Performance Analysis of Half-Hyperbolic Convolution (HHC)-Type Operators via Regression-Based Metrics

In this paper, we first introduce the adjustable half-hyperbolic (adj HH) tangent function as an activation function. We then establish both quantitative and qualitative convergence results for HH-activated convolution-type positive linear operators (PLOs) acting on the space of bounded and continuous functions on the real line. The theoretical convergence results are numerically validated by means of error decay plots obtained using Python (version 3.13). Moreover, we compare three different classes of HHC-type operators in terms of their convergence behavior and approximation performance. Finally, we conclude by discussing several potential application areas that illustrate the relevance of the presented theoretical framework.