Approximation results by multivariate Kantorovich-type neural network sampling operators in Lebesgue spaces with variable exponents (Q6562343)

The problem of the convergence of a family of neural network operators of the type Kantorovich with sigmoidal activation functions in Lebesgue spaces \(L^{p(\cdot)}(\mathfrak{R})\) with variable exponents is studied. It is assumed that \(\mathfrak{R}:=[c_1,d_1]\times \cdots \times [c_m,d_m] \subset \mathbb{R}^m\) and \(1\leq p^-\leq p(x)\leq p^+ <+\infty,\ x\in \mathfrak{R}\). Also, the pointwise and uniform convergence for functions belonging to suitable spaces are proved. Note that the \(L^{p(\cdot)}\) convergence is established by using the density of the set of functions in such spaces. Some examples of sigmoidal activation functions for which the present theory can be applied are presented. The convergence of the Kantorovich approximations of \(L^{p(\cdot)}\) functions are illustrated by graphical representation.