The approximation capabilities of Durrmeyer-type neural network operators

In this paper, a new family of neural network (NN) operators is introduced. The idea is to consider a Durrmeyer-type version of the widely studied discrete NN operators by Costarelli and Spigler (Neural Netw 44:101–106, 2013). Such operators are constructed using special density functions generated from suitable sigmoidal functions, while the reconstruction coefficients are based on a convolution between a general kernel function \(\chi \) $\chi$ and the function being reconstructed, f. Here, we investigate their approximation capabilities, establishing both pointwise and uniform convergence theorems for continuous functions. We also provide quantitative estimates for the approximation order thanks to the use of the modulus of continuity of f; this turns out to be strongly influenced by the asymptotic behaviour of the sigmoidal function \(\sigma \) $\sigma$ . Our study also shows that the estimates we provide are, under suitable assumptions, the best possible. Finally, \(L^p\) $L^p$ -approximation is also established. At the end of the paper, examples of activation functions are discussed.