arXiv:2107.04093v1 [math.FA] 8 Jul 2021 Estimates for entropy numbers of multiplier operators of multiple series Sergio A. Córdoba * , Jéssica Milaré † and Sergio A. Tozoni ‡ ∗ Departamento de Vías y Transporte, Universidad del Cauca, Calle 5 Nº 4-70, Popayán - Cauca, CEP 190003, Colombia †,‡ Instituto de Matemática, Universidade Estadual de Campinas, Rua Sérgio Buarque de Holanda 651, Campinas- SP, CEP 13083-859, Brazil Abstract The asymptotic behavior for entropy numbers of general Fourier multiplier operators of multiple series with respect to an abstract complete orthonormal system {φ m } m∈N d 0 on a probability space and bounded in L ∞ , is studied. The orthonormal system can be of the type φ m (x)= φ (1) m1 (x 1 ) ··· φ (d) m d (x d ), where each  φ (j) l  l∈N0 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. MSC2020: 41A46, 42C10, 47B06 Keywords: Vilenkin series, multipliers operators, entropy numbers, approximation theory. 1 Introduction In [2, 9, 10, 18], the asymptotic behavior of entropy numbers of multiplier operators was studied, using dif- ferent techniques. Estimates were obtained for entropy numbers of sets of finitely and infinitely differentiable functions and sets of analytic functions, on homogeneous spaces and on the torus. In this paper, we continue these studies considering now general Fourier multiplier operators of multiple series with respect to an abstract complete orthonormal system. Recently, much attention has been devoted to the study of entropy numbers of different sets of functions. This problem has a long history and some fundamental problems in this area are still open. Estimates for entropy numbers of different operators and embeddings between function spaces of the types Besov and Sobolev with mixed smoothness into Lebesgue have been studied [8, 11, 12, 19, 21]. In the papers [10, 18] sharp estimates were obtained for entropy numbers of sets of infinitely differentiable functions and of analytic functions, on two-points homogeneous spaces and on the torus, in several cases. We do not know other papers where studies of this type have been carried out. Our approach is based on estimates for Levy means of norms constructed using an orthonormal system of multiple functions on a probability space. To prove the results for general multiplier operators we make use of * email: sergiocordoba@unicauca.edu.co † email: jessymilare@gmail.com ‡ email: tozoni@unicamp.br 1