Strong and weak sharp bounds for neural network operators in Sobolev-Orlicz spaces and their quantitative extensions to Orlicz spaces

In this paper, we establish sharp bounds for a family of Kantorovich-type neural network operators within the general frameworks of Sobolev-Orlicz and Orlicz spaces. We establish both strong (in terms of the Luxemburg norm) and weak (in terms of the modular functional) estimates, using different approaches. The strong estimates are derived for spaces generated by gyp-functions that are N-functions or satisfy the O '-condition. Such estimates also lead to convergence results with respect to the Luxemburg norm in several instances of Orlicz spaces, including the exponential case. Meanwhile, the weak estimates are achieved under less restrictive assumptions on the involved gyp-function. To obtain these results, we introduce some new tools and techniques in Orlicz spaces. Central to our approach is the Orlicz Minkowski inequality, which allows us to obtain unified strong estimates for the operators. We also present a weak (modular) version of this inequality holding under weaker conditions. Additionally, we introduce a novel notion of discrete absolute gyp-moments of the hybrid type, and we employ the Hardy-Littlewood maximal operator within Orlicz spaces for the asymptotic analysis. Furthermore, we introduce the new space W1,phi(I), which is embedded in the Sobolev-Orlicz space W1,phi (I) and modularly dense in L phi (I). This allows to achieve asymptotic estimates for a wider class of gyp-functions, including those that do not meet the O2-condition. For the extension to the whole Orliczsetting, we generalize a Sobolev-Orlicz density result given by H. Musielak using Steklov functions, providing a modular counterpart. Moreover, we explore the relationships between weak and strong Orlicz-Lipschitz classes, corresponding to the above moduli of smoothness, providing qualitative results on the rate of convergence of the operators. Finally, a (Luxemburg norm) inverse approximation theorem in Orlicz spaces has been established, from which we deduce a characterization of the corresponding Lipschitz classes in terms of the order of convergence of the operators. The latter result shows that some of the achieved estimates are sharp. (c) 2025 The Author(s). Published by Elsevier Masson SAS. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).