Extension of approximation capability of three layered neural networks to derivatives

The author considers the problem of approximating arbitrary differentiable functions defined on compact sets of R^d, as well as their derivatives, by finite sums of the form a₀ +Σ_i=1^p a_ig(W_i×x +b), where W₁ are vectors of R^d and g is an arbitrary nonpolynomial C^∞-function fixed beforehand. If f is a polynomial of order n, the upper bound of p is n _n+d-1C_n. The linear combinations can be realized by three-layer neural networks