A criterion for the~almost-everywhere convergence of Fourier--Walsh square partial sums of integrable functions

S. F. Lukomskii

Sbornik. Mathematics, Tome 186 (1995) no. 7, pp. 1057-1070

Résumé

S. V. Konyagin showed that if the one-dimensional Lebesgue constants $L_{n_k}$ for the Walsh–Paley system are unbounded, then the square partial sums $S_{n_k,n_k}(f)$ of some integrable function $f({x})=f(x_1,x_2)$ diverge almost everywhere. On the other hand the author constructed an example of sequence $\{n_k\}$ for which, sup $\sup L_{n_k}$ is finite, but for some integrable function $f({x})=f(x_1,x_2)$ the partial sums $S_{n_k,n_k}(f)$ diverge almost everywhere. Thus boundedness of the Lebesgue constants $L_{n_k}$ is not a necessary and sufficient condition for the convergence almost everywhere of the partial sums $S_{n_k,n_k}(f)$ of any integrable function. In this article we find such a necessary and sufficient condition.