The Faculty of Education for Pure Sciences in the Department of Mathematics researched a master’s thesis on two-dimensional effects of the Szászsas type
The thesis presented by the researcher (Shahd Khalaf Kazem) aims to
The Szasz and Szasz-Kantorovich effects are considered essential tools in approximation theory due to their pivotal role in approximating functions and analyzing convergence properties. Based on the need to improve the accuracy of the approximation and build more flexible generalizations, this thesis presents four new modified operators based on the Szasz and Szasz-Kantorowicz effects in two-dimensional space. These modifications are organized into two categories. The first is based on the half-bounds (even terms) of both the Szasz and Szasz-Kantorowicz operators, while the second class employs generalized r-bounds for the former. Theoretical construction begins with the derivation of basic priors in the one-dimensional space, which are systematically extended to establish the two-dimensional framework. All operators are defined in the space C_∝ ((0,∞)×(0,∞)), ensuring their applicability to infinite domains. Convergence is proven using Korvikin's theorem, and Voronovskaya-type formulas are derived to characterize the behavior of the approximation when n→∞. Furthermore, error estimates are obtained using a continuity measure. Theoretical analysis proved that the modified effects show a clear improvement in the convergence rank compared to their classical counterparts. Numerical examples are also provided for verification
Theoretical results, which showed that the proposed modifications give lower error bounds and faster convergence for different choices of the parameter r. Overall, this work provides mathematically accurate and practically efficient generalizations of the Szasz and Szasz-Kantorowicz effects in two dimensions, contributing to the development of constructive approximation theory

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