Multivariate Gudermannian function based neural network approximation

Abstract

Here we present multivariate quantitative approximations of Banach space valued continuous multivariate functions on a box or R
N ; N 2 N, by the multivariate normalized, quasi-interpolation, Kantorovich type and quadrature type neural network operators. We examine also the case of approximation by iterated operators of the last four types. These approximations are achieved by establishing multidimensional Jackson type inequalities involving the multivariate modulus of continuity of the engaged function or its high order FrÈchet derivatives. Our multivariate operators are deÖned by using a multidimensional density function induced by the Gudermannian sigmoid function. The approximations are pointwise and uniform. The related feed-forward neural network is with one hidden layer