A Generalization of Blending-Type Szasz–Mirakjan Operators and Their Approximation Properties

Abstract

This paper introduces a new generalization of blending-type Szász-Mirakjan operators via an additional parameter $\alpha$. We investigate fundamental approximation properties including moment estimates, central moments, and local approximation results. Korovkin-type theorems are established to prove uniform convergence, while various tools are employed to study rates of convergence. Weighted approximation properties are examined in depth, analyzing the behavior of operators in weighted spaces and establishing convergence results for functions with polynomial growth. Furthermore, A-statistical approximation properties are thoroughly investigated, providing convergence results under weaker conditions than classical approaches. The theoretical findings are supported by comprehensive numerical and graphical analyses, demonstrating the effectiveness of the proposed operators. Error analysis confirms that approximation quality improves significantly as the parameter increases, with visual evidence showing uniform convergence behavior. Both global and local approximation properties are examined using moduli of smoothness and Peetre's $K$-functional in different function spaces. The results confirm that our operators provide enhanced approximation capabilities compared to existing ones.

Keywords

• Blending-Type operators
• Korovkin Theorem
• Sz\
• Local approximation.

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