Positive Linear Framework for Neural Networks with Generalized Logistic Structure: Multivariate and Finite Domain Case

In this work, we examine a family of abstract neural network operators in a multivariate setting, defined on compact subsets of R^N and taking values in general Banach spaces. These operators are constructed using a symmetric density derived from a modified logistic-type activation function with adjustable parameters. Within the framework of positive linear operator theory, we provide quantitative convergence estimates towards the identity operator. Our results are formulated using tools such as the modulus of continuity and Frechet derivatives. Furthermore, we offer a detailed discussion on the behavior of the operators when applied to convex functions. This article reflects both the author's contributions to the field over the past 30 years and some recent developments in neural network approximation theory.