Hardy and Bellman transformations of series with respect to multiplicative systems

The well-known Hardy transformation by the method of arithmetic means of Fourier series with respect to multiplicative systems and the Bellman transformation dual to it are investigated. An integral representation of the Hardy operator is given; it is proved that spaces in a certain class that possess a majorant of the modulus of continuity in $L_p[0,1)$, $1\leq p\leq \infty$, $\mathrm{BMO}(\mathbf P,[0,1))$ or $H(\mathbf P,[0,1))$ are stable under the Hardy and Bellman transformations. Criteria for functions with generalized monotonic Fourier coefficients to belong to certain spaces are obtained; these are given in terms of their Fourier coefficients and their Hardy and Bellman transformations.
Bibliography: 30 titles.