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, on a probability
space and uniformly bounded, is studied. For example, the orthonormal system can be
obtained as the product of the functions of the Vilenkin system, Walsh system on a real sphere
or of 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.