General sigmoid based Banach space valued neural network multivariate approximations
Keywords:
General sigmoid function, multivariate neural network approximation, quasi-interpolation operator, Kantorovich type operator, quadrature type operator, multivariate modulus of continuity, abstract approximation, iterated approximation.Abstract
Here we expose 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 treat also the case of approximation by iterated operators of the last four types. These approximations are derived 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 a general sigmoid function. The approximations are pointwise and uniform. The related feed-forward neural network is with one hidden layer
