Multivariate sigmoidal neural network approximation

Here we study the multivariate quantitative constructive approximation of real and complex valued continuous multivariate functions on a box or RNRN, N∈NN∈N, by the multivariate quasi-interpolation sigmoidal neural network operators. The “right” operators for our goal are fully and precisely described. This approximation is derived by establishing multidimensional Jackson type inequalities involving the multivariate modulus of continuity of the engaged function or its high order partial derivatives. Our multivariate operators are defined by using a multidimensional density function induced by the logarithmic sigmoidal function. The approximations are pointwise and uniform. The related feed-forward neural network is with one hidden layer.