Approximation by Neural Network operators in $L^p$ spaces associated with an arbitrary measure

In this paper, we investigate the approximation behavior of both one and multidimensional neural network type operators for functions in $L^p(I^d,\rho)$, where $1\leq p<\infty$, associated with a general measure $\rho$ defined over a hypercube. First, we prove the uniform approximation for a continuous function and the $L^p$ approximation theorem by the NN operators in one and multidimensional settings. In addition, we also obtain the $L^p$ error bounds in terms of $\mathcal{K}$-functionals for these neural network operators. Finally, we consider the logistic and tangent hyperbolic activation functions and verify the hypothesis of the theorems. We also show the implementation of continuous and integrable functions by NN operators with respect to the Lebesgue and Jacobi measures defined on $[0,1]\times[0,1]$ with logistic and tangent hyperbolic activation functions.