Simultaneous L/sup p/-approximations of polynomials and derivatives on the whole space

We have obtained a sufficient condition that a linear sum of an activation function can simultaneously approximate polynomials and their derivatives in the sense of L/sup p/(R/sup d/, /spl mu/). If the probability measure /spl mu/ is rapidly decreasing, a wide variety of differentiable functions satisfy this condition. For the Gaussian measure, rapidly increasing functions such as e/sup t/, exp(t/sup 2//2) and others can be activation functions. The proof is constructive and elementary, which enables the approximation formulas to be written explicitly.