Estimates for entropy numbers of multiplier operators of multiple series

The asymptotic behavior for entropy numbers of general Fourier multiplier operators of multiple series with respect to an abstract complete orthonormal system $\{\phi_{\textbf{m}}\}_{\textbf{m}\in \mathbb{N}^d_0}$ on a probability space and bounded in $L^{\infty}$, is studied. The orthonormal system can be of the type $\phi_{\textbf{m}}(\textbf{x}) = \phi_{m_1}^{(1)}(x_1)\cdots \phi_{m_d}^{(d)}(x_d)$, where each $\left\{\phi_{l}^{(j)}\right\}_{l\in \mathbb{N}_0}$ is an orthonormal system, that can be different for each $j$, for example, it can be a Vilenkin system, a Walsh system on a real sphere or the trigonometric system on the unit circle. General upper and lower bounds for the entropy numbers are established by using Levy means of norms constructed using the orthonormal system. These results are applied to get upper and lower bounds for entropy numbers of specific multiplier operators, which generate, in particular cases, sets of finitely and infinitely differentiable functions, in the usual sense and in the dyadic sense. It is shown that these estimates have order sharp in various important cases.