate convergence and divergence of specific subsequences of partial sums with respect to the Walsh system on martingale Hardy spaces. By using these results we obtain a relationship of the ratio of convergence of the partial sums of the Walsh series and the modulus of continuity of the martingale. These conditions are in a sense necessary and sufficient.