Adjustable Half-Hyperbolic Convolution (HHC)-Type Operators with Symmetrized Kernel for Multivariate Approximation

In this study, we develop a class of multivariate HHC-type operators generated by adjustable half-hyperbolic tangent activation functions and a symmetrized kernel structure. The analysis is carried out within the classical framework of positive linear operators (PLOs), which allows a deeper investigation of their approximation behavior. By means of the modulus of continuity, we obtain quantitative convergence estimates toward the identity operator together with explicit bounds for the approximation error. The proposed construction is developed in a multivariate setting, where both simultaneous approximation with respect to several variables and iterated applications of the operators are examined. It is shown that the convergence properties remain stable under iteration, which further strengthens the analytical framework. The proposed operators also preserve important structural features of the approximated functions, including convexity and differentiability, and therefore provide a mathematically controlled approach to multivariate approximation. In contrast to earlier univariate HHC-type studies, the present work introduces a symmetry-enhanced multivariate operator structure together with explicit multivariate error analysis and a broader comparative numerical investigation. The numerical study, supported by Python 3.13 computations, combines regression-based error metrics with graphical analysis in order to illustrate convergence and compare the behavior of the classical, Kantorovich-type, and quadrature-type forms of the operators. Overall, the results contribute to the theory of convolution-type PLOs and provide a meticulous approximation framework with potential relevance to computational mathematics and learning-oriented operator design.