@article{Boccali_Costarelli_Vinti_2024, title={A Jackson-type estimate in terms of the \(\tau\)-modulus for neural network operators in \(L^{p}\)-spaces}, volume={2}, url={https://modernmathmeth.com/index.php/pub/article/view/42}, abstractNote={<p>In this paper, we study the order of approximation with respect to the \(L^{p}\)-norm for the (shallow) neural network (NN) operators. We establish a Jackson-type estimate for the considered family of discrete approximation operators using the averaged modulus of smoothness introduced by Sendov and Popov, also known by the name of \(\tau\)-modulus, in the case of bounded and measurable functions on the interval \([-1,1]\). The results here proved, improve those given by Costarelli (J. Approx. Theory 294:105944, 2023), obtaining a sharper approximation. In order to provide quantitative estimates in this context, we first establish an estimate in the case of functions belonging to Sobolev spaces. In the case \(1 &lt; p &lt;+\infty\), a crucial role is played by the so-called Hardy-Littlewood maximal function. The case of \(p=1\) is covered in case of density functions with compact support.</p>}, number={2}, journal={Modern Mathematical Methods}, author={Boccali, Lorenzo and Costarelli, Danilo and Vinti, Gianluca}, year={2024}, month={Aug.}, pages={90–102} }