Parametrized hyperbolic tangent based Banach space valued multivariate multi layer neural network approximations

Here we examine the multivariate quantitative approximations of Banach space valued continuous multivariate functions on a box or R N , N ∈ N, by the multivariate normalized, quasi-interpolation, Kantorovich type and quadrature type neural network operators. We research also the case of approximation by iterated operators of the last four types, that is multi hidden layer approximations. These approximations are achieved by establishing multidimensional Jackson type inequalities involving the multivariate modulus of continuity of the engaged function or its high order Fr´echet derivatives. Our multivariate operators are defined by using a multidimensional density function induced by a parametrized hyperbolic tangent sigmoid function. The approximations are pointwise, uniform and Lp. The related feed-forward neural networks are with one or multi hidden layers.