The simultaneous approximation order of multivariate function by neural networks

In this document, we first give the order of simultaneous approximation of multivariate functions defined in simplex by multivariate Bernstein polynomials. Second, we prove that multivariate polynomials defined in simplex can be simultaneously approximated arbitrarily by a sigmoidal neural networks. At last, we give the order of simultaneous approximation of multivariate functions in simplex by neural networks.