On Orthogonal Systems with Extremely Large $L_2$-Norm of the Maximal Operator

The problem of constructing examples that establish the sharpness of the Menchoff–Rademacher theorem on the Weyl multiplier for the almost everywhere convergence of series in general orthogonal systems is considered. We construct an example of a discrete orthonormal system based on $4\times 4$ blocks such that the partial sums of the series in this system has majorants whose $L_2$-norm increases as $\log_2N$. This orthonormal system is generated by an orthogonal matrix that has improved characteristics, as compared to the Hilbert matrix. We continue the study of B. S. Kashin, who constructed an example: of an orthonormal system, based on binary blocks, with majorant of partial sums increasing as $\sqrt{\log_2N}$.