In the present paper, asymptotic expansion and Voronovskaja type theorem for the neural network operators have been proved. The above results are based on the computation of the algebraic truncated moments of the density functions generated by suitable sigmoidal functions, such as the logistic functions, sigmoidal functions generated by splines and other. Further, operators with high-order convergence are also studied by considering finite linear combination of the above neural network type operators and Voronovskaja type theorems are again proved. At the end of the paper, numerical results are provided.

中文翻译：

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具有S形函数的Voronovskaja型定理和高阶收敛神经网络算符

本文证明了神经网络算子的渐近展开和Voronovskaja型定理。以上结果基于对由合适的S型函数（例如逻辑函数，样条线生成的S型函数）生成的密度函数的代数截断矩的计算。此外，还通过考虑上述神经网络类型算子的有限线性组合来研究具有高阶收敛性的算子，并再次证明了Voronovskaja型定理。在本文的最后，提供了数值结果。