Adaptive Voronovskaya-Type Expansions and Sobolev-Santos Uniform Convergence for Symmetrized Hyperbolic Tangent Neural Networks

This work introduces a novel class of multivariate neural network operators activated by symmetrized and perturbed hyperbolic tangent functions, with a focus on the \textbf{Sobolev-Santos Uniform Convergence Theorem}. The operators basic, Kantorovich, and quadrature types are analyzed through Voronovskaya-type asymptotic expansions, providing rigorous convergence rates for approximating continuous functions and their derivatives in Sobolev spaces \(W^{s,p}(\mathbb{R}^N)\). The proposed symmetrization method enhances both approximation power and regularity, enabling precise asymptotic descriptions as the network size increases. The study establishes uniform convergence rates in \(L^p\) and Sobolev norms, explicitly quantifying the impact of smoothness, dimensionality, and grid parameters. The \textbf{Sobolev-Santos Theorem} ensures uniform stability of these expansions under parametric variations of the activation function, guaranteeing robustness across different configurations. The results highlight the superior performance of these operators in high-dimensional approximation problems, with implications for artificial intelligence, data analytics, and numerical analysis. The explicit constants and uniform bounds provided offer a solid foundation for both theoretical and applied research in neural network-based function approximation.