Best approximation and inverse results for neural network operators (Q6595826)

In this article, approximations by neural network operators are studied and their errors are estimated from above. The neural networks are of a classical type using sigmoidal functions from standard classes. The theorems are restricted to the one-dimensional setting.\N\NThe error estimates are formulated in terms of moduli of smoothness and some are shown to be optimal. The approximation rates are studied with respect to the uniform norm for example under the condition that the sigmoidal function which defines the neuron, call it \(\sigma\), satisfies (for \(x\to-\infty\)) the bound \(\sigma(x)=O(|x|^{-\alpha})\) with an exponent \(\alpha\) greater than one. The error bounds in the Chebyshev norm are shown to be optimal in various subcases. Also inverse theorems are established for some classes of Lipschitz functions.